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Spora A Journal of Biomathematics Spora A Journal of Biomathematics
Volume 6 Article 3
2020
Modeling the Effects of Passive Immunity in Birds for the Disease Modeling the Effects of Passive Immunity in Birds for the Disease
Dynamics of West Nile Virus Dynamics of West Nile Virus
Noelle West Dixie State University d00355490dmaildixieedu
Vinodh K Chellamuthu Dixie State University vinodhchellamuthudixieedu
Follow this and additional works at httpsirlibraryillinoisstateeduspora
Part of the Ordinary Differential Equations and Applied Dynamics Commons
Recommended Citation Recommended Citation West Noelle and Chellamuthu Vinodh K (2020) Modeling the Effects of Passive Immunity in Birds for the Disease Dynamics of West Nile Virus Spora A Journal of Biomathematics Vol 6 16ndash25 DOI httpsdoiorg1030707SPORA61PAMG6515 Available at httpsirlibraryillinoisstateedusporavol6iss13
This Mathematics Research is brought to you for free and open access by ISU ReD Research and eData It has been accepted for inclusion in Spora A Journal of Biomathematics by an authorized editor of ISU ReD Research and eData For more information please contact ISUReDilstuedu
ISSN 2473-3067 (Print)
ISSN 2473-5493 (Online)
Modeling the Effects of Passive Immunity in Birds for theDisease Dynamics of West Nile Virus
Noelle West1 Vinodh K Chellamuthu1
CorrespondenceDr Vinodh ChellamuthuMathematics DepartmentDixie State University225 South 700 EastSt George UT 84770-3876USAvinodhchellamuthu
dixieedu
13
Abstract
West Nile Virus (WNV) is a mosquito-borne virus that circulates among birds but also affectshumans Migrating birds carry these viruses from one place to another each year WNV hasspread rapidly across the continental United States resulting in numerous human infectionsand deaths Several studies suggest that larval mosquito control measures should be takenas early as possible in a season to control the mosquito population size Also adult mosquitocontrol measures are necessary to prevent the transmission of WNV from mosquitoes tobirds and humans To better understand the effective strategy for controlling affected larvaemosquito population we have developed a mathematical model using a system of first orderdifferential equations to investigate the transmission dynamics of WNV in a mosquito-bird-human community We also incorporated vertical transmission in mosquitoes and passiveimmunity in birds to more accurately simulate the spread of the disease
Keywords SIR model passive immunity West Nile Virus
1 Introduction
West Nile Virus is a vector transmitted disease trans-ferred from infected mosquitoes to other animals such asbirds and humans From 1999 to 2012 the virus causedover 1500 deaths [23] and the most dramatic outbreak ofthe virus in North America occurred during the summerof 2002 [25] The virus crossed the Mississippi River andinfected the Pacific Coast region of the United Statesbringing the total number of infected human cases to4156 and resulting in 284 deaths [25] The economicburden of West Nile Virus is significant [4 9] the cost oftreating and maintaining patients with West Nile Viruswas $778 million from 1999 to 2012 [4] Thus it is im-perative that we contain this disease
Mathematicians have modeled the spread of West NileVirus using SIR and SEIR models [9 17 19 22] Mostnotably they have simulated the use of pesticides onmosquito populations to test the effect of West Nile Viruson birds and humans [22] The CDC recommends us-ing multiple abatement strategies to control the localmosquito population One such strategy incorporates theuse of two different insecticides adulticides (targets ma-ture mosquitoes) and larvicides (targets mosquito larvae)[7] Adulticides on their own are shown to have a greateffect on lowering the mosquito populations [22] Natu-rally when sprayed more often such as weekly instead ofmonthly the mosquito population falls more drastically
1Mathematics Department Dixie State University St GeorgeUT
and is controlled faster [22] Since mosquitoes are theprimary carrier of the disease when their populations arelowered we see a lower spread of the virus [22 24] Birdscirculate the disease amongst themselves and mosquitoes[22 24] An infected bird flies into a new area and sus-ceptible mosquitoes bite the infected bird transmittingthe disease to the mosquito [22 24] Infected birds willalso fly to other areas spreading the disease across conti-nents while some will remain in the same area infectingmore mosquitoes in the community [5] Those mosquitoeswill go on to bite a susceptible bird giving them thedisease and the cycle repeats itself perpetually [22 24]Mosquitoes are also able to bite humans and transmitWest Nile Virus to them [22 24] All of these elementsare necessary to create a model of how the disease spreadsHowever there are two important elements that previousmodels have overlooked vertical transmission and pas-sive immunity [1 2 17]
Most mathematical models of West Nile Virus have fo-cused on how mosquitoes spread the virus themselves andhow birds are quick to fall ill to the virus [9 17 19 22]Yet mosquitoes can spread the disease not only by in-fecting birds and humans but also by passing the diseaseonto their larvae a process known as vertical transmission[2 17 18] On the other hand once a bird recovers fromWest Nile Virus it is immune to the disease The bird hasa high chance of passing its immunity down to its younga process known as passive immunity [1 20] Many birdspecies develop passive immunity including house spar-rows [21] flamingos [3] rock pigeons [14] and Eastern
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 1 Schematic diagram of the model featuring alleighteen populations subdivided into three categoriesmosquitoes represented by circles birds represented by di-amonds and humans represented by squares The dashedlines showcase interactions between species while the solidlines showcase interactions within the same species Notethe passive immunity cycle in the bird category
screech-owls [15] Both vertical transmission and passiveimmunity are essential parts of simulating the dynamicsof West Nile Virus precisely Vertical transmission de-pends on the overall mosquito population and increasesthe spread of disease while passive immunity depends onboth the bird population and a birdrsquos chances of recover-ing from the disease slowing disease spread The goal ofour work is to give insight to the importance of passiveimmunity in birds and to show how it is incorporated inthe disease dynamics
2 Model
Our model utilizes a system of eighteen ordinary differ-ential equations (ODEs) based off of [22] that act as amodified SEIR model explained in the schematic dia-gram in Figure 1 There are three main categories birdsmosquitoes and humans The birds category highlightspassive immunity the mosquitoes category highlights ver-tical transmission and these important additions to themodel affect the humans category Every category inter-acts with at least one other category in order to properlysimulate the spread of West Nile Virus
The bird category contains seven sections suscepti-ble eggs (ES) immune eggs (ER) immune birds (BM )susceptible birds (BS) infected birds (BI) recovered
birds (BR) and dead birds (BD) Susceptible and in-fected birds lay susceptible eggs These eggs hatch andgrow into susceptible birds that are capable of developingWest Nile Virus Birds that have been infected with andsubsequently become immune to the virus lay recoveredeggs The process of adult bird populations passing theirimmunity onto their young is passive immunity recoveredeggs hatch into the temporary state of immune birds [20]This state makes the birds immune to West Nile Virus fora specific amount of time before they join the suscepti-ble bird category Susceptible birds are not infected withthe disease but are capable of developing West Nile Virus[22] Infected birds are currently suffering from the dis-ease and are capable of spreading the virus to susceptiblemosquitoes [22] Birds move into the recovered categoryafter they are no longer hosting the disease [22] Once abird is recovered it is permanently immune to the disease[22] Recovered birds are also able to pass their immunityonto their young if they reproduce The dead birds cat-egory records only birds that have died from West NileVirus [22]
The mosquito category contains five sections sus-ceptible larvae (LS) infected larvae (LI) suscepti-ble mosquitoes (MS) exposed mosquitoes (ME) andinfected mosquitoes (MI) Susceptible and exposedmosquitoes lay susceptible larvae These larvae havea chance of maturing into susceptible healthy adultmosquitoes Infected mosquitoes lay infected larvae [2]Due to vertical transmission infected mosquitoes havethe ability to pass West Nile Virus onto their young[2] In turn these larvae will mature into infectedmosquitoes The difference between the categories ofmature mosquitoes is that susceptible mosquitoes arehealthy but biting infected birds will infect them exposedmosquitoes bit an infected bird but are not yet infectiousand infected mosquitoes have West Nile Virus and areable to pass the disease onto birds and humans [22] In-fected mosquitoes are also able to vertically transmit theirdisease to their young should they reproduce
The human category contains six sections suscepti-ble humans (HS) exposed humans (HE) humans withWest Nile Fever (HF ) humans with neuroinvasive dis-ease (HN ) recovered humans (R) and dead humans (D)Susceptible humans are healthy but able to catch WestNile Virus should they be bitten Infected mosquitoesbite humans exposing them to the disease There is achance that exposed humans recover before developingany symptoms of West Nile Fever or neuroinvasive dis-ease However humans are considered dead-end hostsand are unable to infect mosquitoes even if they are in-fected [19] Recovered humans have recovered from eitherWest Nile Fever or neuroinvasive disease and are immuneto catching West Nile Virus again Dead humans refersto humans who have perished from neuroinvasive disease
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
dLS
dt= b(MS +ME)minusmLS minus δLLS
dLI
dt= bMI minusmLI minus δLLI
dMS
dt= mLS minus
αMβMSBI
NTotalminus δMMS minus T (t)MS
dME
dt=αMβMSBI
NTotalminus ηME minus δMME minus T (t)ME
dMI
dt= mLI + ηME minus δMMI minus T (t)MI
dES
dt= φS(BS +BI) + (1minus micro)φRBR minus θES minus ψES
dER
dt= microφRBR minus θER minus ψER
dBS
dt= Λminus αBβMIBS
NTotal+ pSBM + ψES minus τBS
dBI
dt=αBβMIBS
NTotalminus δBBI minus τBI
dBR
dt= (1minus σ)δBBI + pRBM minus τBR
dBM
dt= ψER minus (pS + pR)BM minus τBM
dBD
dt= σδBBI
dHS
dt= minusαHβMIHS
NTotal
dHE
dt=αHβMIHS
NTotalminus δEHE
dHF
dt= (1minus γ minus κ)δEHE minus δFHF
dHN
dt= κδEHE minus δNHN
dR
dt= δFHF + (1minus ω)δNHN + γδEHE
dD
dt= ωδNHN
The parameter NTotal includes the total blood supplysuch that NTotal = BS(t) + BI(t) + BR(t) + BM (t) +HS(t) + HE(t) + HF (t) + HN (t) + R(t) Also note thatthere is no birth nor natural death rate for humans likethere is for birds and mosquitoes Since the model takesplaces over a three month period the death rate for hu-mans is negligible The other parameter values for thismodel are listed in Table 1
The interaction between the mosquito bird and hu-man populations is the base of our model of the spreadof the virus Infected mosquitoes lay eggs at a rate of bthat become infected larvae Once hatched they matureat growth rate m into infected mosquitoes continuingthe cycle Similarly susceptible mosquitoes lay eggs ata rate of b which become susceptible larvae that grow
into susceptible mosquitoes at a rate of m Once adultssusceptible mosquitoes bite at a rate of β the averagebiting rate per day We assume the mosquito biting ratefor both humans and birds are the same [22] Dependingon the amount of infected birds as αMβNTotal modelsthere is a chance that a susceptible mosquito bites an in-fected bird exposing the mosquito to the virus Once thevirus enters their system the susceptible mosquito movesinto the exposed mosquito category Exposed mosquitoestransition to infected mosquitoes at a rate of η and arethen able to transmit West Nile Virus to birds and hu-mans as well as participate in vertical transmission
Infected mosquitoes are also able to bite suscep-tible birds and change them into infected birds asαBβNTotal models After a bird is infected it has achance to recover which (1minus σ)δB models and a chanceto die from the virus which σδB models Recovered birdspass their immunity down to their young which microφR sim-ulates These eggs hatch and grow into birds at a rate ofψ There is also a chance that they do not pass theirimmunity down to their young which (1minus micro)φR modelsIf this were to happen they would lay susceptible eggsinstead of recovered eggs Susceptible eggs grow into sus-ceptible birds at a rate of ψ These birds lay susceptibleeggs which φS models
West Nile Virus similarly affects humans αHβNTotal
represents the infected mosquitoes that bite susceptiblehumans and expose them to the disease [22] Once a hu-man is exposed to the disease there is a chance that theywill recover immediately from the disease at a rate of γδE However they may suffer from West Nile Fever at a rateof (1 minus γ minus κ)δE Most people recover from West NileFever at a rate of δF while others develop neuroinvasivedisease at a rate of κδE Once somebody has neuroinva-sive disease they either recover at a rate of (1minus ω)δN orthey die at a rate of ωδN
Several studies suggest the use of adulticides in orderto control the mosquito population [7 9 12 22] T (t)models the adulticides Within this function s equals theeffectiveness of the adulticide being used Our model usess-values of 0 20 40 and 80 treatment effective-ness as given by example in Figure 3 Simulating spray-ing at certain intervals such as every seven fourteen ortwenty-four days makes this function dependent on timeFor example if we were to treat weekly T (t) = s whent equiv 0 mod 7 and T (t) = 0 when t 6equiv 0 mod 7
We also set our immune birds to lose or permanentlymaintain their immunity at various time intervals basedon their species (ie every 14 days for chickens 10 daysfor owls and 9 days for house sparrows [20]) After eachtime interval pS percent of the immune bird populationmoves to the susceptible bird category and pR percent ofthe immune bird population moves to the recovered birdcategory [20]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Table 1 Parameters and their units values and sources for the system of ordinary differential equations that modelsthe disease dynamics
Parameter Definitions Units Value Source
b Mosquito Birth Rate Larvae(Day middot Adults) 0045 [22]
m Mosquito Maturation Rate Adults(Larvae middot Day) 007 [23 27]
δL Natural Larval Death Rate Dayminus1 0027 [11]
αM Probability of Transmission from Birdsto Mosquitoes
mdash 023 [12]
β Bite Rate Dayminus1 25 [22]
δM Natural Mosquito Death Rate Dayminus1 0031 [11]
T (t) T (t) = s Success Rate of Adulticides Dayminus1 varies [22]
η Virus Incubation Rate in Mosquitoes Dayminus1 01 [10]
φS Egg Laying Rate for Susceptible Birds Dayminus1 varies [15 19 21]
micro Percent of Eggs Receiving Passive Immunity mdash varies [15 19 21]
φR Egg Laying Rate for Recovered Birds Dayminus1 varies [15 19 21]
θ Natural Death Rate of Bird Eggs Dayminus1 045 [15 19 21]
ψ Maturation Rate of Bird Eggs Dayminus1 varies [15 19 21]
Λ Recruitment Rate of Birds BirdsDay varies [22]
αB Probability of Transmission from Mosquitoesto Birds
mdash 027 [12]
Ï„ Natural Bird Death Rate Dayminus1 varies [26]
δB Rate of Recovery in Birds Dayminus1 1(45) [16]
σ Fraction of WNV Infected Birds Dyingfrom the Disease
Dayminus1 072 [16]
αH Probability of Transmission from Mosquitoesto Humans
mdash 006 [6]
δE Incubation Period in Humans Dayminus1 14 [22]
γ Fraction of Human Population thatis Asymptomatic
Dayminus1 075 [8]
κ Fraction of Human Population thatCan Develop Neuroinvasive Disease
Dayminus1 001 [8]
δF Rate of Recovery for WNV Fever in Humans Dayminus1 114 [6]
δN Rate of Recovery for Neuroinvasive Diseasein Humans
Dayminus1 1(375) [13]
ω Fraction of Humans Dyingfrom Neuroinvasive Disease
Dayminus1 01 [8]
pS Percent of Immune Birds that Losetheir Immunity
mdash varies [15 19 20 21]
pR Percent of Immune Birds that Keeptheir Immunity
mdash varies [15 19 20 21]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
ISSN 2473-3067 (Print)
ISSN 2473-5493 (Online)
Modeling the Effects of Passive Immunity in Birds for theDisease Dynamics of West Nile Virus
Noelle West1 Vinodh K Chellamuthu1
CorrespondenceDr Vinodh ChellamuthuMathematics DepartmentDixie State University225 South 700 EastSt George UT 84770-3876USAvinodhchellamuthu
dixieedu
13
Abstract
West Nile Virus (WNV) is a mosquito-borne virus that circulates among birds but also affectshumans Migrating birds carry these viruses from one place to another each year WNV hasspread rapidly across the continental United States resulting in numerous human infectionsand deaths Several studies suggest that larval mosquito control measures should be takenas early as possible in a season to control the mosquito population size Also adult mosquitocontrol measures are necessary to prevent the transmission of WNV from mosquitoes tobirds and humans To better understand the effective strategy for controlling affected larvaemosquito population we have developed a mathematical model using a system of first orderdifferential equations to investigate the transmission dynamics of WNV in a mosquito-bird-human community We also incorporated vertical transmission in mosquitoes and passiveimmunity in birds to more accurately simulate the spread of the disease
Keywords SIR model passive immunity West Nile Virus
1 Introduction
West Nile Virus is a vector transmitted disease trans-ferred from infected mosquitoes to other animals such asbirds and humans From 1999 to 2012 the virus causedover 1500 deaths [23] and the most dramatic outbreak ofthe virus in North America occurred during the summerof 2002 [25] The virus crossed the Mississippi River andinfected the Pacific Coast region of the United Statesbringing the total number of infected human cases to4156 and resulting in 284 deaths [25] The economicburden of West Nile Virus is significant [4 9] the cost oftreating and maintaining patients with West Nile Viruswas $778 million from 1999 to 2012 [4] Thus it is im-perative that we contain this disease
Mathematicians have modeled the spread of West NileVirus using SIR and SEIR models [9 17 19 22] Mostnotably they have simulated the use of pesticides onmosquito populations to test the effect of West Nile Viruson birds and humans [22] The CDC recommends us-ing multiple abatement strategies to control the localmosquito population One such strategy incorporates theuse of two different insecticides adulticides (targets ma-ture mosquitoes) and larvicides (targets mosquito larvae)[7] Adulticides on their own are shown to have a greateffect on lowering the mosquito populations [22] Natu-rally when sprayed more often such as weekly instead ofmonthly the mosquito population falls more drastically
1Mathematics Department Dixie State University St GeorgeUT
and is controlled faster [22] Since mosquitoes are theprimary carrier of the disease when their populations arelowered we see a lower spread of the virus [22 24] Birdscirculate the disease amongst themselves and mosquitoes[22 24] An infected bird flies into a new area and sus-ceptible mosquitoes bite the infected bird transmittingthe disease to the mosquito [22 24] Infected birds willalso fly to other areas spreading the disease across conti-nents while some will remain in the same area infectingmore mosquitoes in the community [5] Those mosquitoeswill go on to bite a susceptible bird giving them thedisease and the cycle repeats itself perpetually [22 24]Mosquitoes are also able to bite humans and transmitWest Nile Virus to them [22 24] All of these elementsare necessary to create a model of how the disease spreadsHowever there are two important elements that previousmodels have overlooked vertical transmission and pas-sive immunity [1 2 17]
Most mathematical models of West Nile Virus have fo-cused on how mosquitoes spread the virus themselves andhow birds are quick to fall ill to the virus [9 17 19 22]Yet mosquitoes can spread the disease not only by in-fecting birds and humans but also by passing the diseaseonto their larvae a process known as vertical transmission[2 17 18] On the other hand once a bird recovers fromWest Nile Virus it is immune to the disease The bird hasa high chance of passing its immunity down to its younga process known as passive immunity [1 20] Many birdspecies develop passive immunity including house spar-rows [21] flamingos [3] rock pigeons [14] and Eastern
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 1 Schematic diagram of the model featuring alleighteen populations subdivided into three categoriesmosquitoes represented by circles birds represented by di-amonds and humans represented by squares The dashedlines showcase interactions between species while the solidlines showcase interactions within the same species Notethe passive immunity cycle in the bird category
screech-owls [15] Both vertical transmission and passiveimmunity are essential parts of simulating the dynamicsof West Nile Virus precisely Vertical transmission de-pends on the overall mosquito population and increasesthe spread of disease while passive immunity depends onboth the bird population and a birdrsquos chances of recover-ing from the disease slowing disease spread The goal ofour work is to give insight to the importance of passiveimmunity in birds and to show how it is incorporated inthe disease dynamics
2 Model
Our model utilizes a system of eighteen ordinary differ-ential equations (ODEs) based off of [22] that act as amodified SEIR model explained in the schematic dia-gram in Figure 1 There are three main categories birdsmosquitoes and humans The birds category highlightspassive immunity the mosquitoes category highlights ver-tical transmission and these important additions to themodel affect the humans category Every category inter-acts with at least one other category in order to properlysimulate the spread of West Nile Virus
The bird category contains seven sections suscepti-ble eggs (ES) immune eggs (ER) immune birds (BM )susceptible birds (BS) infected birds (BI) recovered
birds (BR) and dead birds (BD) Susceptible and in-fected birds lay susceptible eggs These eggs hatch andgrow into susceptible birds that are capable of developingWest Nile Virus Birds that have been infected with andsubsequently become immune to the virus lay recoveredeggs The process of adult bird populations passing theirimmunity onto their young is passive immunity recoveredeggs hatch into the temporary state of immune birds [20]This state makes the birds immune to West Nile Virus fora specific amount of time before they join the suscepti-ble bird category Susceptible birds are not infected withthe disease but are capable of developing West Nile Virus[22] Infected birds are currently suffering from the dis-ease and are capable of spreading the virus to susceptiblemosquitoes [22] Birds move into the recovered categoryafter they are no longer hosting the disease [22] Once abird is recovered it is permanently immune to the disease[22] Recovered birds are also able to pass their immunityonto their young if they reproduce The dead birds cat-egory records only birds that have died from West NileVirus [22]
The mosquito category contains five sections sus-ceptible larvae (LS) infected larvae (LI) suscepti-ble mosquitoes (MS) exposed mosquitoes (ME) andinfected mosquitoes (MI) Susceptible and exposedmosquitoes lay susceptible larvae These larvae havea chance of maturing into susceptible healthy adultmosquitoes Infected mosquitoes lay infected larvae [2]Due to vertical transmission infected mosquitoes havethe ability to pass West Nile Virus onto their young[2] In turn these larvae will mature into infectedmosquitoes The difference between the categories ofmature mosquitoes is that susceptible mosquitoes arehealthy but biting infected birds will infect them exposedmosquitoes bit an infected bird but are not yet infectiousand infected mosquitoes have West Nile Virus and areable to pass the disease onto birds and humans [22] In-fected mosquitoes are also able to vertically transmit theirdisease to their young should they reproduce
The human category contains six sections suscepti-ble humans (HS) exposed humans (HE) humans withWest Nile Fever (HF ) humans with neuroinvasive dis-ease (HN ) recovered humans (R) and dead humans (D)Susceptible humans are healthy but able to catch WestNile Virus should they be bitten Infected mosquitoesbite humans exposing them to the disease There is achance that exposed humans recover before developingany symptoms of West Nile Fever or neuroinvasive dis-ease However humans are considered dead-end hostsand are unable to infect mosquitoes even if they are in-fected [19] Recovered humans have recovered from eitherWest Nile Fever or neuroinvasive disease and are immuneto catching West Nile Virus again Dead humans refersto humans who have perished from neuroinvasive disease
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
dLS
dt= b(MS +ME)minusmLS minus δLLS
dLI
dt= bMI minusmLI minus δLLI
dMS
dt= mLS minus
αMβMSBI
NTotalminus δMMS minus T (t)MS
dME
dt=αMβMSBI
NTotalminus ηME minus δMME minus T (t)ME
dMI
dt= mLI + ηME minus δMMI minus T (t)MI
dES
dt= φS(BS +BI) + (1minus micro)φRBR minus θES minus ψES
dER
dt= microφRBR minus θER minus ψER
dBS
dt= Λminus αBβMIBS
NTotal+ pSBM + ψES minus τBS
dBI
dt=αBβMIBS
NTotalminus δBBI minus τBI
dBR
dt= (1minus σ)δBBI + pRBM minus τBR
dBM
dt= ψER minus (pS + pR)BM minus τBM
dBD
dt= σδBBI
dHS
dt= minusαHβMIHS
NTotal
dHE
dt=αHβMIHS
NTotalminus δEHE
dHF
dt= (1minus γ minus κ)δEHE minus δFHF
dHN
dt= κδEHE minus δNHN
dR
dt= δFHF + (1minus ω)δNHN + γδEHE
dD
dt= ωδNHN
The parameter NTotal includes the total blood supplysuch that NTotal = BS(t) + BI(t) + BR(t) + BM (t) +HS(t) + HE(t) + HF (t) + HN (t) + R(t) Also note thatthere is no birth nor natural death rate for humans likethere is for birds and mosquitoes Since the model takesplaces over a three month period the death rate for hu-mans is negligible The other parameter values for thismodel are listed in Table 1
The interaction between the mosquito bird and hu-man populations is the base of our model of the spreadof the virus Infected mosquitoes lay eggs at a rate of bthat become infected larvae Once hatched they matureat growth rate m into infected mosquitoes continuingthe cycle Similarly susceptible mosquitoes lay eggs ata rate of b which become susceptible larvae that grow
into susceptible mosquitoes at a rate of m Once adultssusceptible mosquitoes bite at a rate of β the averagebiting rate per day We assume the mosquito biting ratefor both humans and birds are the same [22] Dependingon the amount of infected birds as αMβNTotal modelsthere is a chance that a susceptible mosquito bites an in-fected bird exposing the mosquito to the virus Once thevirus enters their system the susceptible mosquito movesinto the exposed mosquito category Exposed mosquitoestransition to infected mosquitoes at a rate of η and arethen able to transmit West Nile Virus to birds and hu-mans as well as participate in vertical transmission
Infected mosquitoes are also able to bite suscep-tible birds and change them into infected birds asαBβNTotal models After a bird is infected it has achance to recover which (1minus σ)δB models and a chanceto die from the virus which σδB models Recovered birdspass their immunity down to their young which microφR sim-ulates These eggs hatch and grow into birds at a rate ofψ There is also a chance that they do not pass theirimmunity down to their young which (1minus micro)φR modelsIf this were to happen they would lay susceptible eggsinstead of recovered eggs Susceptible eggs grow into sus-ceptible birds at a rate of ψ These birds lay susceptibleeggs which φS models
West Nile Virus similarly affects humans αHβNTotal
represents the infected mosquitoes that bite susceptiblehumans and expose them to the disease [22] Once a hu-man is exposed to the disease there is a chance that theywill recover immediately from the disease at a rate of γδE However they may suffer from West Nile Fever at a rateof (1 minus γ minus κ)δE Most people recover from West NileFever at a rate of δF while others develop neuroinvasivedisease at a rate of κδE Once somebody has neuroinva-sive disease they either recover at a rate of (1minus ω)δN orthey die at a rate of ωδN
Several studies suggest the use of adulticides in orderto control the mosquito population [7 9 12 22] T (t)models the adulticides Within this function s equals theeffectiveness of the adulticide being used Our model usess-values of 0 20 40 and 80 treatment effective-ness as given by example in Figure 3 Simulating spray-ing at certain intervals such as every seven fourteen ortwenty-four days makes this function dependent on timeFor example if we were to treat weekly T (t) = s whent equiv 0 mod 7 and T (t) = 0 when t 6equiv 0 mod 7
We also set our immune birds to lose or permanentlymaintain their immunity at various time intervals basedon their species (ie every 14 days for chickens 10 daysfor owls and 9 days for house sparrows [20]) After eachtime interval pS percent of the immune bird populationmoves to the susceptible bird category and pR percent ofthe immune bird population moves to the recovered birdcategory [20]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Table 1 Parameters and their units values and sources for the system of ordinary differential equations that modelsthe disease dynamics
Parameter Definitions Units Value Source
b Mosquito Birth Rate Larvae(Day middot Adults) 0045 [22]
m Mosquito Maturation Rate Adults(Larvae middot Day) 007 [23 27]
δL Natural Larval Death Rate Dayminus1 0027 [11]
αM Probability of Transmission from Birdsto Mosquitoes
mdash 023 [12]
β Bite Rate Dayminus1 25 [22]
δM Natural Mosquito Death Rate Dayminus1 0031 [11]
T (t) T (t) = s Success Rate of Adulticides Dayminus1 varies [22]
η Virus Incubation Rate in Mosquitoes Dayminus1 01 [10]
φS Egg Laying Rate for Susceptible Birds Dayminus1 varies [15 19 21]
micro Percent of Eggs Receiving Passive Immunity mdash varies [15 19 21]
φR Egg Laying Rate for Recovered Birds Dayminus1 varies [15 19 21]
θ Natural Death Rate of Bird Eggs Dayminus1 045 [15 19 21]
ψ Maturation Rate of Bird Eggs Dayminus1 varies [15 19 21]
Λ Recruitment Rate of Birds BirdsDay varies [22]
αB Probability of Transmission from Mosquitoesto Birds
mdash 027 [12]
Ï„ Natural Bird Death Rate Dayminus1 varies [26]
δB Rate of Recovery in Birds Dayminus1 1(45) [16]
σ Fraction of WNV Infected Birds Dyingfrom the Disease
Dayminus1 072 [16]
αH Probability of Transmission from Mosquitoesto Humans
mdash 006 [6]
δE Incubation Period in Humans Dayminus1 14 [22]
γ Fraction of Human Population thatis Asymptomatic
Dayminus1 075 [8]
κ Fraction of Human Population thatCan Develop Neuroinvasive Disease
Dayminus1 001 [8]
δF Rate of Recovery for WNV Fever in Humans Dayminus1 114 [6]
δN Rate of Recovery for Neuroinvasive Diseasein Humans
Dayminus1 1(375) [13]
ω Fraction of Humans Dyingfrom Neuroinvasive Disease
Dayminus1 01 [8]
pS Percent of Immune Birds that Losetheir Immunity
mdash varies [15 19 20 21]
pR Percent of Immune Birds that Keeptheir Immunity
mdash varies [15 19 20 21]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 1 Schematic diagram of the model featuring alleighteen populations subdivided into three categoriesmosquitoes represented by circles birds represented by di-amonds and humans represented by squares The dashedlines showcase interactions between species while the solidlines showcase interactions within the same species Notethe passive immunity cycle in the bird category
screech-owls [15] Both vertical transmission and passiveimmunity are essential parts of simulating the dynamicsof West Nile Virus precisely Vertical transmission de-pends on the overall mosquito population and increasesthe spread of disease while passive immunity depends onboth the bird population and a birdrsquos chances of recover-ing from the disease slowing disease spread The goal ofour work is to give insight to the importance of passiveimmunity in birds and to show how it is incorporated inthe disease dynamics
2 Model
Our model utilizes a system of eighteen ordinary differ-ential equations (ODEs) based off of [22] that act as amodified SEIR model explained in the schematic dia-gram in Figure 1 There are three main categories birdsmosquitoes and humans The birds category highlightspassive immunity the mosquitoes category highlights ver-tical transmission and these important additions to themodel affect the humans category Every category inter-acts with at least one other category in order to properlysimulate the spread of West Nile Virus
The bird category contains seven sections suscepti-ble eggs (ES) immune eggs (ER) immune birds (BM )susceptible birds (BS) infected birds (BI) recovered
birds (BR) and dead birds (BD) Susceptible and in-fected birds lay susceptible eggs These eggs hatch andgrow into susceptible birds that are capable of developingWest Nile Virus Birds that have been infected with andsubsequently become immune to the virus lay recoveredeggs The process of adult bird populations passing theirimmunity onto their young is passive immunity recoveredeggs hatch into the temporary state of immune birds [20]This state makes the birds immune to West Nile Virus fora specific amount of time before they join the suscepti-ble bird category Susceptible birds are not infected withthe disease but are capable of developing West Nile Virus[22] Infected birds are currently suffering from the dis-ease and are capable of spreading the virus to susceptiblemosquitoes [22] Birds move into the recovered categoryafter they are no longer hosting the disease [22] Once abird is recovered it is permanently immune to the disease[22] Recovered birds are also able to pass their immunityonto their young if they reproduce The dead birds cat-egory records only birds that have died from West NileVirus [22]
The mosquito category contains five sections sus-ceptible larvae (LS) infected larvae (LI) suscepti-ble mosquitoes (MS) exposed mosquitoes (ME) andinfected mosquitoes (MI) Susceptible and exposedmosquitoes lay susceptible larvae These larvae havea chance of maturing into susceptible healthy adultmosquitoes Infected mosquitoes lay infected larvae [2]Due to vertical transmission infected mosquitoes havethe ability to pass West Nile Virus onto their young[2] In turn these larvae will mature into infectedmosquitoes The difference between the categories ofmature mosquitoes is that susceptible mosquitoes arehealthy but biting infected birds will infect them exposedmosquitoes bit an infected bird but are not yet infectiousand infected mosquitoes have West Nile Virus and areable to pass the disease onto birds and humans [22] In-fected mosquitoes are also able to vertically transmit theirdisease to their young should they reproduce
The human category contains six sections suscepti-ble humans (HS) exposed humans (HE) humans withWest Nile Fever (HF ) humans with neuroinvasive dis-ease (HN ) recovered humans (R) and dead humans (D)Susceptible humans are healthy but able to catch WestNile Virus should they be bitten Infected mosquitoesbite humans exposing them to the disease There is achance that exposed humans recover before developingany symptoms of West Nile Fever or neuroinvasive dis-ease However humans are considered dead-end hostsand are unable to infect mosquitoes even if they are in-fected [19] Recovered humans have recovered from eitherWest Nile Fever or neuroinvasive disease and are immuneto catching West Nile Virus again Dead humans refersto humans who have perished from neuroinvasive disease
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
dLS
dt= b(MS +ME)minusmLS minus δLLS
dLI
dt= bMI minusmLI minus δLLI
dMS
dt= mLS minus
αMβMSBI
NTotalminus δMMS minus T (t)MS
dME
dt=αMβMSBI
NTotalminus ηME minus δMME minus T (t)ME
dMI
dt= mLI + ηME minus δMMI minus T (t)MI
dES
dt= φS(BS +BI) + (1minus micro)φRBR minus θES minus ψES
dER
dt= microφRBR minus θER minus ψER
dBS
dt= Λminus αBβMIBS
NTotal+ pSBM + ψES minus τBS
dBI
dt=αBβMIBS
NTotalminus δBBI minus τBI
dBR
dt= (1minus σ)δBBI + pRBM minus τBR
dBM
dt= ψER minus (pS + pR)BM minus τBM
dBD
dt= σδBBI
dHS
dt= minusαHβMIHS
NTotal
dHE
dt=αHβMIHS
NTotalminus δEHE
dHF
dt= (1minus γ minus κ)δEHE minus δFHF
dHN
dt= κδEHE minus δNHN
dR
dt= δFHF + (1minus ω)δNHN + γδEHE
dD
dt= ωδNHN
The parameter NTotal includes the total blood supplysuch that NTotal = BS(t) + BI(t) + BR(t) + BM (t) +HS(t) + HE(t) + HF (t) + HN (t) + R(t) Also note thatthere is no birth nor natural death rate for humans likethere is for birds and mosquitoes Since the model takesplaces over a three month period the death rate for hu-mans is negligible The other parameter values for thismodel are listed in Table 1
The interaction between the mosquito bird and hu-man populations is the base of our model of the spreadof the virus Infected mosquitoes lay eggs at a rate of bthat become infected larvae Once hatched they matureat growth rate m into infected mosquitoes continuingthe cycle Similarly susceptible mosquitoes lay eggs ata rate of b which become susceptible larvae that grow
into susceptible mosquitoes at a rate of m Once adultssusceptible mosquitoes bite at a rate of β the averagebiting rate per day We assume the mosquito biting ratefor both humans and birds are the same [22] Dependingon the amount of infected birds as αMβNTotal modelsthere is a chance that a susceptible mosquito bites an in-fected bird exposing the mosquito to the virus Once thevirus enters their system the susceptible mosquito movesinto the exposed mosquito category Exposed mosquitoestransition to infected mosquitoes at a rate of η and arethen able to transmit West Nile Virus to birds and hu-mans as well as participate in vertical transmission
Infected mosquitoes are also able to bite suscep-tible birds and change them into infected birds asαBβNTotal models After a bird is infected it has achance to recover which (1minus σ)δB models and a chanceto die from the virus which σδB models Recovered birdspass their immunity down to their young which microφR sim-ulates These eggs hatch and grow into birds at a rate ofψ There is also a chance that they do not pass theirimmunity down to their young which (1minus micro)φR modelsIf this were to happen they would lay susceptible eggsinstead of recovered eggs Susceptible eggs grow into sus-ceptible birds at a rate of ψ These birds lay susceptibleeggs which φS models
West Nile Virus similarly affects humans αHβNTotal
represents the infected mosquitoes that bite susceptiblehumans and expose them to the disease [22] Once a hu-man is exposed to the disease there is a chance that theywill recover immediately from the disease at a rate of γδE However they may suffer from West Nile Fever at a rateof (1 minus γ minus κ)δE Most people recover from West NileFever at a rate of δF while others develop neuroinvasivedisease at a rate of κδE Once somebody has neuroinva-sive disease they either recover at a rate of (1minus ω)δN orthey die at a rate of ωδN
Several studies suggest the use of adulticides in orderto control the mosquito population [7 9 12 22] T (t)models the adulticides Within this function s equals theeffectiveness of the adulticide being used Our model usess-values of 0 20 40 and 80 treatment effective-ness as given by example in Figure 3 Simulating spray-ing at certain intervals such as every seven fourteen ortwenty-four days makes this function dependent on timeFor example if we were to treat weekly T (t) = s whent equiv 0 mod 7 and T (t) = 0 when t 6equiv 0 mod 7
We also set our immune birds to lose or permanentlymaintain their immunity at various time intervals basedon their species (ie every 14 days for chickens 10 daysfor owls and 9 days for house sparrows [20]) After eachtime interval pS percent of the immune bird populationmoves to the susceptible bird category and pR percent ofthe immune bird population moves to the recovered birdcategory [20]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Table 1 Parameters and their units values and sources for the system of ordinary differential equations that modelsthe disease dynamics
Parameter Definitions Units Value Source
b Mosquito Birth Rate Larvae(Day middot Adults) 0045 [22]
m Mosquito Maturation Rate Adults(Larvae middot Day) 007 [23 27]
δL Natural Larval Death Rate Dayminus1 0027 [11]
αM Probability of Transmission from Birdsto Mosquitoes
mdash 023 [12]
β Bite Rate Dayminus1 25 [22]
δM Natural Mosquito Death Rate Dayminus1 0031 [11]
T (t) T (t) = s Success Rate of Adulticides Dayminus1 varies [22]
η Virus Incubation Rate in Mosquitoes Dayminus1 01 [10]
φS Egg Laying Rate for Susceptible Birds Dayminus1 varies [15 19 21]
micro Percent of Eggs Receiving Passive Immunity mdash varies [15 19 21]
φR Egg Laying Rate for Recovered Birds Dayminus1 varies [15 19 21]
θ Natural Death Rate of Bird Eggs Dayminus1 045 [15 19 21]
ψ Maturation Rate of Bird Eggs Dayminus1 varies [15 19 21]
Λ Recruitment Rate of Birds BirdsDay varies [22]
αB Probability of Transmission from Mosquitoesto Birds
mdash 027 [12]
Ï„ Natural Bird Death Rate Dayminus1 varies [26]
δB Rate of Recovery in Birds Dayminus1 1(45) [16]
σ Fraction of WNV Infected Birds Dyingfrom the Disease
Dayminus1 072 [16]
αH Probability of Transmission from Mosquitoesto Humans
mdash 006 [6]
δE Incubation Period in Humans Dayminus1 14 [22]
γ Fraction of Human Population thatis Asymptomatic
Dayminus1 075 [8]
κ Fraction of Human Population thatCan Develop Neuroinvasive Disease
Dayminus1 001 [8]
δF Rate of Recovery for WNV Fever in Humans Dayminus1 114 [6]
δN Rate of Recovery for Neuroinvasive Diseasein Humans
Dayminus1 1(375) [13]
ω Fraction of Humans Dyingfrom Neuroinvasive Disease
Dayminus1 01 [8]
pS Percent of Immune Birds that Losetheir Immunity
mdash varies [15 19 20 21]
pR Percent of Immune Birds that Keeptheir Immunity
mdash varies [15 19 20 21]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
dLS
dt= b(MS +ME)minusmLS minus δLLS
dLI
dt= bMI minusmLI minus δLLI
dMS
dt= mLS minus
αMβMSBI
NTotalminus δMMS minus T (t)MS
dME
dt=αMβMSBI
NTotalminus ηME minus δMME minus T (t)ME
dMI
dt= mLI + ηME minus δMMI minus T (t)MI
dES
dt= φS(BS +BI) + (1minus micro)φRBR minus θES minus ψES
dER
dt= microφRBR minus θER minus ψER
dBS
dt= Λminus αBβMIBS
NTotal+ pSBM + ψES minus τBS
dBI
dt=αBβMIBS
NTotalminus δBBI minus τBI
dBR
dt= (1minus σ)δBBI + pRBM minus τBR
dBM
dt= ψER minus (pS + pR)BM minus τBM
dBD
dt= σδBBI
dHS
dt= minusαHβMIHS
NTotal
dHE
dt=αHβMIHS
NTotalminus δEHE
dHF
dt= (1minus γ minus κ)δEHE minus δFHF
dHN
dt= κδEHE minus δNHN
dR
dt= δFHF + (1minus ω)δNHN + γδEHE
dD
dt= ωδNHN
The parameter NTotal includes the total blood supplysuch that NTotal = BS(t) + BI(t) + BR(t) + BM (t) +HS(t) + HE(t) + HF (t) + HN (t) + R(t) Also note thatthere is no birth nor natural death rate for humans likethere is for birds and mosquitoes Since the model takesplaces over a three month period the death rate for hu-mans is negligible The other parameter values for thismodel are listed in Table 1
The interaction between the mosquito bird and hu-man populations is the base of our model of the spreadof the virus Infected mosquitoes lay eggs at a rate of bthat become infected larvae Once hatched they matureat growth rate m into infected mosquitoes continuingthe cycle Similarly susceptible mosquitoes lay eggs ata rate of b which become susceptible larvae that grow
into susceptible mosquitoes at a rate of m Once adultssusceptible mosquitoes bite at a rate of β the averagebiting rate per day We assume the mosquito biting ratefor both humans and birds are the same [22] Dependingon the amount of infected birds as αMβNTotal modelsthere is a chance that a susceptible mosquito bites an in-fected bird exposing the mosquito to the virus Once thevirus enters their system the susceptible mosquito movesinto the exposed mosquito category Exposed mosquitoestransition to infected mosquitoes at a rate of η and arethen able to transmit West Nile Virus to birds and hu-mans as well as participate in vertical transmission
Infected mosquitoes are also able to bite suscep-tible birds and change them into infected birds asαBβNTotal models After a bird is infected it has achance to recover which (1minus σ)δB models and a chanceto die from the virus which σδB models Recovered birdspass their immunity down to their young which microφR sim-ulates These eggs hatch and grow into birds at a rate ofψ There is also a chance that they do not pass theirimmunity down to their young which (1minus micro)φR modelsIf this were to happen they would lay susceptible eggsinstead of recovered eggs Susceptible eggs grow into sus-ceptible birds at a rate of ψ These birds lay susceptibleeggs which φS models
West Nile Virus similarly affects humans αHβNTotal
represents the infected mosquitoes that bite susceptiblehumans and expose them to the disease [22] Once a hu-man is exposed to the disease there is a chance that theywill recover immediately from the disease at a rate of γδE However they may suffer from West Nile Fever at a rateof (1 minus γ minus κ)δE Most people recover from West NileFever at a rate of δF while others develop neuroinvasivedisease at a rate of κδE Once somebody has neuroinva-sive disease they either recover at a rate of (1minus ω)δN orthey die at a rate of ωδN
Several studies suggest the use of adulticides in orderto control the mosquito population [7 9 12 22] T (t)models the adulticides Within this function s equals theeffectiveness of the adulticide being used Our model usess-values of 0 20 40 and 80 treatment effective-ness as given by example in Figure 3 Simulating spray-ing at certain intervals such as every seven fourteen ortwenty-four days makes this function dependent on timeFor example if we were to treat weekly T (t) = s whent equiv 0 mod 7 and T (t) = 0 when t 6equiv 0 mod 7
We also set our immune birds to lose or permanentlymaintain their immunity at various time intervals basedon their species (ie every 14 days for chickens 10 daysfor owls and 9 days for house sparrows [20]) After eachtime interval pS percent of the immune bird populationmoves to the susceptible bird category and pR percent ofthe immune bird population moves to the recovered birdcategory [20]
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Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Table 1 Parameters and their units values and sources for the system of ordinary differential equations that modelsthe disease dynamics
Parameter Definitions Units Value Source
b Mosquito Birth Rate Larvae(Day middot Adults) 0045 [22]
m Mosquito Maturation Rate Adults(Larvae middot Day) 007 [23 27]
δL Natural Larval Death Rate Dayminus1 0027 [11]
αM Probability of Transmission from Birdsto Mosquitoes
mdash 023 [12]
β Bite Rate Dayminus1 25 [22]
δM Natural Mosquito Death Rate Dayminus1 0031 [11]
T (t) T (t) = s Success Rate of Adulticides Dayminus1 varies [22]
η Virus Incubation Rate in Mosquitoes Dayminus1 01 [10]
φS Egg Laying Rate for Susceptible Birds Dayminus1 varies [15 19 21]
micro Percent of Eggs Receiving Passive Immunity mdash varies [15 19 21]
φR Egg Laying Rate for Recovered Birds Dayminus1 varies [15 19 21]
θ Natural Death Rate of Bird Eggs Dayminus1 045 [15 19 21]
ψ Maturation Rate of Bird Eggs Dayminus1 varies [15 19 21]
Λ Recruitment Rate of Birds BirdsDay varies [22]
αB Probability of Transmission from Mosquitoesto Birds
mdash 027 [12]
Ï„ Natural Bird Death Rate Dayminus1 varies [26]
δB Rate of Recovery in Birds Dayminus1 1(45) [16]
σ Fraction of WNV Infected Birds Dyingfrom the Disease
Dayminus1 072 [16]
αH Probability of Transmission from Mosquitoesto Humans
mdash 006 [6]
δE Incubation Period in Humans Dayminus1 14 [22]
γ Fraction of Human Population thatis Asymptomatic
Dayminus1 075 [8]
κ Fraction of Human Population thatCan Develop Neuroinvasive Disease
Dayminus1 001 [8]
δF Rate of Recovery for WNV Fever in Humans Dayminus1 114 [6]
δN Rate of Recovery for Neuroinvasive Diseasein Humans
Dayminus1 1(375) [13]
ω Fraction of Humans Dyingfrom Neuroinvasive Disease
Dayminus1 01 [8]
pS Percent of Immune Birds that Losetheir Immunity
mdash varies [15 19 20 21]
pR Percent of Immune Birds that Keeptheir Immunity
mdash varies [15 19 20 21]
wwwsporajournalorg 2020 Volume 6(1) page 19
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
wwwsporajournalorg 2020 Volume 6(1) page 20
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
wwwsporajournalorg 2020 Volume 6(1) page 21
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
wwwsporajournalorg 2020 Volume 6(1) page 22
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Table 1 Parameters and their units values and sources for the system of ordinary differential equations that modelsthe disease dynamics
Parameter Definitions Units Value Source
b Mosquito Birth Rate Larvae(Day middot Adults) 0045 [22]
m Mosquito Maturation Rate Adults(Larvae middot Day) 007 [23 27]
δL Natural Larval Death Rate Dayminus1 0027 [11]
αM Probability of Transmission from Birdsto Mosquitoes
mdash 023 [12]
β Bite Rate Dayminus1 25 [22]
δM Natural Mosquito Death Rate Dayminus1 0031 [11]
T (t) T (t) = s Success Rate of Adulticides Dayminus1 varies [22]
η Virus Incubation Rate in Mosquitoes Dayminus1 01 [10]
φS Egg Laying Rate for Susceptible Birds Dayminus1 varies [15 19 21]
micro Percent of Eggs Receiving Passive Immunity mdash varies [15 19 21]
φR Egg Laying Rate for Recovered Birds Dayminus1 varies [15 19 21]
θ Natural Death Rate of Bird Eggs Dayminus1 045 [15 19 21]
ψ Maturation Rate of Bird Eggs Dayminus1 varies [15 19 21]
Λ Recruitment Rate of Birds BirdsDay varies [22]
αB Probability of Transmission from Mosquitoesto Birds
mdash 027 [12]
Ï„ Natural Bird Death Rate Dayminus1 varies [26]
δB Rate of Recovery in Birds Dayminus1 1(45) [16]
σ Fraction of WNV Infected Birds Dyingfrom the Disease
Dayminus1 072 [16]
αH Probability of Transmission from Mosquitoesto Humans
mdash 006 [6]
δE Incubation Period in Humans Dayminus1 14 [22]
γ Fraction of Human Population thatis Asymptomatic
Dayminus1 075 [8]
κ Fraction of Human Population thatCan Develop Neuroinvasive Disease
Dayminus1 001 [8]
δF Rate of Recovery for WNV Fever in Humans Dayminus1 114 [6]
δN Rate of Recovery for Neuroinvasive Diseasein Humans
Dayminus1 1(375) [13]
ω Fraction of Humans Dyingfrom Neuroinvasive Disease
Dayminus1 01 [8]
pS Percent of Immune Birds that Losetheir Immunity
mdash varies [15 19 20 21]
pR Percent of Immune Birds that Keeptheir Immunity
mdash varies [15 19 20 21]
wwwsporajournalorg 2020 Volume 6(1) page 19
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
wwwsporajournalorg 2020 Volume 6(1) page 20
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
wwwsporajournalorg 2020 Volume 6(1) page 21
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
wwwsporajournalorg 2020 Volume 6(1) page 22
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 2 Susceptible and immune egg populations withan 80 treatment effectiveness against mosquitoes No-tice the treatment frequency increasing from no treatmentto bi-weekly treatment to weekly treatment
3 Results
31 Birds and Eggs
To show the effects of different hatching rates growthrates laying rates and passive immunity rates we se-lected three bird species (due to there being pre-existingliterature of these species) house sparrows [21] screechowls [15] and chickens [19] We then plotted them againstdifferent treatment effectiveness rates and treatment oc-currence rates Some differences of these species includechickens overall having the highest egg laying rate due tothem being able to lay one egg per day However thismeans that their clutch size is only one much smallerthan both house sparrows which have a clutch size any-where between four and six eggs and screech owls whichhave on average a clutch size of 3 eggs Sparrows laya clutch every 3ndash4 weeks while screech owls lay a clutchabout once per month The hatching rate for every birdspecies is also different It is important to consider theaverage rates for birds in the specific community to obtainthe most accurate results
As seen in Figure 2 our simulation suggests that theamount of eggs that develop an immunity to West Nile
Figure 3 Infected and immune bird populations as af-fected by different ψ-values (00 03 06 09) with treat-ment frequency being bi-weekly The treatment effective-ness increases left to right at 20 increments Note thatthese oscillations are from a bi-weekly treatment and frombirds leaving the immunity category
Virus through passive immunity is inversely proportionalto the frequency of the treatment Treating the mosquitopopulation with adulticides leads to a smaller recoveredegg population This is due to the decrease in birds be-coming infected with the virus in the first place and thusleading to less birds becoming recovered So it followsthat there would be less birds that become immune be-cause there are less recovered birds laying immune eggsSo the more rampant West Nile Virus the more of aneffect passive immunity has on bird and egg populations
The amount of susceptible eggs increases with the fre-quency of treatment This is because of the lack of dis-ease in the area similar to how the lack of disease showsa decrease in the recovered egg category Birds are lesslikely to become infected with West Nile Virus and aretherefore remaining in the susceptible category to lay sus-ceptible eggs with no chance to develop an immunity tothe disease However this is not the only cause Withthe lack of disease birds are less likely to die creating anoverall increase in the bird population [17]
The parameter ψ is the maturation rate from eggs to
wwwsporajournalorg 2020 Volume 6(1) page 20
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
wwwsporajournalorg 2020 Volume 6(1) page 21
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
wwwsporajournalorg 2020 Volume 6(1) page 22
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
birds From Figure 3 we can see that as ψ increases sodoes the overall bird populations since more birds arein the population Also as treatment goes up boththe infected and immune bird populations decrease be-cause fewer birds are becoming infected since the infectedmosquitoes are being killed off Immune birds decreaseoverall since there are fewer birds becoming infected thatrecover and are laying eggs This growth rate for eggscan be adjusted based on the species of birds in onersquoslocal community
The parameter pR is the percent of immune birds thatretain their passive immunity throughout their entireadulthood This varies based on the species of birdsFrom Figure 4 infected bird population increases as pRdecreases When there is a smaller percent of birds retain-ing their immunity there are more birds that become sus-ceptible after their temporary period of being an immunebird Thus there are more birds that can become in-fected with the disease The population of immune birdsincreases as the percent of birds retaining their immunityincreases This is attributed to an increase in recoveredbirds that will lay eggs that will eventually hatch intoimmune birds
As expected the less treatment available the more in-fected birds there are as we see in Figure 5 We canalso see that as treatment effectiveness increases themore dramatically the infected bird populations drop be-cause the treatment kills of a larger population of infectedmosquitoes drastically lowering the chance of birds be-coming infected with the disease later on
32 Mosquitoes and Larvae
The bird population also affects the mosquitoes FromFigure 6 the less effective the treatment is the morethe population of birds affects the mosquito populationThe infected mosquito and larvae populations grow as ψincreases because there is an increase of birds that thedisease can infect We can also see from Figure 6 thatthe infected mosquito population does indeed decrease asthe treatment becomes more effective
The different rates that different bird species have im-pacts the mosquito populations So in Figure 7 we com-pare the infected mosquito population with different birdspecies that have different hatching rates clutch sizespassive immunity rates and growth rates
Infected mosquitoes decrease with more treatment asdo all mosquito populations The more often the treat-ment the more the graph oscillates This is due to spray-ing in different intervals and killing a large portion ofmosquitoes off at a time However as seen in the con-trol case if we do not treat at all the infected mosquitopopulation will actually increase posing a health hazard
Figure 4 Infected and immune bird populations as af-fected by different pR-values (00 02 04 06 08) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andbirds leaving the immunity category
33 Humans
Infected humans are composed of both humans with neu-roinvasive disease and humans with West Nile Fever InFigure 8 we see that infected humans diminish as fewerhumans are exposed to the disease As expected the morewe treat the fewer humans that become infected with thedisease
4 Discussion and Conclusion
Modeling West Nile Virus is crucial to simulating thespread of the disease in local communities to prepare fordisease outbreaks in both human and bird populationsSince these models are used to improve public health it isparamount to give the most precise model Since passiveimmunity and vertical transmission are both phenomenaobserved in the natural world it is necessary to includethem within the disease dynamics [1 2] Our model im-proved upon past models by exploring the importance ofpassive immunity in birds showing that passive immunity
wwwsporajournalorg 2020 Volume 6(1) page 21
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
wwwsporajournalorg 2020 Volume 6(1) page 22
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 5 The infected bird population represented bythree species of birds house sparrows (blue) screech owls(red) and chickens (yellow) From left to right treatmenteffectiveness increases in 20 increments and from top tobottom treatment frequency increases from no treatmentto bi-weekly treatment to weekly treatment Note thatthese oscillations are from various treatment frequenciesand birds leaving the immunity category
affects both the bird and mosquito populations Passiveimmunity increases bird immunity and lowers the diseasespread overall
The more factors that are considered the more ac-curate our estimations of disease spread will be Usingpassive immunity requires a new category of birds alsoknown as immune birds These are different from recov-ered birds in that after the small period of immunity thebirds will be able to get infected again This is essen-tial knowledge to apply to mathematical modeling as itchanges the outcome and results of other mathematicalmodels Vertical transmission increases the rate of infec-tion and thus is a potential threat to human and birdhealth Thus it is important to utilize these in math-ematical models Our work has laid the foundation ofutilizing both of passive immunity and vertical transmis-sion in West Nile Virus modeling yet there is still moreto be done For instance it would be interesting to lookat the stability analysis of the model as well as a mix ofbird populations This would provide accurate results fora specific area when considering the makeup of the birdpopulation by species It would also be compelling to
Figure 6 Infected mosquitoes and larvae populations asaffected by different ψ-values (00 03 06 09) withtreatment frequency being bi-weekly The treatment ef-fectiveness increases left to right at 20 increments Notethat these oscillations are from a bi-weekly treatment andfrom birds leaving the immunity category
consider temperature patterns in future work as temper-ature affects a multitude of parameters such as the ratemosquitoes lay their eggs the rate birds enter and leavethe community and the mosquito biting rate
In order to more accurately simulate the disease spreadin a specific location scientists need data for birds in thatlocation such as the average bird egg laying rates theaverage egg growth rates and the average rates of passiveimmunity for the specific bird species in that area Thiswill give the most accurate results for the disease spreadin any given community Instead of using specific birdsin the model these values can be replaced for the averagevalues for passerine birds in a local area
Since West Nile Virus mainly pertains to the summermonths in terms of disease spread it is also importantto consider the variance of bird populations over severalyears An expansion of this model could consider thelong-term ramifications of West Nile Virus in a certainarea Overall our work serves as a template for modelingWest Nile Virus in any of the diverse environments inwhich West Nile Virus is extant
wwwsporajournalorg 2020 Volume 6(1) page 22
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
Figure 7 The infected mosquito population using threedifferent species of birds house sparrows (blue) screechowls (red) and chickens (yellow) From left to righttreatment effectiveness increases in 20 increments andfrom top to bottom treatment frequency increases fromno treatment to bi-weekly treatment to weekly treatment
Author Contributions
Noelle West (undergraduate) and Vinodh Chellamuthu(faculty) designed the model performed the numericalsimulations and analyzed model output Noelle West(undergraduate) implemented the model using MatLaband did the literature review
References
[1] Ahlers L R H amp Goodman A G (2018) The Im-mune Responses of the Animal Hosts of West NileVirus A Comparison of Insects Birds and Mam-mals Frontiers in cellular and infection microbiol-ogy 8 96 doi 103389fcimb201800096
[2] Anderson J F Main A J Cheng G FerrandinoF J amp Fikrig E (2012) Horizontal and verticaltransmission of West Nile Virus genotype NY99 byCulex salinarius and genotypes NY99 and WN02by Culex tarsalis The American journal of trop-ical medicine and hygiene 86 (1) 134ndash139 doi104269ajtmh201211-0473
Figure 8 The infected human (humans with WNV Feverand humans with neuroinvasive disease) population usingthree different species of birds house sparrows (blue)screech owls (red) and chickens (yellow) From left toright treatment effectiveness increases in 20 incrementsand from top to bottom treatment frequency increasesfrom no treatment to bi-weekly treatment to weekly treat-ment
[3] Baitchman E J Tlusty M F amp Murphy H W(2007) Passive Transfer Of Maternal Antibod-ies To West Nile Virus In Flamingo Chicks(Phoenicopterus Chilensis And PhoenicopterusRuber Ruber) Journal of Zoo and WildlifeMedicine 38 (2) 337ndash340 doi 1016381042-7260(2007)038[0337ptomat]20co2
[4] Barrett A (2014) Economic burden of West NileVirus in the United States The American journal oftropical medicine and hygiene 90 (3) 389ndash390 doi104269ajtmh14-0009
[5] Bergsman L D Hyman J M amp Manore C A(2015) A mathematical model for the spread of WestNile Virus in migratory and resident birds Math-ematical Biosciences and Engineering 13 (2) 401ndash424 doi 103934mbe2015009
[6] Bowman C Gumel A B van den Driessche PWu J amp Zhu H (2005) A mathematical model forassessing control strategies against West Nile VirusBulletin of Mathematical Biology 67 1107ndash1133
wwwsporajournalorg 2020 Volume 6(1) page 23
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[7] The Centers for Disease Control and Preven-tion (2018 December 10) Integrated MosquitoManagement url httpswwwcdcgov
westnilevectorcontrolintegrated_mosquito_
managementhtml
[8] The Center for Disease Control and Prevention(2019 November 12) West Nile Virus url https
wwwcdcgovwestnileindexhtml
[9] Chen Longbin (2017) Mathematical and StatisticalModels of Culex Mosquito Abundance and Trans-mission Dynamics of West Nile Virus with WeatherImpact (Doctoral dissertation) York Space Insti-tutional Repository url httphdlhandlenet
1031534496
[10] Darensburg T amp Kocic V L (2004) On the dis-crete model of West Nile-like epidemics Proc DynSys Appl 4 358ndash366
[11] Daszak P (2000) Emerging Infectious Diseasesof Wildlifendash Threats to Biodiversity and Hu-man Health Science 287 (5452) 443ndash449 doi101126science2875452443
[12] Diekmann O amp Heesterbeek J A P (2000) Math-ematical epidemiology of infectious diseases modelbuilding analysis and interpretation ChichesterWiley
[13] Flores Anticona E M Zainah H Ouellette D Ramp Johnson L E (2012) Two casereports of neuroin-vasive West Nile Virus infection in the critical careunit Case Reports in Infectious Diseases 2012 Ar-ticle ID 839458 doi 1011552012839458
[14] Gibbs S E Hoffman D M Stark L M Marle-nee N L Blitvich B J Beaty B J amp StallknechtD E (2005) Persistence of antibodies to West NileVirus in naturally infected rock pigeons (Columbalivia) Clinical and diagnostic laboratory immunol-ogy 12 (5) 665ndash667 doi 101128CDLI125665-6672005
[15] Hahn D C Nemeth N M Edwards E BrightP R amp Komar N (2006) Passive West Nile Virusantibody transfer from maternal Eastern screech-owls (Megascops asio) to progeny Avian Dis 50454ndash455
[16] Komar N Langevin S Hinten S Nemeth NEdwards E Hettler D Davis B Bowen R ampBunning M (2003) Experimental Infection of NorthAmerican Birds with the New York 1999 Strain ofWest Nile Virus Emerging Infectious Diseases 9 (3)311ndash322 doi 103201eid0903020628
[17] Maidana N A amp Yang H M (2011) Dynamicof West Nile Virus transmission considering sev-eral coexisting avian populations Mathematicaland Computer Modelling 53 (5ndash6) 1247ndash1260 doi101016jmcm201012008
[18] Miller B R Nasci R S Savage H M Lut-wama J J Peters C J Lanciotti R S amp God-sey M S (2000) First field evidence for naturalvertical transmission of West Nile Virus in Culexunivittatus complex mosquitoes from Rift Valleyprovince Kenya The American Journal of Trop-ical Medicine and Hygiene 62 (2) 240ndash246 doi104269ajtmh200062240
[19] Nemeth N M amp Bowen R A (2007) Dynamicsof passive immunity to West Nile Virus in domesticchickens (Gallus gallus domesticus) The AmericanJournal of Tropical Medicine and Hygiene 76 310ndash317
[20] Nemeth N M Bowen R A amp Oesterle P T(2008) Passive Immunity to West Nile Virus Pro-vides Limited Protection in a Common Passer-ine Species The American Journal of Tropi-cal Medicine and Hygiene 79 (2) 283ndash290 doi104269ajtmh200879283
[21] Nemeth N M Oesterle P T amp Bowen R A(2009) Humoral immunity to West Nile Virus islong-lasting and protective in the house sparrow(Passer domesticus) The American journal of trop-ical medicine and hygiene 80 (5) 864ndash869
[22] Pawelek K A Niehaus P Salmeron C HagerE J amp Hunt G J (2014) Modeling Dynamics ofCulex pipiens Complex Populations and AssessingAbatement Strategies for West Nile Virus PLoSONE 9 (9) doi 101371journalpone0108452
[23] Roehrig J T (2013) West Nile Virus in the UnitedStates - a historical perspective Viruses 5 (12)3088ndash3108 doi103390v5123088
[24] Rossi S L Ross T M amp Evans J D (2010) WestNile Virus Clinics in laboratory medicine 30 (1) 47ndash65 doi 101016jcll200910006
[25] Sejvar J J (2003) West Nile Virus an historicaloverview The Ochsner journal 5 (3) 6ndash10
[26] Simpson J E Hurtado P J Medlock J MolaeiG Andreadis T G Galvani A P amp Diuk-WasserM A (2011) Vector host-feeding preferences drivetransmission of multi-host pathogens West NileVirus as a model system Proceedings of the RoyalSociety B Biological Sciences 279 925ndash933 doi101098rspb20111282
wwwsporajournalorg 2020 Volume 6(1) page 24
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
Modeling the Effects of Passive Immunity in Birds for WNV West Chellamuthu
[27] Wonham M J Lewis M A Renclawowicz J ampvan den Driessche P (2006) Transmission assump-tions generate conflicting predictions in host-vectordisease models a case study in West Nile VirusEcology Letters 9 (6) 706ndash725 doi 101111j1461-0248200600912x
wwwsporajournalorg 2020 Volume 6(1) page 25
