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Modeling Ebola at theMathematical andTheoretical BiologyInstitute (MTBI)C. Castillo-Chavez, K. Barley, D. Bichara, D. Chowell,E. Diaz Herrera, B. Espinoza, V. Moreno, S. Towers,and K. E. YongCommunicated by Steven J. Miller
AbstractWork with STEM students at MTBI advanced thepossibility that quarantine can cause increasedlevels of Ebola transmission.1
C. Castillo-Chavez is regents professor and the JoaquínBustoz, Jr. Professor of Mathematical Biology at ArizonaState University, as well as the executive director of theMathematical and Theoretical Biology Institute. His emailaddress is [email protected]. Barley is a graduate student at the Simon A. LevinMathematical, Computational and Modeling Sciences Cen-ter. His email address is [email protected]. Bichara is a postdoctoral fellow at the Simon A. LevinMathematical, Computational and Modeling Sciences Cen-ter. His email address is [email protected]. Chowell is a graduate student at the Simon A. LevinMathematical, Computational and Modeling SciencesCenter. His email address is [email protected] research has been partially supported by NSF-DMD1263374, NSF-IRES 1261211 and DODH98230-15-1-0021.The views expressed are the sole responsibility of the au-thors and not the funding agencies.1Write-up of a Centennial Lecture by Castillo-Chavez atthe 2015 MathFest of the Mathematical Association ofAmerica.For permission to reprint this article, please contact:[email protected].
DOI: http://dx.doi.org/10.1090/noti1364
Disease, Epidemics, and ModelsSince twenty-first-century problems stretchacross disciplines, they require interdisciplinaryapproaches. As Steve Strogatz [1] observes:
Cancer will not be cured by biologists work-ing alone. Its solutionwill require ameldingof both great discoveries of 1953 [the Fermi-Pasta-Ulam computer experiment and theWatson and Crick discovery of the chemicalstructure of DNA]. Many cancers, perhapsmost of them, involve the derangement ofbiochemical networks that choreograph theactivity of thousands of genes and proteins.As Fermi and his colleagues taught us, a
E. Diaz Herrera is a CONACYT Research Fellow at theInstituto Nacional de Salud Pública. His email address [email protected]. Espinoza is a graduate student at the Simon A. LevinMathematical, Computational and Modeling SciencesCenter. His email address is [email protected]. Moreno is a graduate student at the Simon A. LevinMathematical, Computational and Modeling SciencesCenter. His email address is [email protected]. Towers is professor of mathematics at the Simon A.Levin Mathematical, Computational and Modeling Sci-ences Center. Her email address is [email protected]. E. Yong is professor of mathematics at the Univer-sity of Hawaiì-West Oàhu. His email address is [email protected].
366 Notices of the AMS Volume 63, Number 4
complex system like this can’t be under-stood merely by cataloging its parts andthe rules governing their interactions. Thenonlinear logic of cancer will be fathomedonly through the collaborative efforts ofmolecular biologists—the heirs to Dr. Wat-son and Dr. Crick—and mathematicianswho specialize in complex systems—theheirs to Fermi, Pasta and Ulam.
Per Strogatz, the future scientists at the Mathe-matical and Theoretical Biology Institute (MTBI ),established at Cornell University in 1996, learnthe problem-solving ropes by engaging in collabo-rative, cross-discipline efforts—often enhanced bythe systematic use of computer experiments anddata science—to answer focused scientific ques-tions. The interdisciplinary research described inthis note involves the dynamics of deadly diseasessuch as Ebola.
Germ Theory and Epidemic ModelsAristotle’s hypothesis on the existence of invisi-ble microorganisms; Antonie van Leeuwenhoek’sdiscovery of such germs; and the germ-theoryframework advanced by Jacob Henle, Robert Koch,Joseph Lister, and Louis Pasteur served as the foun-dation for the study of infectious diseases. DanielBernoulli in 1760 introduced a mathematicalmodel used immediately to assess the effec-tiveness of inoculation against smallpox virus.Much later, W. N. Hamer developed and ana-lyzed a discrete-time mathematical model to helpunderstand the recurrence of measles epidemics.
In 1902 Sir Ronald Ross was awarded the NobelPrize for tying malaria to Plasmodium parasitescarried by mosquitoes. He formulated a nonlinearsystem of differential equations that captured itstransmission dynamics. His clear understandingof host-vector-pathogen dynamics drove him toconsider the impact of such interactions at thepopulation level. Ross concluded from his analy-sis that reducing the vector (mosquito) populationbelow some threshold would drastically decreasemalaria’s devastating impact on the health andsurvival of individuals living in malaria-infestedregions. Following Ross’s legacy, Kermack andMcKendrick advanced the concept of epidemicthreshold in the context of communicable dis-eases such as influenza or tuberculosis. Theideas of these pioneers were later popularizedby M. Gladwell in his best-selling book The Tip-ping Point. Important expansions and variationson these foundational models have given rise tothe field of mathematical epidemiology and itsapplications (see e.g. [2]).
Ross also highlighted in his research the powerof abstraction inherent in mathematics by explic-itly alluding to the applicability of his malariaframework to the study of sexually transmitted
diseases (STD). Cooke and Yorke followed throughon Ross’s ideas and produced one of the firstSTD models in 1973. Hethcote and Yorke laterintroduced the concept of core group, a conceptwhich has had a tremendous impact on policiesaimed at reducing gonorrhea incidence.
Ebola Control and Ebola Spatial DynamicsAs the Ebola outbreak spread acrossWest Africa in2014–2015, 24/7 cable news coverage ensured thatfear of the disease spread among large subpop-ulations of individuals living far outside affectedareas.
Whether or not Ebola-infected individuals canbe detected before becoming infectious turnedout to be a central question. MTBI alumnus DiegoChowell wondered how effective Ebola detectionusing Polymerase Chain Reaction (PCR) would beif deployed at the presymptomatic stage of thedisease. Following the research carried out byMTBIalumni during the 2003 Canada SARS outbreak,Chowell et al. [3] proposed the following singleoutbreak epidemic model; see also Figure 1, andTables 1 and 2.
S E1
E2
I J P
Figure 1. Modeling the effect of early detectionof Ebola.
𝑆 = −𝛽𝑆(𝐼 + 𝑙𝐽𝑁 ) ,
𝐸1 = 𝛽𝑆(𝐼 + 𝑙𝐽𝑁 )− 𝜅1𝐸1,
𝐸2 = 𝜅1𝐸1 −𝜅2𝐸2 − 𝑓𝑇𝐸2,𝐼 = 𝜅2𝐸2 − (𝛼+𝛾)𝐼,𝐽 = 𝛼𝐼 + 𝑓𝑇𝐸2 −𝛾𝑟𝐽,
�� = 𝛾(1 − 𝛿)𝐼 + 𝛾𝑟(1 − 𝛿)𝐽,�� = 𝛾𝛿𝐼 + 𝛾𝑟𝛿𝐽,𝑁 = 𝑆+ 𝐸1 +𝐸2 + 𝐼+ 𝐽+𝑅.
Table 1. Variables of the Model.
Class Description𝑆 Susceptible𝐸1 Latent undetectable𝐸2 Latent detectable𝐼 Infectious and symptomatic𝐽 Isolated𝑅 Recovered𝐷 Ebola-induced death𝑃 𝐷+𝑅
April 2016 Notices of the AMS 367
Table 2. Parameters of the Model.
Parameter Description Value𝛽 Mean transmission rate 0.3331/𝑘1 Mean period between undetectable state and latent detectable state 4 days1/𝑘2 Mean period between latent detectable state to infectious state 3 days1/𝛼 Mean period between infectious state and isolation state 3 days1/𝛾 Mean period between infectious state and recovery or Ebola-induced death state 6 days1/𝛾𝑟 Mean period between isolation state and recovery or Ebola-induced death state 7 days𝑓𝑇 Fraction of latent individuals diagnosed before onset of symptoms (0.1)𝛿 Fatality rate 0.7𝑙 Relative transmissibility of isolated individuals (0.1)
This transmission model was parameterizedusing prior MTBI research on Ebola dynamics. Itwas determined, via simulations, that the use ofmodern technologies (PCR) are indeed insufficientto stop an Ebola outbreak unless enough facilitiesare available to isolate a significant proportion ofEbola-diagnosed individuals (see Figure 2). More-over, the DNApolymerase used in the PCR reactionis prone to errors and can lead to mutations in thefragment generated. Additionally, the specificityof the generated PCR product may be changed bynonspecific binding of the primers to other similarsequences on the template DNA. Furthermore, inorder to design primers to generate a PCR product,some prior information on the sequence is usuallynecessary.
80
0 5 10 15 20 25 30 35 40 45 50
70
60
50
40
30
10
20
0
Proportion of latent individuals diagnosed before onset of symptoms (%)
Atta
ck ra
te (%
)
Effectiveness of isolation 50%
Effectiveness of isolation 60%Effectiveness of isolation 65%
Figure 2. Modeling the effect of early detectionof Ebola. The attack rate is defined as the totalnumber of Ebola cases per population size.
The research carried out in response to the2002–2003 SARS outbreak identified specific ap-proaches that made it possible to assess thepotential of a disease outbreak with limited data.Today, epidemiologists using single outbreak epi-demiologicalmodels in conjunctionwith incomingreported outbreak data can estimate the basic re-production numberℛ0 and often also the typicallydecreasing effective reproductive number ℛeff(𝑡).That is, it is possible to estimate the initial epidemicgrowth rate andassociated typicallydeclining ratesof growth over time, the result of the depletionof susceptible individuals over the course of anoutbreak. Towers et al. [4] found out that the rateof growth inWest Africa was in fact not decreasingover time but increasing (see Figure 3). This groupof researchers used changing epidemic growthestimates to forecast 6, 800 cases by the end of
September 2014, quite close to 6,553, the actualnumber of cases reported by the World HealthOrganization.
Figure 3. Time series of recorded averagenumber of new EVD cases per day during theinitial phase of the 2014 West Africanoutbreak for Guinea, Sierra Leone, and Liberia(dots). The green lines show a selection of thepiecewise exponential fits to the data. The redlines show moving-window exponential fitsfrom July 1 onwards.
Our epidemiologically model-driven data anal-ysis drove us to consider the possibility that theenforced “cordons sanitaires” mass quarantinerestrictions in West Africa in late summer 2014,which forcefully restricted the movement of indi-viduals within the most affected Ebola areas, mayhave in fact contributed to increased levels ofEbola transmission due to overcrowding and poorsanitation [4].
MTBI alumnus Edgar Diaz Herrera in his PhDdissertation [5] looked at the dynamics of visiblysymptomatic diseases such as leprosy or Ebolaon the spatial dynamics of epidemics. He askedwhether individuals will aggregate in particularspatial patterns as a result of physically visiblesymptoms. Diaz Herrera introduced the epidemi-ological classes of susceptible 𝑆(𝑥, 𝑡), infectious
368 Notices of the AMS Volume 63, Number 4
asymptomatic 𝐼1(𝑥, 𝑡), and infectious visibly symp-tomatic 𝐼2(𝑥, 𝑡) via the following nonlinear systemof partial differential equations in response to hisquestion:
𝜕𝑆𝜕𝑡 = − 𝛽
1+ 𝐼2𝑆𝐼1 +𝛼𝐼2 +𝐷𝑆
𝜕2𝑆𝜕𝑥2 ,
𝜕𝐼1𝜕𝑡 = 𝛽
1+ 𝐼2𝑆𝐼1 −𝛿𝐼1 +𝐷1
𝜕2𝐼1𝜕𝑥2 ,
𝜕𝐼2𝜕𝑡 = 𝛿𝐼1 −𝛼𝐼2 +𝐷2
𝜕2𝐼2𝜕𝑥2 .
Diaz Herrera’s model incorporated a modifiedmass-action law, where the transmission coeffi-cient 𝛽 was taken to be a decreasing function
𝛽1+ 𝐼2
of 𝐼2. The growing population of the visi-bly infectious is assumed to decrease the contactrates between individuals. We found [6] that theabove reaction-diffusion system was capable ofsupporting diffusive instabilities when 𝐼2 individ-uals are not infectious for too long. In this case,aggregation occurs when 𝐷2 > 𝐷1, rapidly when𝐷2 ≫ 𝐷1; see Figure 4.
Lagrangian Approach to Epidemic Modelsand EbolaLagrangian models (see e.g. [7]) keep track of eachindividual at all times. A general Susceptible-Infectious-Susceptible, or SIS, model involving𝑛-patches is given by the following system ofnonlinear equations:
⎧⎪⎨⎪⎩
𝑆𝑖 = 𝑏𝑖−𝑑𝑖𝑆𝑖+𝛾𝑖𝐼𝑖−∑𝑛𝑗=1(𝑆𝑖 infected in Patch 𝑗)
𝐼𝑖 = ∑𝑛𝑗=1(𝑆𝑖 infected in Patch 𝑗) − 𝛾𝑖𝐼𝑖 −𝑑𝑖𝐼𝑖
��𝑖 = 𝑏𝑖 −𝑑𝑖𝑁𝑖,where 𝑏𝑖, 𝑑𝑖, and 𝛾𝑖 denote the per capita birth,natural death, and recovery rates respectively.Infection is modeled as follows:[𝑆𝑖 infected in Patch 𝑗] = 𝛽𝑗⏟
the risk of infection in Patch 𝑗
× 𝑝𝑖𝑗𝑆𝑖⏟susceptible from Patch 𝑖 who are currently in Patch 𝑗
× ∑𝑛𝑘=1 𝑝𝑘𝑗𝐼𝑘
∑𝑛𝑘=1 𝑝𝑘𝑗𝑁𝑘⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
proportion of infected in Patch 𝑗
,
where the last term accounts for the effectiveinfection proportion in Patch 𝑗 at time. The modelreduces to the 𝑛-dimensional system
𝐼𝑖 =𝑛
∑𝑗=1
(𝛽𝑗𝑝𝑖𝑗(𝑏𝑖𝑑𝑖
− 𝐼𝑖) ∑𝑛𝑘=1 𝑝𝑘𝑗𝐼𝑘
∑𝑛𝑘=1𝑝𝑘𝑗
𝑏𝑘𝑑𝑘
)−(𝛾𝑖+𝑑𝑖)𝐼𝑖,
𝑖 = 1, 2,… ,𝑛,with a basic reproduction number ℛ0 that is afunction of the risk vectorℬ = (𝛽1, 𝛽2,… ,𝛽𝑛)𝑡 andthe residence timesmatrixℙ = (𝑝𝑖,𝑗), 𝑖, 𝑗 = 1,… ,𝑛,
where 𝑝𝑖,𝑗 denotes the proportion of the timethat an 𝑖-resident spends visiting Patch 𝑗. Weshow that when ℙ is irreducible (patches arestrongly connected), the Disease Free State isglobally asymptotically stable (g.a.s.) if ℛ0 ≤ 1,while whenever ℛ0 > 1 there exists a uniqueinterior equilibrium which is g.a.s.
The patch-specific basic reproduction numberis given by
ℛ𝑖0(ℙ) = ℛ𝑖
0 ×𝑛
∑𝑗=1
(𝛽𝑗
𝛽𝑖)𝑝𝑖𝑗 (
𝑝𝑖𝑗𝑏𝑖𝑑𝑖
∑𝑛𝑘=1𝑝𝑘𝑗
𝑏𝑘𝑑𝑘
) ,
from where we see, for example, that if 𝑝𝑘𝑗 =0 for all 𝑘 = 1,… ,𝑛, and 𝑘 ≠ 𝑖, then the diseasedies out in Patch 𝑖.
Figure 4. The densities for the asymptomaticinfectious 𝐼1 are plotted for 𝛼 = 0.05, 𝛽 = 1 and𝛿 = 1.3 for the Diaz Herrera model. Slowspatial aggregation of individuals (top) occurswhen the coefficients 𝐷2 > 𝐷1(𝐷1 = 10,𝐷2 = 20), and fast spatial aggregation(bottom) occurs when 𝐷2 ≫ 𝐷1(𝐷1 = 10,𝐷2 = 80).
Ongoing MTBI alumni work [7] uses this modelto testwhethermovement restrictions reduce over-all transmission. We embed susceptible, exposed,infectious, dead, and removed categories into thesimple and basic single-patch (S E I D R) model
April 2016 Notices of the AMS 369
Table 3. Variables and Parameters of the Contagion Model.
Parameter Description Base model values𝛼 Rate at which asymptomatic individuals recover
(not infectious due to acquired immunity) 0 − 0.458 day−1
𝛽 Per susceptible infection rate 0.3056 day−1
𝛾 Rate at which an infected recovers or dies 16.5 day−1
𝜅 Per capita progression rate 17 day−1
𝜈 Per capita body disposal rate 0.5 day−1
𝑓dead Proportion of infected who die due to infection 0.708𝜀 Scale: Ebola infectiousness of dead bodies 1.5
P120 0.2 0.4 0.6 0.8 1
# 105
0
1
2
3
4
5
6
7
8
9
10Final size
Patch 1Patch 2CumulativeNo mobility
Figure 5. Dynamics of maximum final size inthe one-way case with parameters:𝜀1,2 = 1,𝛽1 = 0.305,𝛽2 = 0.1, 𝑓death = 0.708,𝑘 = 1/7,𝛼 = 0,𝜈 = 1/2,𝛾 = 1/6.5. The initialpopulations are 𝑁1 = 𝑁2 = 1,000, 000. For mostvalues of the one-way mobility from Patch 1(high risk) to Patch 2 (low risk), the cumulativefinal size of the epidemic is smaller than forno mobility.
given by the following set of equations:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
𝑁 = 𝑆+𝐸+ 𝐼+𝐷+𝑅
𝑆=−𝛽𝑆 𝐼𝑁 −𝜀𝛽𝑆𝐷
𝑁
𝐸 = 𝛽𝑆 𝐼𝑁 + 𝜀𝛽𝑆𝐷
𝑁 − (𝜅+𝛼)𝐸
𝐼 = 𝜅𝐸−𝛾𝐼
�� = 𝑓dead𝛾𝐼 − 𝜈𝐷
�� = (1 − 𝑓dead)𝛾𝐼 + 𝜈𝐷+𝛼𝐸.
See Table 3 for a description of the parameters.The transition from 𝐷 to 𝑅 represents the properand appropriate handling of dead bodies duringthe burial or ritual services.
In the simplest version of our Lagrangian frame-work, there is a two-patchworld of “residents” and“visitors”, connected by a residence time matrix
ℙ = ( 𝑝11 𝑝12𝑝21 𝑝22
) .
P120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
# 106
0.8
0.9
1
1.1
1.2
1.3
1.4Final size
Beta2=0.122Beta2=0.12Beta2=0.118No mobility
Figure 6. Dynamics of maximum final size inthe one-way case with parameters: 𝜀1,2 =1,𝛽1 = 0.305,𝛽2 = 0.122, 0.12, 0.118, 𝑓death =0.708, 𝑘 = 1/7,𝛼 = 0,𝜈 = 1/2,𝛾 = 1/6.5. Theinitial populations are 𝑁1 = 𝑁2 = 1,000, 000. Asthe level of risk 𝛽2 in the low-risk patchincreases, the value of positive mobilitydisappears.
We explore, via simulations, the role of restric-tions in movement between patches. Preliminarysimulations assume that the patches have equalpopulations, but the high-risk area is assumedto have a high population density, poor healthfacilities, and limited resources, while the low-riskarea is assumed to host a sparsely distributed pop-ulation and solid health facilities and resources.This is a standard hypothesis for communicablediseases in general, and it applies to Ebola inparticular. The parts of West Africa where Ebolaexploded are poverty-stricken areas, with muchhigher density than middle or upper middle classneighborhoods, so the hypothesis is reasonable.Figures 5 and 6 show how mobility and risk af-fect the final size of the epidemic. Sometimesquarantining the high-risk area yields a largerepidemic.
All differences are assumed to be captured bythe assigned 𝛽-values.
Figure 5 and Figure 6 provide some ideas of therole that mobility and risk play in the dynamicsof Ebola in connected high/low-risk environments.Figure 5 plots both the global and the per patchfinal epidemic size as a function of mobility fromPatch 1 (high risk) to Patch 2, exclusively. We see,
370 Notices of the AMS Volume 63, Number 4
for example, that high single-direction mobilityleads to a smaller cumulative final epidemic sizein this case.
Of course, Figure 5 provides only part of thestory. In Figure 6 we plot global epidemic size asa function of the level of risk in the low-risk patch(𝛽2). We see from simulations that the quality ofservices in Patch 2, as defined by 𝛽2, would leadto a beneficial global result only if 𝛽2 ≤ 0.12. Also,simulations suggest that the choice 𝛼 > 0 bringsthe reproduction number ℛ0 below one muchfaster for a wide range of residence times.
Final Remarks and AcknowledgementsThis article has provided the opportunity to show-case the type of research carried out by MTBIstudents over the past twenty years. Members ofthe Simon A. Levin Mathematical, ComputationalandModeling Sciences Center, many of them exter-nal faculty members, join forces in addressing theresearch agenda proposed by undergraduate stu-dents. The research has been funded via NSF, NIH,DOD, and Sloan grants over the past two decades.We want to particularly thank Fred Brauer, TedGreenwood, James Schatz, andour friends and sup-porters atNSF for their continuous support for thisexperimental model of education and mentorshipthrough research, a model that has contributed tothe education of a significant number of minoritystudents and women.
References[1] Steven Strogatz, The real scientific hero of 1953,
The New York Times, November 4, 2003.[2] Herbert W. Hethcote, The mathematics of infec-
tious diseases, SIAM Review 42 (2000), no. 4, 599–653.MR1814049 (2002c:92034)
[3] Diego Chowell, Carlos Castillo-Chavez, Sri Kr-ishna, Xiangguo Qiu, and Karen S. Anderson,Modelling the effect of early detection of Ebola, TheLancet Infectious Diseases 15 (2015), no. 2, 148–149.
[4] Sherry Towers, Oscar Patterson-Lomba, andCarlos Castillo-Chavez, Temporal variations inthe effective reproduction number of the 2014 WestAfrica Ebola Outbreak, PLoS currents 6 (2014).
[5] Edgar Diaz Herrera, Diffusive Instability and Ag-gregation in Epidemics, Arizona State University, 2010.MR2792949
[6] Kamuela E. Yong, Edgar Díaz Herrera, and Car-los Castillo-Chavez, From bee species aggregationto models of disease avoidance: The Ben-Hur effect,preprint, arXiv:1510.04364, 2015.
[7] Derdei Bichara, Yun Kang, Carlos Castillo-Chavez, Richard Horan, and Charles Perrings,SIS and SIR epidemic models under virtual disper-sal, Bulletin of Mathematical Biology (2015), DOI10.1007/s11538-015-0113-5. MR3427113
April 2016 Notices of the AMS 371
