View
223
Download
0
Tags:
Embed Size (px)
Citation preview
EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS
By
Abdul-Aziz Yakubu
Howard University
Epidemics In Strongly Fluctuating Populations: Constant Environments
Barrera et al. MTBI Cornell University Technical Report (1999). Valezquez et al. MTBI Cornell University Technical Report (1999). Arreola, R. MTBI Cornell University Technical Report (2000). Gonzalez, P. A. MTBI Cornell University Technical Report (2000). Castillo-Chavez and Yakubu, Contemporary Mathematics, Vol 284 (2001). Castillo-Chavez and Yakubu, Math. Biosciences, Vol 173 (2001). Castillo-Chavez and Yakubu, Non Linear Anal TMA, Vol 47 (2001). Castillo-Chavez and Yakubu, IMA (2002). Yakubu and Castillo-Chavez J. Theo. Biol. (2002). K. Rios-Soto, Castillo-Chavez, E. Titi, &A. Yakubu, AMS (In press). Abdul-Aziz Yakubu, JDEA (In press).
Epidemics In Strongly Fluctuating Populations: Periodic Environments
Franke & Yakubu : JDEA (2005) Franke & Yakubu : SIAM Journal of
Applied Mathematics (2006) Franke & Yakubu : Bulletin of
Mathematical Biology ( In press) Franke & Yakubu : Mathematical
Biosciences (In press)
Epidemics In Strongly Fluctuating Populations: Almost Periodic Environments
T. Diagana, S. Elaydi and Yakubu (Preprint)
Demographic Equation
Examples Of Demography In Constant Environments
Asymptotically Bounded Growth Demographic Equation (1) with constant rate Λ and
initial condition N(0) gives rise to the following N(t+1)= N(t)+Λ, N(0)=N0
Since
N(1)= N0 +Λ,
N(2)=2 N0 +(+1) Λ,
N(3)=3 N0 +(2 ++1) Λ, ...,
N(t)=t N0 +(t-1+t-2+...++1) Λ
Asymptotically Bounded Growth(Constant Environment)
N(t)1,0 Since
.1 if
11
1, if )(
. t as 1
0
0
Nt
tNtN
Geometric Growth(constant environment)
If new recruits arrive at the positive per-capita rate pergeneration, that is, if f(N(t))=N(t) then
N(t+1)=( + )N(t).That is, N(t)= ( +)t N(0).
The demographic basic reproductive number is
Rd=/(1-)
Rd, a dimensionless quantity, gives the average number of descendants produced by a small pioneer population (N(0)) over its life-time.
• Rd>1 implies that the population invades at a geometric rate.• Rd<1 leads to extinction.
Density-Dependent Growth Rate
If f(N(t))=N(t)g(N(t)), then
N(t+1)=N(t)g(N(t))+ N(t).That is, N(t+1)=N(t)(g(N(t))+).
• Demographic basic reproductive number is
Rd=g(0)/(1-)
The Beverton-Holt Model: Compensatory Dynamics
The Beverton-Holt Model:Compensatory Dynamics
Beverton-Holt Model With The Allee Effect
The Allee effect, a biological phenomenon named after W. C. Allee, describes a positive relation between population density and the per capita growth rate of species.
Effects Of Allee Effects On Exploited Stocks
The Ricker Model: Overcompensatory Dynamics
g(N)=exp(p-N)
The Ricker Model: Overcompensatory Dynamics
Are population cycles globally stable?
In constant environments, population cycles are not globally stable (Elaydi-Yakubu, 2002).
Constant Recruitment In Periodic Environments
Constant Recruitment In Periodic Environment
Periodic Beverton-Holt Recruitment Function
Signature Functions For Classical Population Models In Periodic Environments: R. May, (1974, 1975, etc) Franke and Yakubu : Bulletin of Mathematical Biology
(In press) Franke and Yakubu: Periodically Forced Leslie Matrix
Models (Mathematical Biosciences, In press) Franke and Yakubu: Signature function for the Smith-
Slatkin Model (JDEA, In press)
Geometric Growth In Periodic Environment
SIS Epidemic Model
Disease Persistence Versus Extinction
Asymptotically Cyclic Epidemics
Example
Example
Epidemics and Geometric Demographics
Persistence and Geometric Demographics
Cyclic Attractors and Geometric Demographics
Multiple Attractors
Question
• Are disease dynamics driven by demographic dynamics?
S-Dynamics Versus I-Dynamics
(Constant Environment)
SIS Models In
Constant Environments
In constant environments, the demographic dynamics drive both the susceptible and infective dynamics whenever the disease is not fatal.
Periodic Constant Demographics Generate Chaotic Disease
Dynamics
Periodic Beverton-Holt Demographics Generate Chaotic
Disease Dynamics
Periodic Geometric Demographics Generate Chaotic
Disease Dynamics
Conclusion
• We analyzed a periodically forced discrete-time SIS model via the epidemic threshold parameter R0
• We also investigated the relationship between pre-disease invasion population dynamics and disease dynamics• Presence of the Allee effect in total population implies its presence in the infective population.• With or without the infection of newborns, in constant environments the demographic dynamics drive the disease dynamics•Periodically forced SIS models support multiple attractors• Disease dynamics can be chaotic where demographic dynamics are non-chaotic
S-E-I-S MODEL
Other Models
1. Malaria in Mali (Bassidy Dembele …Ph. D. Dissertation)
2. Epidemic Models With Infected Newborns (Karen Rios-Soto… Ph. D.
Dissertation)
Dynamical Systems Theory
Equilibrium Dynamics, Oscillatory Dynamics, Stability Concepts, etc Attractors and repellors (Chaotic attractors) Basins of Attraction Bifurcation Theory (Hopf, Period-doubling and saddle-node bifurcations) Perturbation Theory (Structural Stability)
Animal Diseases
Diseases in fish populations (lobster, salmon, etc)
Malaria in mosquitoes Diseases in cows, sheep, chickens, camels,
donkeys, horses, etc.