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EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS By Abdul-Aziz Yakubu Howard University ayakubu@howard.edu
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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).
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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)
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Epidemics In Strongly Fluctuating Populations: Almost Periodic Environments T. Diagana, S. Elaydi and Yakubu (Preprint)
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Demographic Equation
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Examples Of Demography In Constant Environments
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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)=N 0 Since N(1)= N 0 +Λ, N(2)= 2 N 0 +( +1) Λ, N(3)= 3 N 0 +( 2 + +1) Λ,..., N(t)= t N 0 +( t-1 + t-2 +...+ +1) Λ
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Asymptotically Bounded Growth (Constant Environment)
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Geometric Growth (constant environment) If new recruits arrive at the positive per-capita rate per generation, 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 R d = /(1- ) R d, a dimensionless quantity, gives the average number of descendants produced by a small pioneer population (N(0)) over its life-time. R d >1 implies that the population invades at a geometric rate. R d <1 leads to extinction.
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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 R d =g(0)/(1- )
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The Beverton-Holt Model: Compensatory Dynamics
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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.
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Effects Of Allee Effects On Exploited Stocks
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The Ricker Model: Overcompensatory Dynamics g(N)=exp(p-N)
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The Ricker Model: Overcompensatory Dynamics
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Are population cycles globally stable? In constant environments, population cycles are not globally stable (Elaydi-Yakubu, 2002).
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Constant Recruitment In Periodic Environments
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Constant Recruitment In Periodic Environment
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Periodic Beverton-Holt Recruitment Function
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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)
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Geometric Growth In Periodic Environment
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SIS Epidemic Model
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Disease Persistence Versus Extinction
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Asymptotically Cyclic Epidemics
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Example
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Epidemics and Geometric Demographics
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Persistence and Geometric Demographics
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Cyclic Attractors and Geometric Demographics
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Multiple Attractors
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Question Are disease dynamics driven by demographic dynamics?
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S-Dynamics Versus I-Dynamics (Constant Environment)
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SIS Models In Constant Environments In constant environments, the demographic dynamics drive both the susceptible and infective dynamics whenever the disease is not fatal.
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Periodic Constant Demographics Generate Chaotic Disease Dynamics
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Periodic Beverton-Holt Demographics Generate Chaotic Disease Dynamics
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Periodic Geometric Demographics Generate Chaotic Disease Dynamics
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Conclusion We analyzed a periodically forced discrete-time SIS model via the epidemic threshold parameter R 0 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
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S-E-I-S MODEL
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Other Models 1.Malaria in Mali (Bassidy Dembele …Ph. D. Dissertation) 2. Epidemic Models With Infected Newborns (Karen Rios-Soto… Ph. D. Dissertation)
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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)
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Animal Diseases Diseases in fish populations (lobster, salmon, etc) Malaria in mosquitoes Diseases in cows, sheep, chickens, camels, donkeys, horses, etc.
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