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EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS

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## EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS

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**EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS**By Abdul-Aziz Yakubu Howard University ayakubu@howard.edu**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)**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**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 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 populationinvades 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-)**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.**The Ricker Model: Overcompensatory Dynamics**g(N)=exp(p-N)**Are population cycles globally stable?**In constant environments, population cycles are not globally stable (Elaydi-Yakubu, 2002).**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)**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 alsoinvestigated 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**Other Models**• Malaria in Mali (Bassidy Dembele …Ph. D. Dissertation) • 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.