EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS By AbdulAziz Yakubu

EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS By Abdul-Aziz Yakubu Howard University ayakubu@howard. edu

Epidemics In Strongly Fluctuating Populations: Constant Environments q Barrera et al. MTBI Cornell University Technical Report (1999). q Valezquez et al. MTBI Cornell University Technical Report (1999). q Arreola, R. MTBI Cornell University Technical Report (2000). q Gonzalez, P. A. MTBI Cornell University Technical Report (2000). q Castillo-Chavez and Yakubu, Contemporary Mathematics, Vol 284 (2001). q Castillo-Chavez and Yakubu, Math. Biosciences, Vol 173 (2001). q Castillo-Chavez and Yakubu, Non Linear Anal TMA, Vol 47 (2001). q Castillo-Chavez and Yakubu, IMA (2002). q Yakubu and Castillo-Chavez J. Theo. Biol. (2002). q K. Rios-Soto, Castillo-Chavez, E. Titi, &A. Yakubu, AMS (In press). q Abdul-Aziz Yakubu, JDEA (In press).

Epidemics In Strongly Fluctuating Populations: Periodic Environments q Franke & Yakubu : JDEA (2005) q Franke & Yakubu : SIAM Journal of Applied Mathematics (2006) q Franke & Yakubu : Bulletin of Mathematical Biology ( In press) q Franke & Yakubu : Mathematical Biosciences (In press)

Epidemics In Strongly Fluctuating Populations: Almost Periodic Environments q 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)=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

Asymptotically Bounded Growth (Constant Environment)

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 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: q R. May, (1974, 1975, etc) q Franke and Yakubu : Bulletin of Mathematical Biology (In press) q Franke and Yakubu: Periodically Forced Leslie Matrix Models (Mathematical Biosciences, In press) q 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 R • 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 0 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 q q q 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 q Diseases in fish populations (lobster, salmon, etc) q Malaria in mosquitoes q Diseases in cows, sheep, chickens, camels, donkeys, horses, etc.