EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS

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

2. 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).

3. 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)

4. Epidemics In Strongly Fluctuating Populations: Almost Periodic Environments • T. Diagana, S. Elaydi and Yakubu (Preprint)

5. Demographic Equation

6. Examples Of Demography In Constant Environments

7. 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

8. Asymptotically Bounded Growth(Constant Environment)

9. 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.

10. 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-)

11. The Beverton-Holt Model: Compensatory Dynamics

12. The Beverton-Holt Model:Compensatory Dynamics

13. 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.

14. Effects Of Allee Effects On Exploited Stocks

15. The Ricker Model: Overcompensatory Dynamics g(N)=exp(p-N)

16. The Ricker Model: Overcompensatory Dynamics

17. Are population cycles globally stable? In constant environments, population cycles are not globally stable (Elaydi-Yakubu, 2002).

18. Constant Recruitment In Periodic Environments

19. Constant Recruitment In Periodic Environment

20. Periodic Beverton-Holt Recruitment Function

21. 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)

22. Geometric Growth In Periodic Environment

24. SIS Epidemic Model

25. Disease Persistence Versus Extinction

26. Asymptotically Cyclic Epidemics

27. Example

28. Example

29. Epidemics and Geometric Demographics

30. Persistence and Geometric Demographics

31. Cyclic Attractors and Geometric Demographics

32. Multiple Attractors

33. Question • Are disease dynamics driven by demographic dynamics?

34. S-Dynamics Versus I-Dynamics (Constant Environment)

35. SIS Models In Constant Environments In constant environments, the demographic dynamics drive both the susceptible and infective dynamics whenever the disease is not fatal.

36. Periodic Constant Demographics Generate Chaotic Disease Dynamics

37. Periodic Beverton-Holt Demographics Generate Chaotic Disease Dynamics

38. Periodic Geometric Demographics Generate Chaotic Disease Dynamics

39. 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

40. S-E-I-S MODEL

41. Other Models • Malaria in Mali (Bassidy Dembele …Ph. D. Dissertation) • Epidemic Models With Infected Newborns (Karen Rios-Soto… Ph. D. Dissertation)

42. 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)

43. Animal Diseases • Diseases in fish populations (lobster, salmon, etc) • Malaria in mosquitoes • Diseases in cows, sheep, chickens, camels, donkeys, horses, etc.