Epidemic Attractors I n Periodic Environments

1. Epidemic Attractors In Periodic Environments • Abdul-Aziz Yakubu • Howard University • (ayakubu@howard.edu)

2. Question • Do basic tenets in theoretical ecology and epidemiology remain true when parameters oscillate or do they need modification?

3. Demographic Equations in Constant Environments N(t) is total population size in generation t. N(t+1) = f(N(t))+gN(t), (1) where gin (0,1) is the constant "probability" of surviving per generation, and f : R+→ R+ models the birth or recruitment process. Howard HOWARD

4. Examples Of Recruitment Functions In Constant Environments • If the recruitment rate is constant per generation, then the total population is asymptotically constant. • If the recruitment rate is f(N(t))=mN(t), then N(t)=(m+g)tN(0) and Rd=m/(1-g). • If the recruitment rate is density dependent via the Beverton-Holt model, then the total population is asymptotically constant. • If the recruitment rate is density dependent via the Ricker model, then the total population is cyclic or chaotic. • If the recruitment rate is density dependent via either a “modified” Beverton-Holt’s model or a “modified” Ricker’s model, then the demographic equation exhibits the Allee effect. • References: May (1974), Hassell (1976), Castillo-Chavez-Yakubu (2000, 2001), Franke-Yakubu (2005, 2006), Yakubu (2007). Howard

5. Other important aspects of realistic demographic equations include…. • Delays and Periodic (seasonal) effects. • Age structure and related effects. • Genetic variations etc • Multi-species and ecosystem effects. • Spatial effects and diffusion (S. Levin, Amer. Nat. 1974). • Deterministic versus stochastic effects, • ….

6. Demographic EquationIn Periodic Environments If the recruitment function is p - periodically forced, then the p - periodic demographic equation is N(t+1) = f(t,N(t))+LN(t), (2) where 0p5 N such that f(t,N(t)) = f(t+p,N(t))-t5Z+. We assume throughout that f(t,N)5C²(Z+× R+ , R+ ) and L 5 (0,1).

7. Examples Of Recruitment Functions In Periodic Environments • Periodic constant recruitment functionf(t,N(t)) = kt(1-L). • Periodic Beverton-Holt recruitment function • kt is ap-periodic the carrying capacity. • kt = kt+p-t5Z+.

8. Asymptotically Cyclic Demographics Theorem 1 (2005): Model (2) with f(t,N(t)) = kt(1-L) has a globally attracting positive s - periodic cycle, where s divides p, that starts at Theorem 2 (2005): Model (2) has a globally attracting positive s - periodic cycle when

9. Question • Are oscillations in the carrying capacity deleterious to a population? • Jillson, D.: Nature 1980 (Experimental results) • Cushing, J.: Journal of Mathematical Biology (1997). • May, R. M.: Stability & Complexity in Model Ecosystems (2001).

10. Resonance Versus Attenuance • When the recruitment function is the period constant, then the average total biomass remains the same as the average carrying capacity (the globally attracting cycle is neither resonant nor attenuant). • When the recruitment function is the periodic Beverton-Holt model, then periodic environments are always disadvantageous for our population (the globally attracting cycle is attenuant, Cushing et al., JDEA 2004, Elaydi & Sacker, JDEA 2005). • When all parameters are periodically forced, then attenuance and resonance depends on the model parameters (Franke-Yakubu, Bull. Math. Biol. 2006).

11. The Ricker Model in Periodic Environments In periodic environments, the Ricker Model exhibits multiple (coexisting) attractors via cusp bifurcation. Reference: Franke-Yakubu JDEA 2005

12. S-I-S Epidemic Models In Seasonal Environments

13. SIS Epidemic Model Using S(t) = N(t) - I(t)the I - equation becomes

14. Infective Density Sequence Let Then I(t+1) = FN(t)(I(t)), and the iterates of the nonautonomous map FN(t)is the set of density sequences generated by the infective equation.

15. Persistence and Uniform Persistence Definition: The total population in Model (2) is persistent if whenever N(0) > 0. The total population is uniformly persistent if there exists a positive constant Rsuch that whenever N(0) > 0. Periodic constant or Beverton-Holt recruitment functions give uniformly persistent total populations.

16. Basic Reproductive Number R0 • In constant environments f(t,N(t))=f(N(t)), and • R0 = -JLd′(0)/(1 – La). • Reference: Castillo Chavez and Yakubu (2001). • In constant environments, the presence of the Allee effect in the total population implies its presence in the infective population whenever R0>1. Reference: Yakubu (2007). • In periodic environments, if the total population is uniformly persistent then the disease goes extinct whenever R0<1. However, the disease persists uniformly whenever R0>1. • Reference:Franke-Yakubu (2006).

17. Question • What is the nature and structure of the basins of attraction of epidemic attractors in periodically forced discrete-time models?

18. Attractors

19. N-I System

20. Limiting Systems Theory(CCC, H. Thieme, and Zhao)

21. Compact Attractors

22. Period-Doubling Bifurcations and Chaos

23. Question • Are disease dynamics driven by demographic dynamics? (Castillo-Chavez & Yakubu, 2001-2002)

24. Illustrative Examples: Cyclic and Chaotic Attractors

25. Multiple Attractors

26. Illustrative Example: Multiple Attractors

27. Basins of Attraction

28. Basins Of Attraction

29. Periodic S-I-S Epidemic Models With Delay S(t+1)=f(t,N(t-k))+S(t)G(I(t)/N(t))+I(t)(1-), I(t+1)=(1-G(I(t)/N(t)))S(t)+  I(t) Demographic equation becomes N(t+1)=f(t, N(t-k))+ N(t)

30. S-E-I-S MODEL

31. Malaria in Seasonal Environments ……Bassidy Dembele and Avner Friedman

32. Malaria • Malaria is one of the most life threatening tropical diseases for which no successful vaccine has been developed (UNICEF 2006 REPORT).

33. Malaria • How effective is Sulfadoxine Pyrimethane (SP) as a temporary vaccine?

34. Malaria In Bandiagara-Mali Am. J. Trop. Med. Hyg. 2002: Coulibaly et al. Bassidy, Friedman and Yakubu, 2007

35. Drug Administration

36. Protocols 2 and 3 versus 1 • Protocol 2 shows no significant advantage over 1 in reducing the first malaria episode. • Protocol 3 reduces the first episode of malaria significantly. • Both Protocol 2 and 3 may significantly reduce the side effects of drugs because of sufficient spacing of drug administration.

37. Question • What are the effects of almost periodic environments on disease persistence and control? • Diagana-Elaydi-Yakubu, JDEA 2007

38. Conclusion • In constant environments, the demographic dynamics drive the disease dynamics (CC-Y, 2001). However, in periodic environments disease dynamics are independent of the demographic dynamics. • In constant environments, simple SIS models do not exhibit multiple attractors. However, in periodic environments the corresponding simple SIS models exhibit multiple attractors with complicated basins of attraction. • In periodic environments, simple SIS models with no Allee effect exhibit extreme dependence of long-term dynamics on initial population sizes. What are the implications on the persistence and control of infectious diseases?