Howard Epidemic Attractors I n Periodic Environments Abdul-Aziz Yakubu Howard University

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

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

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))=  N(t), then N(t)=(  t N(0) and Rd=  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

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, ….

Howard Demographic Equation In 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 0 p 5 N such that f(t,N(t)) = f(t+p,N(t)) -t 5 Z +. We assume throughout that f(t,N) 5 C²(Z + × R +, R + ) and L 5 (0,1).

Howard Examples Of Recruitment Functions In Periodic Environments A.Periodic constant recruitment function f(t,N(t)) = k t (1- L). B.Periodic Beverton-Holt recruitment function 1.k t is a p-periodic the carrying capacity. 2.k t = k t+p -t 5 Z +.

Howard Asymptotically Cyclic Demographics Theorem 1 (2005): Model (2) with f(t,N(t)) = k t (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

Howard 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).

Howard 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).

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

Howard S-I-S Epidemic Models In Seasonal Environments

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

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

Howard 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 R such that whenever N(0) > 0. Periodic constant or Beverton-Holt recruitment functions give uniformly persistent total populations. Persistence and Uniform Persistence

Howard Basic Reproductive Number R 0 In constant environments f(t,N(t))=f(N(t)), and R 0 = -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. Reference: Franke-Yakubu (2006).

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

Howard Attractors

Howard N-I System

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

Howard Compact Attractors

Howard Period-Doubling Bifurcations and Chaos

Howard Question Are disease dynamics driven by demographic dynamics? (Castillo- Chavez & Yakubu, )

Howard Illustrative Examples: Cyclic and Chaotic Attractors

Howard Multiple Attractors

Howard Illustrative Example: Multiple Attractors

Howard Basins of Attraction

Howard Basins Of Attraction

Howard 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)

Howard S-E-I-S MODEL

Howard Malaria in Seasonal Environments …… Bassidy Dembele and Avner Friedman

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

Howard Malaria How effective is Sulfadoxine Pyrimethane (SP) as a temporary vaccine?

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

Howard Drug Administration

Howard 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.

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

Howard 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?