Collaborators and references

1. Collaborators and references Coupling ecology and evolution: malaria and the S-gene across time scalesZhilan Feng, Department of Mathematics, Purdue University • Zhilan Feng, David Smith, F. Ellis McKenzie, Simon LevinMathematical Biosciences(2004) • Zhilan Feng, Yingfei Yi, Huaiping Zhu J. Dynamics and Differential Equations(2004) • Zhilan Feng, Carlos Castillo-Chavez Mathematical Biosciences and Engineering(2006)

2. Outline • Malaria epidemiology and the sickle-cell gene • An endemic model of malaria without genetics • A population genetics model without epidemics • A model coupling epidemics and S-gene dynamics • Analysis of the model • Discussion

3. Malaria and the sickle-cell gene • Malaria has long been a scourge to humans. The exceptionally high mortality in some regions has led to strong selection for resistance, even at the cost of increased risk of potentially fatal red blood cell deformities in some offspring. • Genes that confer resistance to malaria when they appear in heterozygous • individuals are known to lead to sickle-cell anemia, or other blood diseases, • when they appear in homozygous form. • Thus, there is balancing selection against the evolution of resistance, with the strength of that selection dependent upon malaria prevalence. • Over longer time scales, the increased frequency of resistance may decrease the prevalence of malaria and reduce selection for resistance • However, possession of the sickle-cell gene leads to longer-lasting parasitaemia in heterozygote individuals, and therefore the presence of resistance may actually increase infection prevalence We explore the interplay among these processes, operating over very different time scales

4. A simple SIS model with a vector (mosquito) (1) b(N) : growth rate of hosts bh : infection rate of hosts bm : infection rate of mosquitoes g : recovery rate of hosts a : malaria-related death rate mh : per capita natural death rate of hosts mm : infection rate of mosquitoes S: susceptible hosts I: infected hosts N=S+I: total number of hosts z: fraction of infected mosquitoes

5. The basic reproductive number is • The disease dies out if R0<1 • A unique endemic equilibrium E* = (S*, I*, z*) exists and is l.a.s. if R0>1 Dynamics of system (1)

6. Assume that aa is lethal so Naa=0. Ni : number of type i individuals (i=AA, Aa, aa) : frequency of A alleles q=1-p :frequency of a alleles m, n :per capita natural, extra (due to S-gene) death rate respectively A simple model of population genetics (2)

7. Dynamics of system (2) Note from the equation for the a gene: Thus, the gene frequency q converges to zero.

8. A model coupling dynamics of malaria and the S-gene (3) i=1, 2 (AA, Aa)

9. Analysis of model (3) Introduce fractions: ( i=1,2 ) Note that Then system (3) is equivalent to: A measure of S-gene frequency (4)

10. Fast and slow time scales Note: b, mi , aiare on the order of 1/decades bhi , bi , gmi , mm are on the order of 1/days Rescale the parameters: e > 0 is small

11. Separation of fast and slow dynamics Then system (4) w.r.t. the fast time variables: (5) and w.r.t. the slow time variables (Andreasen and Christiansen, 1993): (6)

12. y1 1 (0.3, 0.58) 0 0 1 w Geometric theory of singular perturbations N. Fenichel. Geometric singular perturbation theory for ordinary differential equations Let be a set of stable equilibria of (5) with e=0. Then in terms of (6) M is a 2-D slow manifold. The slow dynamics on M is described by (7) If the slow dynamics of (7) can be characterized via bifurcations, then the bifurcating dynamics on M are structurally stable hence robust to perturbations

13. w is the S-gene frequency Malaria disease dynamics on the fast time scale The reproductive number of malaria is On the fast time-scale, if R0 > 1 then all solutions are hyperbolically asymptotic to the endemic equilibrium Em* = (y1*, y2*, z*) where and z* > 0 is a solution to a quadratic equation with ki=……

14. Define the fitness of the S-gene to be then s2 Note: is the death rate weighted by malaria related Wi s2 = s1 S-gene cannot invade Population extinction E*=(w*, N*) Global interior attractor s2 = h(s1) s1 S-gene dynamics on the slow time scale where • Fitness F = s1 - s2 determines • The slow dynamics Bi-stable equilibria possible

15. Possible equilibria of the slow system N N H1 (1,K) (1,K) H1 H2 H2 0 w 0 w 1 1 w* w1* w2*

16. N (1,K) H1 H2 w 0 1 w* Global dynamics of the slow manifold • The slow system (7) has no periodic solution or homoclinic orbit. Suppose there is a closed orbit around E*=(w*,N*). Construct Q1(w), Q2(N) and Q(w,N)=Q1+Q2 as: Note that and Contradiction 2 Q(w,N) 0 -2 (w*, N*) 0 N 2000 w

17. N Stable Unstable w s2 N N N s2 = s1 S-gene cannot invade Population extinction E*=(w*, N*) Global interior attractor s2 = h(s1) w w w s1 S-gene dynamics on the slow time scale Bistability

18. R0 g2 w (c) g2=0.09 (b) g2=0.06 y1+ y2 y1+ y2 time time Effect of S-gene dynamics on malaria prevalence w :S-gene frequency 1/gi : Infectious period Possession of the S-geneleads to longer-lasting parasitaemia (1/g2) in heterozygote individuals, and therefore the presence of resistance may actually increase infection prevalence

19. Fitness F =s1 -s2 =- n + a1W1 - a2W2 s2 s2 = s1 S-gene cannot invade Population extinction E*=(w*, N*) Global interior attractor s2 = h(s1) s1 Influence of malaria on population genetics A balancing selection against the evolution of resistance, with the strength of selection dependent upon malaria prevalence. n: Death due to S-gene ai:Death due to malaria Wi: Malaria parameters

20. Conclusion • By coupling malaria epidemics and the S-gene dynamics, our model allows • for a joint investigation of • influence of malaria on population genetic composition • effect of the S-gene dynamics on the prevalence of malaria, and • coevolution of host and parasite These results cannot be obtained from epidemiology models without genetics or genetic models without epidemics.

21. Acknowledgements National Science Foundation Jams S. McDonnell Foundation