DIMACS 10/9/06 Zhilan Feng Collaborators and references Zhilan Feng, David Smith, F. Ellis McKenzie, Simon Levin Mathematical Biosciences (2004) Zhilan Feng, Yingfei Yi, Huaiping Zhu J. Dynamics and Differential Equations (2004) Zhilan Feng, Carlos Castillo-Chavez Mathematical Biosciences and Engineering (2006) Coupling ecology and evolution: malaria and the S-gene across time scales Zhilan Feng, Department of Mathematics, Purdue University

DIMACS 10/9/06 Zhilan Feng 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

DIMACS 10/9/06 Zhilan Feng 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

DIMACS 10/9/06 Zhilan Feng A simple SIS model with a vector (mosquito) b(N) : growth rate of hosts  h : infection rate of hosts  m : infection rate of mosquitoes  : recovery rate of hosts  : malaria-related death rate   : per capita natural death rate of hosts    infection rate of mosquitoes S: susceptible hosts I: infected hosts N=S+I: total number of hosts z: fraction of infected mosquitoes (1)

DIMACS 10/9/06 Zhilan Feng Dynamics of system (1) The basic reproductive number is  The disease dies out if R 0 <1  A unique endemic equilibrium E* = (S*, I*, z*) exists and is l.a.s. if R 0 >1

DIMACS 10/9/06 Zhilan Feng A simple model of population genetics (2)  ,  : per capita natural, extra (due to S-gene) death rate respectively : frequency of A alleles q=1-p : frequency of a alleles Assume that aa is lethal so N aa =0. N i : number of type i individuals (i=AA, Aa, aa)

DIMACS 10/9/06 Zhilan Feng Dynamics of system (2) Note from the equation for the a gene: Thus, the gene frequency q converges to zero.

DIMACS 10/9/06 Zhilan Feng A model coupling dynamics of malaria and the S-gene (3) i  1, 2 (AA, Aa)

DIMACS 10/9/06 Zhilan Feng Analysis of model (3) Introduce fractions: Note that ( i  1,2 ) Then system (3) is equivalent to: (4) A measure of S-gene frequency

DIMACS 10/9/06 Zhilan Feng Fast and slow time scales Note: b, m i,  i are on the order of 1/decades  hi,  i,  mi,  m are on the order of 1/days Rescale the parameters:  > 0 is small

DIMACS 10/9/06 Zhilan Feng Separation of fast and slow dynamics Then system (4) w.r.t. the fast time variables: (6) and w.r.t. the slow time variables (Andreasen and Christiansen, 1993): (5)

DIMACS 10/9/06 Zhilan Feng N. Fenichel. Geometric singular perturbation theory for ordinary differential equations Geometric theory of singular perturbations Let be a set of stable equilibria of (5) with  =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 y1y1 w (0.3, 0.58)

DIMACS 10/9/06 Zhilan Feng On the fast time-scale, if R 0 > 1 then all solutions are hyperbolically asymptotic to the endemic equilibrium E m * = (y 1 *, y 2 *, z*) Malaria disease dynamics on the fast time scale The reproductive number of malaria is and z* > 0 is a solution to a quadratic equation with k i =…… where w is the S-gene frequency

DIMACS 10/9/06 Zhilan Feng S-gene dynamics on the slow time scale Define the fitness of the S-gene to be then where Note: is the death rate weighted by malaria related W i  Fitness F =  1 -  2 determines The slow dynamics E*=(w*, N*) Global interior attractor  2 =  1  2 = h(  1 ) 11 22 S-gene cannot invade Population extinction Bi-stable equilibria possible

DIMACS 10/9/06 Zhilan Feng Possible equilibria of the slow system 1 0w N H2H2 w* H1H1 (1,K) w 1 0 N H2H2 H1H1 w1*w1*w2*w2*

DIMACS 10/9/06 Zhilan Feng N Q(w,N) 22 2 (w*, N*) w Global dynamics of the slow manifold Suppose there is a closed orbit around E*  (w*,N*). Construct Q 1 (w), Q 2 (N) and Q(w,N)=Q 1 +Q 2 as: 1 0 w N H2H2 w* H1H1 (1,K)  The slow system (7) has no periodic solution or homoclinic orbit. Note thatand Contradiction

DIMACS 10/9/06 Zhilan Feng S-gene dynamics on the slow time scale E*=(w*, N*) Global interior attractor  2 =  1  2 = h(  1 ) 11 22 S-gene cannot invade Population extinction N w NN ww N w Stable Unstable Bistability

DIMACS 10/9/06 Zhilan Feng Effect of S-gene dynamics on malaria prevalence R0R0 w  y 1 + y 2 (c)    =0.09 (b)    =0.06 time Possession of the S-gene leads to longer-lasting parasitaemia (1/   ) in heterozygote individuals, and therefore the presence of resistance may actually increase infection prevalence w : S-gene frequency 1 /  i : Infectious period

DIMACS 10/9/06 Zhilan Feng Influence of malaria on population genetics : Death due to S-gene  i : Death due to malaria W i : Malaria parameters E*=(w*, N*) Global interior attractor  2 =  1  2 = h(  1 ) 11 22 S-gene cannot invade Population extinction A balancing selection against the evolution of resistance, with the strength of selection dependent upon malaria prevalence. Fitness F   1   2    W 1   W 2

DIMACS 10/9/06 Zhilan Feng 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. Conclusion

DIMACS 10/9/06 Zhilan Feng Acknowledgements National Science Foundation Jams S. McDonnell Foundation