Coupling Ecology and Evolution: Malaria and the S-Gene Across Time Scales
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Coupling ecology and evolution: malaria and the S-gene across time scales Zhilan Feng, Department of Mathematics, Purdue University 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) 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) (1) Infection Susceptible S Quarantined Q Early Medical Encounter ME Late Medical Encounter ML Recovery Prodromal Symptoms IP Progression Presentation Diagnosis Recovery Waning Presentation 3 Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Recovery S: susceptible hosts I: infected hosts N=S+I: total number of hosts z: fraction of infected mosquitoes 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 DIMACS 10/9/06 Zhilan Feng Dynamics of system (1) 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 DIMACS 10/9/06 Zhilan Feng A simple model of population genetics (2) Infection Susceptible S Quarantined Q Presentation Recovery 4 Recovery Prodromal Symptoms IP Progression 22 Late Medical Encounter ML Recovery Waning Presentation Early Medical Encounter ME Diagnosis Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R 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 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) Infection Susceptible S Quarantined Q Presentation Recovery 4 Recovery Prodromal Symptoms IP Progression 22 Late Medical Encounter ML Recovery Waning Presentation Early Medical Encounter ME Diagnosis Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R DIMACS 10/9/06 Zhilan Feng Analysis of model (3) Introduce fractions: ( i =1,2 ) Note that Then system (3) is equivalent to: A measure of S-gene frequency Infection Susceptible S Quarantined Q Early Medical Encounter ME Late Medical Encounter ML Recovery Prodromal Symptoms IP Progression Presentation Diagnosis Recovery Waning Presentation 3 Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Recovery (4) DIMACS 10/9/06 Zhilan Feng Fast and slow time scales Note: b, mi , ai are on the order of 1/decades bhi , bi , gmi , mm are on the order of 1/days Rescale the parameters: Infection Susceptible S Quarantined Q Early Medical Encounter ME Late Medical Encounter ML Recovery Prodromal Symptoms IP Progression Presentation Diagnosis Recovery Waning Presentation 3 Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Recovery > 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: (5) and w.r.t. the slow time variables (Andreasen and Christiansen, 1993): Infection Susceptible S Quarantined Q Early Medical Encounter ME Late Medical Encounter ML Recovery Prodromal Symptoms IP Progression Presentation Diagnosis Recovery Waning Presentation 3 Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Recovery (6) DIMACS 10/9/06 Zhilan Feng 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 =0. Then in terms of (6) M is a 2-D slow manifold. The slow dynamics on M is described by y1 1 (0.3, 0.58) (7) 0 w 0 1 If the slow dynamics of (7) can be characterized via bifurcations, then the bifurcating dynamics on M are structurally stable hence robust to perturbations DIMACS 10/9/06 Zhilan Feng Malaria disease dynamics on the fast time scale The reproductive number of malaria is w is the S-gene frequency On the fast time-scale, if R0 > 1 then all solutions are hyperbolically Infection Susceptible S Quarantined Q Early Medical Encounter ME Late Medical Encounter ML Recovery Prodromal Symptoms IP Progression Presentation Diagnosis Recovery Waning Presentation 3 Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Recovery asymptotic to the endemic equilibrium Em* = (y1*, y2*, z*) where and z* > 0 is a solution to a quadratic equation with ki=…… 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 Infection Susceptible S Quarantined Q Presentation Recovery 4 Recovery Prodromal Symptoms IP Progression 22 Late Medical Encounter ML Recovery Waning Presentation Early Medical Encounter ME Diagnosis Progression Progression Exposed E Quarantine Medical Practitioners Hospitalized P NotQ Hospitalized R HP Hospitalized R NotHP Respiratory Symptoms IR Recovery Progression Health Authorities Hospitalized PQ Recovered R Note: is the death rate weighted by malaria related Wi s2 Fitness F = s1 - s2 determines s 2 = s1 S-gene cannot invade The slow dynamics Bi-stable equilibria possible Population extinction E*=(w*, N*) Global interior attractor s2 = h(s1) s1 DIMACS 10/9/06 Zhilan Feng Possible equilibria of the slow system N N H1 (1,K) H1 H2 0 w* (1,K) H2 1 w 0 w1* w2* 1 w DIMACS 10/9/06 Zhilan Feng 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 N 500 1000 1500 2000 2 N 5 2.5 Q(w,N) 0 -2 0 Q H2 -2.5 (w*, N*) 0 N 2000 0.2 0 0.6 0.4 w w 0.8 1 (1,K) H1 w* 1 w DIMACS 10/9/06 Zhilan Feng S-gene dynamics on the slow time scale s2 N s2 = s1 S-gene cannot invade Stable Unstable Population extinction s2 = h(s1) E*=(w*, N*) Global interior attractor w s1 N N w N w w DIMACS 10/9/06 Zhilan Feng Effect of S-gene dynamics on malaria prevalence w : S-gene frequency 1/gi : Infectious period R0 Possession of the S-gene leads to longer-lasting parasitaemia (1/g2) in heterozygote individuals, and therefore the presence of resistance may actually increase infection prevalence g2 w (c) g2 =0.09 y1+ y2 y1+ y2 (b) g2 =0.06 time time DIMACS 10/9/06 Zhilan Feng Influence of malaria on population genetics s2 s2 = s1 A balancing selection against the evolution of resistance, with the strength of selection dependent upon malaria prevalence. S-gene cannot invade Population extinction E*=(w*, N*) Global interior attractor Fitness F s2 = h(s1) s1 = s1 - s2 = - n + a1W1 - a2W2 n: Death due to S-gene ai: Death due to malaria Wi: Malaria parameters DIMACS 10/9/06 Zhilan Feng 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. DIMACS 10/9/06 Zhilan Feng Acknowledgements National Science Foundation Jams S. McDonnell Foundation DIMACS 10/9/06 Zhilan Feng
