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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)
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
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 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)
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
, : 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 Naa=0.
Ni : 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
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
(3) i 1, 2 (AA, Aa)
DIMACS 10/9/06
Zhilan Feng
Analysis of model (3)
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
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
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
Note: b, mi , 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
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
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
y1
w
(0.3, 0.58)
00
1
1
DIMACS 10/9/06
Zhilan Feng
On the fast time-scale, if R0 > 1 then all solutions are hyperbolically
asymptotic to the endemic equilibrium Em* = (y1*, y2*, z*)
Malaria disease dynamics on the fast time scale
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
The reproductive number of malaria is
and z* > 0 is a solution to a quadratic equation
with ki=……
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
Health AuthoritiesMedical Practitioners
SusceptibleS
Late MedicalEncounterML RecoveredR
QuarantinedQ
HospitalizedR NotHP
InfectionProgression
Recovery
ProdromalSymptomsIPEarly MedicalEncounterME RespiratorySymptomsIRPresentationDiagnosis Recovery
ExposedEQuarantine
HospitalizedP NotQ Progression
Presentation WaningHospitalizedP Q
HospitalizedR HP
Recovery
Progression
Recovery
Progression
1
11
21
2
()tS
2
1 22
4
3
Note: is the death rate weighted by malaria related Wi
Fitness F = 1 - 2 determines
The slow dynamics
E*=(w*, N*)Global interior attractor
2 = 1
2 = h(1)
1
2
S-gene cannot invade
Population extinction
Bi-stable equilibria possible
DIMACS 10/9/06
Zhilan Feng
Possible equilibria of the slow system
10 w
N
H2
w*
H1
(1,K)
w10
N
H2
H1
(1,K)
w1* w2*
DIMACS 10/9/06
Zhilan Feng
500 1000 1500 2000
N
0.2 0.4 0.6 0.8 1
w
-2.5
0
2.5
5
Q
-2.5
0
2.5
5
Q
N
Q(w,N)
0
2000
0
2
2
(w*, N*)
w
Global dynamics of the slow manifold
Suppose there is a closed orbit around E*(w*,N*). Construct Q1(w), Q2(N) and Q(w,N)=Q1+Q2 as:
10w
N
H2
w*
H1
(1,K)
The slow system (7) has no periodic solution or homoclinic orbit.
Note that and 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
2
S-gene cannot invade
Population extinction
N
w
N N
ww
N
w
Stable
Unstable
Bistab
ility
DIMACS 10/9/06
Zhilan Feng
Effect of S-gene dynamics on malaria prevalence
R0
w
y 1+
y2
y 1+
y2
(c) =0.09(b) =0.06
time 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
Wi: Malaria parameters
E*=(w*, N*)Global interior attractor
2 = 1
2 = h(1)
1
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
W1W2
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