dimacs 10/9/06 zhilan feng collaborators and references zhilan feng, david smith, f. ellis mckenzie,...

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)


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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)


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


SusceptibleS


QuarantinedQ

HospitalizedR NotHP


Recovery


ExposedEQuarantine



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


2 = 1

2 = h(1)

1

2



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


2 = 1

2 = h(1)

1

2



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

DIMACS 10/9/06

Zhilan Feng

Acknowledgements

National Science Foundation

Jams S. McDonnell Foundation

dimacs 10/9/06 zhilan feng collaborators and references zhilan feng, david smith, f. ellis mckenzie,...

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