Modelling Two Host Strains with an Indirectly Transmitted

Pathogen

Angela Giafis

20th April 2005

Motivation

1. Disease can be spread by contact with infectious materials (free stages) in the environment.

2. Interested in what happens when 2 different host types, one more susceptible to infection the other more resistant, are subjected to such an infection.

Structure

• Discuss the differences between two

models.

• Equilibrium solutions, feasibility and

stability.

• Show parameter plots.

• Look at some dynamical illustrations.

Models

22222

22222

2222222

11111

11111

1111111

1

1

WIdtdW

IWSdtdI

IWSKHSr

dtdS

WIdtdW

IWSdtdI

IWSKHSr

dtdS

WIdt

dW

IWSWSdt

dI

WSHSqSrdt

dS

WSHSqSrdt

dS

2211

222222

1111111

2

Model 1:

Model 2:

Both models have demography (births and deaths) and infection but as we see there are details that differ and this turns out to matter.

Much of the behaviour of model 1 is governed by the term D0 which is the basic depression ratio, where

.)(

0W

ISKD

Equilibrium Solutions and Feasibility: Model 1

1. Total extinction (0,0,0,0,0,0)

2. Uninfected coexistence (S1

*,0,0,K-S1*,0,0)

with S1* є [0,K]

3. Strain 1 alone with the pathogen (Ŝ1,Î1,Ŵ1,0,0,0)

4. Strain 2 alone with the pathogen (0,0,0,Ŝ2,Î2,Ŵ2)

1. Feasible

2. Feasible

3. Feasible if K > HT,1

4. Feasible if K > HT,2

Equilibrium Solutions and Feasibility: Model 2

1. Total extinction (0,0,0,0)

2. Strain 1 alone at its carrying

capacity (K1,0,0,0)

3. Strain 2 alone at its carrying

capacity (0,K2,0,0)

4. Strain 1 alone with the pathogen

(Ŝ1,0,Î1,Ŵ1)

5. Strain 2 alone with the pathogen

(0,Ŝ2,Î2,Ŵ2)

6. Coexistence of the strains and

the pathogen (S1*,S2

*,I*,W*)

1. Feasible

2. Feasible

3. Feasible

4. Feasible if K1 > HT,1

5. Feasible if K2 > HT,2

6. Feasible if q1υ2-q2υ1 < 0 and HT,1<K12<HT,2

Stability: Model 1

22

222222

22222

222

2222222

11

111111

111111

11111

111

1

0000

00

1

0000

00

1

SWS

SK

SrW

K

Sr

K

HrS

K

Sr

K

Sr

SSW

SK

Sr

K

SrS

K

SrW

K

Sr

K

Hr

1. (0,0,0,0,0,0) is unstable

2. (S1*,0,0,K-S1

*,0,0) is neutrally stable iff (K-S1*)/HT,2+S1

*/HT,1<1 and given feasibility. For point stability we need ABC-C2-A2D>0, if ABC-C2-A2D<0 we expect limit cycles.

3. (Ŝ1,Î1,Ŵ1,0,0,0) is stable iff D0,1<D0,2 and given feasibility. For point stability we need A1B1-C1>0, if A1B1-C1<0 we expect limit cycles.

4. (0,0,0,Ŝ2,Î2,Ŵ2) is stable iff D0,2<D0,1 and given feasibility. For point stability we need A2B2-C2>0, if A2B2-C2<0 we expect limit cycles.

Stability: Model 2

00

221121

22222222222

11111111111

SSWW

SSqWSqHqrSq

SSqSqWSqHqr

1. (0,0,0,0) is unstable

2. (K1,0,0,0) is stable iff K1<HT,1

3. (0,K2,0,0) is unstable

4. (Ŝ1,0,Î1,Ŵ1) is stable iff HT,1>K12 and given feasibility. For point stability we need A1B1-C1>0, if A1B1-C1<0 we expect limit cycles.

5. (0,Ŝ2,Î2,Ŵ2) is stable iff K12>HT,2 and given feasibility. For point stability we need A2B2-C2>0, if A2B2-C2<0 we expect limit cycles.

6. (S1*,S2

*,I*,W*) is stable given feasibility. For point stability we need ABC-C2-A2D>0, if ABC-C2-A2D<0 we expect limit cycles.

Parameter Plots

• Trade-off : individual hosts pay for their increased resistance to the pathogen by a reduction in the contribution to the overall fitness.

• Our parameter plots are representative of our stability conditions.

• The susceptible strain (strain 1) values are fixed and we will vary two of the resistant strain (strain 2) parameters.

Parameter Plots: Model 1A) Susceptible strain point stable

Parameter Plots: Model 1B) Susceptible strain cyclic stable

Parameter Plots: Model 2A) Susceptible strain point stable

Parameter Plots: Model 2B) Susceptible strain cyclic stable

Dynamical Illustrations: Model 1B

Region A Region B

Region C

Region D

Region D

Region E

Region F

Region F

Region F

Region A

Region B

Region C

Region D Region D

Dynamical Illustrations 2B

Region A Region B

Region C Region C

Region A

Region B

Region B

Region C Region C

Summary

• In both models we considered two cases, one where the more susceptible strain is point stable (A1B1-C1>0) and one where we expect to see limit cycles (A1B1-C1<0).

• Model 1: coexistence not possible. • Model 2: coexistence possible. Indeed cyclic coexistence

of all the populations is possible.

• Outcome depends on balance between costs and benefits.

Future Work

• n-strain models.

• Investigating cycles.