Modelling Two Host Strains with an Indirectly Transmitted Pathogen

1
Modelling Two Host Strains with an Indirectly
Transmitted Pathogen

• Angela Giafis
• 20th April 2005

2
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.

3
Structure

• Discuss the differences between two models.
• Equilibrium solutions, feasibility and stability.
• Show parameter plots.
• Look at some dynamical illustrations.

4
Models
Model 1
Model 2
5

• 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

6
Equilibrium Solutions and Feasibility Model 1

• Total extinction (0,0,0,0,0,0)
• Uninfected coexistence (S1,0,0,K-S1,0,0)
• with S1 ? 0,K
• Strain 1 alone with the pathogen (S1,Î1,W1,0,0,0)
• Strain 2 alone with the pathogen (0,0,0,S2,Î2,W2)

1. Feasible
2. Feasible
3. Feasible if K gt HT,1
4. Feasible if K gt HT,2

7
Equilibrium Solutions and Feasibility Model 2

• Total extinction (0,0,0,0)
• Strain 1 alone at its carrying capacity
  (K1,0,0,0)
• Strain 2 alone at its carrying capacity
  (0,K2,0,0)
• Strain 1 alone with the pathogen (S1,0,Î1,W1)
• Strain 2 alone with the pathogen (0,S2,Î2,W2)
• Coexistence of the strains and the pathogen
  (S1,S2,I,W)

• Feasible
• Feasible
• Feasible
• Feasible if K1 gt HT,1
• Feasible if K2 gt HT,2
• Feasible if q1?2-q2?1 lt 0 and HT,1ltK12ltHT,2

8
Stability Model 1

1. (0,0,0,0,0,0) is unstable
2. (S1,0,0,K-S1,0,0) is neutrally stable iff
   (K-S1)/HT,2S1/HT,1lt1 and given feasibility.
   For point stability we need ABC-C2-A2Dgt0, if
   ABC-C2-A2Dlt0 we expect limit cycles.
3. (S1,Î1,W1,0,0,0) is stable iff D0,1ltD0,2 and
   given feasibility. For point stability we need
   A1B1-C1gt0, if A1B1-C1lt0 we expect limit cycles.
4. (0,0,0,S2,Î2,W2) is stable iff D0,2ltD0,1 and
   given feasibility. For point stability we need
   A2B2-C2gt0, if A2B2-C2lt0 we expect limit cycles.

9
Stability Model 2

1. (0,0,0,0) is unstable
2. (K1,0,0,0) is stable iff K1ltHT,1
3. (0,K2,0,0) is unstable
4. (S1,0,Î1,W1) is stable iff HT,1gtK12 and given
   feasibility. For point stability we need
   A1B1-C1gt0, if A1B1-C1lt0 we expect limit cycles.
5. (0,S2,Î2,W2) is stable iff K12gtHT,2 and given
   feasibility. For point stability we need
   A2B2-C2gt0, if A2B2-C2lt0 we expect limit cycles.
6. (S1,S2,I,W) is stable given feasibility. For
   point stability we need ABC-C2-A2Dgt0, if
   ABC-C2-A2Dlt0 we expect limit cycles.

10
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.

11
Parameter Plots Model 1A) Susceptible strain
point stable
12
Parameter Plots Model 1B) Susceptible strain
cyclic stable
13
Parameter Plots Model 2A) Susceptible strain
point stable
14
Parameter Plots Model 2B) Susceptible strain
cyclic stable
15
Dynamical Illustrations Model 1B
Region A
Region B
16
Region D
Region C
Region D
17
Region F
Region E
Region F
Region F
18
Region B
Region A
Region C
19
Region D
Region D
20
Dynamical Illustrations 2B
Region B
Region A
21
Region C
Region C
22
Region B
Region A
Region B
23
Region C
Region C
24
Summary

• In both models we considered two cases, one where
  the more susceptible strain is point stable
  (A1B1-C1gt0) and one where we expect to see limit
  cycles (A1B1-C1lt0).
• 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.

25
Future Work

• n-strain models.
• Investigating cycles.