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 • Disease can be spread by contact with infectious materials (free stages) in the environment. • 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. Total extinction (0,0,0,0,0,0) Uninfected coexistence (S1*,0,0,K-S1*,0,0) with S1*є [0,K] Strain 1 alone with the pathogen (Ŝ1,Î1,Ŵ1,0,0,0) Strain 2 alone with the pathogen (0,0,0,Ŝ2,Î2,Ŵ2) Feasible Feasible Feasible if K > HT,1 Feasible if K > HT,2 Equilibrium Solutions and Feasibility: Model 1

7. 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 (Ŝ1,0,Î1,Ŵ1) Strain 2 alone with the pathogen (0,Ŝ2,Î2,Ŵ2) Coexistence of the strains and the pathogen (S1*,S2*,I*,W*) Feasible Feasible Feasible Feasible if K1 > HT,1 Feasible if K2 > HT,2 Feasible if q1υ2-q2υ1 < 0 and HT,1<K12<HT,2 Equilibrium Solutions and Feasibility: Model 2

8. Stability: Model 1 • (0,0,0,0,0,0) is unstable • (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. • (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. • (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.

9. Stability: Model 2 • (0,0,0,0) is unstable • (K1,0,0,0) is stable iff K1<HT,1 • (0,K2,0,0) is unstable • (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. • (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. • (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.

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-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.

25. Future Work • n-strain models. • Investigating cycles.