1. Modelling Two Host Strains with an Indirectly Transmitted Pathogen Angela Giafis 20 th 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 D 0 which is the basic depression ratio, where

6. Equilibrium Solutions and Feasibility: Model 1 1.Total extinction (0,0,0,0,0,0) 2.Uninfected coexistence (S 1 *,0,0,K-S 1 *,0,0) with S 1 * є [0,K] 3.Strain 1 alone with the pathogen (Ŝ 1,Î 1,Ŵ 1,0,0,0) 4.Strain 2 alone with the pathogen (0,0,0,Ŝ 2,Î 2,Ŵ 2 ) 1.Feasible 2.Feasible 3.Feasible if K > H T,1 4.Feasible if K > H T,2

7. Equilibrium Solutions and Feasibility: Model 2 1.Total extinction (0,0,0,0) 2.Strain 1 alone at its carrying capacity (K 1,0,0,0) 3.Strain 2 alone at its carrying capacity (0,K 2,0,0) 4.Strain 1 alone with the pathogen (Ŝ 1,0,Î 1,Ŵ 1 ) 5.Strain 2 alone with the pathogen (0,Ŝ 2,Î 2,Ŵ 2 ) 6.Coexistence of the strains and the pathogen (S 1 *,S 2 *,I *,W * ) 1.Feasible 2.Feasible 3.Feasible 4.Feasible if K 1 > H T,1 5.Feasible if K 2 > H T,2 6.Feasible if q 1 υ 2 -q 2 υ 1 < 0 and H T,1 <K 12 <H T,2

8. Stability: Model 1 1.(0,0,0,0,0,0) is unstable 2.(S 1 *,0,0,K-S 1 *,0,0) is neutrally stable iff (K-S 1 * )/H T,2 +S 1 * /H T,1 0, if ABC-C 2 -A 2 D<0 we expect limit cycles. 3.(Ŝ 1,Î 1,Ŵ 1,0,0,0) is stable iff D 0,1 0, if A 1 B 1 -C 1 <0 we expect limit cycles. 4.(0,0,0,Ŝ 2,Î 2,Ŵ 2 ) is stable iff D 0,2 0, if A 2 B 2 -C 2 <0 we expect limit cycles.

9. Stability: Model 2 1.(0,0,0,0) is unstable 2.(K 1,0,0,0) is stable iff K 1 <H T,1 3.(0,K 2,0,0) is unstable 4.(Ŝ 1,0,Î 1,Ŵ 1 ) is stable iff H T,1 >K 12 and given feasibility. For point stability we need A 1 B 1 -C 1 >0, if A 1 B 1 -C 1 <0 we expect limit cycles. 5.(0,Ŝ 2,Î 2,Ŵ 2 ) is stable iff K 12 >H T,2 and given feasibility. For point stability we need A 2 B 2 -C 2 >0, if A 2 B 2 -C 2 <0 we expect limit cycles. 6.(S 1 *,S 2 *,I *,W * ) is stable given feasibility. For point stability we need ABC-C 2 - A 2 D>0, if ABC-C 2 -A 2 D<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 1 A) Susceptible strain point stable

12. Parameter Plots: Model 1 B) Susceptible strain cyclic stable

13. Parameter Plots: Model 2 A) Susceptible strain point stable

14. Parameter Plots: Model 2 B) Susceptible strain cyclic stable

15. Dynamical Illustrations: Model 1B Region ARegion B

16. Region C Region D

17. Region E Region F

18. Region A Region B Region C

19. Region D

20. Dynamical Illustrations 2B Region A Region B

21. Region C

22. Region A Region B

23. Region C

24. Summary In both models we considered two cases, one where the more susceptible strain is point stable (A 1 B 1 -C 1 >0) and one where we expect to see limit cycles (A 1 B 1 -C 1 <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.