1. Community dynamics, invasion criteria and the co-evolution of host and pathogen. Rachel Bennett

2. Contents Understanding the biology Model Equilibria h host strains with p pathogen strains Coexistence of 2 host strains with 2 pathogen strains Future investigations

3. Biological Background Strains Community dynamics Co-evolution not evolution R 0 =. It is known that pathogen virulence evolves to maximise R 0 which yields monomorphism. D 0 =. It is known that host resistance evolves to minimise D 0 which yields monomorphism. Do R 0 and D 0 interact to give polymorphism?

4. Model Ignoring latency, immunity etc.: Where:

5. E.g. Of Model : 2 host strains, 2 pathogen strains

6. Analysis of model Find equilibrium points Feasibility conditions Jacobian Stability- determinant > 0 and trace < 0 (2x2 matrices only) - eigenvalues (Re < 0) Stability conditions Dynamical illustrations by numerical integration

7. Equilibria Uninfected H = K, Y hp = 0 for all h and p. Infected (monomorphic) X h = X * hp = H T,hp, X k = Y kq = 0 for all k  h and q  p. Provided that the threshold density H T,hp < K. What about equilibria for polymorphisms in host and pathogen strains?

8. 1 host strain, 1 pathogen strain Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strain dies out (H T,hp > K) endemic infection (H T,hp < K)

9. 1 host strain, 2 pathogen strains Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strain dies out (H T,hp > K) host strain 1 with pathogen strain 2 host strain 1 with pathogen strain 1

10. 2 host strains, 1 pathogen strain Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strain dies out with either X 1 = K - X 2 and X 2 = X 2, or X 2 = K and X 1 = 0, orX 1 = K and X 2 = 0. (H T,hp > K) host strain 2 with pathogen strain 1 host strain 1 with pathogen strain 1

11. 2 host strains, 2 pathogen strains Equilibrium points with conditions: host and pathogen strains die out (unstable) pathogen strains die out : X 1 = K - X 2 and X 2 = X 2 (H T,hp > K) host strain 1 with pathogen strain 1 (D 0,21 >D 0,11, R 0,11 >R 0,12 ) host strain 1 with pathogen strain 2 (D 0,22 >D 0,12, R 0,12 >R 0,11 ) host strain 2 with pathogen strain 1 (D 0,11 >D 0,21, R 0,21 >R 0,22 ) host strain 2 with pathogen strain 2 (D 0,12 >D 0,22, R 0,22 >R 0,21 ) coexistence/polymorphism…

12. Coexistence of 2 host strains and 2 pathogen strains At equilibrium we have: Feasibility conditions Conjecture: Feasibility conditions ≡ Stability conditions

13. Jacobian diagonalised Jacobian 2 negative eigenvalues so far……

14. There is polymorphism when: Host strain 2 would win with pathogen strain 1 & host strain 1 would win with pathogen strain 2 while, pathogen strain 1 would win with host strain 1 & pathogen strain 2 would win with host strain 2.

15. Summary Co-evolution not evolution Importance of R 0 in pathogen virulence Importance of D 0 in host resistance Determining stability using the Jacobian method Polymorphism of 2 host strains with 2 pathogen strains

16. Future Investigation Complete 2 host, 2 pathogen strain case

17. Future Investigation Complete 2 host, 2 pathogen strain case n host, n pathogen strain case

18. Future Investigation Complete 2 host, 2 pathogen strain case n host, n pathogen strain case Adaptive dynamics

19. Future Investigation Complete 2 host, 2 pathogen strain case n host, n pathogen strain case Adaptive dynamics Evolution of sex