1. How do the basic reproduction ratio and the basic depression ratio determine the dynamics of a system with many host and many pathogen strains? Rachel Bennett and Roger Bowers

2. Contents Understanding the biology Definitions Mathematical Approach Examples n host strains with n pathogen strains

3. Biological Background Strains Community dynamics Co-evolution not evolution

4. is the expected number of secondary cases per primary in a totally susceptible population. is the amount by which the total population is decreased, per infected individual, due to the presence of infection. Definitions

5. Faced with an individual host strain pathogen virulence evolves to maximise which yields monomorphism. (Bremermann & Thieme, 1989) Faced with an individual pathogen strain host resistance evolves to minimise which yields monomorphism. (Bowers, 2001) So, with many host and pathogen strains - how do and interact? - is multi-strain (polymorphism) co-existence possible? - can stable cycles occur? Questions!

6. Model. Where: : susceptible, : infective, K: carrying capacity, : intrinsic growth rate, : transmission rate, : recovery rate, : pathogen induced death rate, : uninfected death rate.

7. Mathematical approach Find equilibrium points Feasibility conditions Jacobian Stability conditions Dynamical illustrations by numerical integration

8. 1 host strain, 1 pathogen strain Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strain dies out (R 0,11 < 1) endemic infection (R 0,11 > 1)

9. 1 host strain, 2 pathogen strains Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strains die out (R 0,11 < 1, R 0,12 < 1) host strain 1 with pathogen strain 2 (R 0,12 > 1, R 0,12 > R 0,11 ) host strain 1 with pathogen strain 1 (R 0,11 > 1, R 0,11 > R 0,12 )

10. 2 host strains, 1 pathogen strain Equilibrium points with conditions: host and pathogen strain die out (unstable) pathogen strain dies out with X 1 + X 2 = K (R 0,11 < 1, R 0,21 < 1) host strain 2 with pathogen strain 1 (R 0,21 > 1, D 0,11 > D 0,21 ) host strain 1 with pathogen strain 1 (R 0,11 > 1, D 0,21 > D 0,11 )

11. 2 host strains, 2 pathogen strains Equilibrium points with conditions: host and pathogen strains die out (unstable) pathogen strains die out : X 1 + X 2 = K (K > X 1 * R 0,11 + X 2 * R 0,21, K > X 1 * R 0,12 + X 2 * R 0,22 ) host strain 1 with pathogen strain 1 (D 0,21 > D 0,11, R 0,11 > R 0,12, R 0,11 > 1) host strain 1 with pathogen strain 2 (D 0,22 > D 0,12, R 0,12 > R 0,11, R 0,12 > 1) host strain 2 with pathogen strain 1 (D 0,11 > D 0,21, R 0,21 > R 0,22, R 0,21 > 1) host strain 2 with pathogen strain 2 (D 0,12 > D 0,22, R 0,22 > R 0,21, R 0,22 > 1)

12. 2 host strain, 2 pathogen strain coexistence Jacobian: diagonalised 3 negative eigenvalues Identical feasibility and stability conditions given that stability changes via a transcritical bifurcation.

13. Point Stable (4 host strains with 4 pathogen strains) R 0,11 > R 0,12 > R 0,13 > R 0,14, 1234 R 0,22 > R 0,21 > R 0,23 > R 0,24, 2134 R 0,33 > R 0,32 > R 0,31 > R 0,34, 3214 R 0,44 > R 0,43 > R 0,42 > R 0,41. 4321 D 0,11 > D 0,21 > D 0,31 > D 0,41, 1234 D 0,22 > D 0,32 > D 0,42 > D 0,12, 2341 D 0,33 > D 0,43 > D 0,13 > D 0,23, 3412 D 0,44 > D 0,14 > D 0,24 > D 0,34. 4123

14. Cyclic stable (4 host strains with 4 pathogen strains) R 0,11 > R 0,12 > R 0,13 > R 0,14, 1234 R 0,22 > R 0,23 > R 0,24 > R 0,21, 2341 R 0,33 > R 0,34 > R 0,31 > R 0,32, 3412 R 0,44 > R 0,41 > R 0,42 > R 0,43. 4123 D 0,11 > D 0,21 > D 0,31 > D 0,41, 1234 D 0,22 > D 0,32 > D 0,42 > D 0,12, 2341 D 0,33 > D 0,43 > D 0,13 > D 0,23, 3412 D 0,44 > D 0,14 > D 0,24 > D 0,34. 4123

15. Possible equilibria for n host strains with n pathogen strains Uninfected:H = K, Y hp = 0 for all h and p. Infected X h = X * hp = H T,hp, (monomorphic): X k = Y kq = 0 for all k  h and q  p. Coexistence States (polymorphic):

16. n host strains with n pathogen strains Smaller n x n coexistence can occur within a larger n x n system e.g. 3 x 3 coexist in a 5 x 5 system. n x m coexistence is not possible, e.g. 2 x 3 cannot coexist in a 7 x 7 system.

17. Summary Co-evolution not evolution. Importance of R 0 in pathogen virulence. Importance of D 0 in host resistance. The interaction of R 0 and D 0. Multi-strain (polymorphism) co-existence is possible. Stable cycles can occur.

18. Other work and future investigation analysed n strain predation, mutualism and competition models. modelling mutation (Adaptive Dynamics) connection between current results and adaptive dynamics results