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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. Contents. Understanding the biology Definitions Mathematical Approach Examples

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  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 • Communitydynamics • Co-evolution not evolution

  4. Definitions 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.

  5. Questions! 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?

  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 (R0,11< 1) • endemic infection (R0,11> 1)

  9. 1 host strain, 2 pathogen strains Equilibrium points with conditions: • host and pathogen strain die out (unstable) • pathogen strains die out (R0,11< 1, R0,12< 1) • host strain 1 with pathogen strain 2 (R0,12> 1, R0,12> R0,11) • host strain 1 with pathogen strain 1 (R0,11> 1, R0,11> R0,12)

  10. 2 host strains, 1 pathogen strain Equilibrium points with conditions: • host and pathogen strain die out (unstable) • pathogen strain dies out with X1 + X2= K (R0,11 < 1, R0,21 < 1) • host strain 2 with pathogen strain 1 (R0,21 > 1, D0,11 > D0,21) • host strain 1 with pathogen strain 1 (R0,11 > 1, D0,21 > D0,11)

  11. Equilibrium points with conditions: host and pathogen strains die out (unstable) pathogen strains die out : X1+X2 = K (K > X1*R0,11 + X2*R0,21, K > X1*R0,12 + X2*R0,22) host strain 1 with pathogen strain 1 (D0,21>D0,11, R0,11> R0,12, R0,11> 1) host strain 1 with pathogen strain 2 (D0,22> D0,12, R0,12> R0,11, R0,12> 1) host strain 2 with pathogen strain 1 (D0,11> D0,21, R0,21> R0,22, R0,21> 1) host strain 2 with pathogen strain 2 (D0,12> D0,22, R0,22> R0,21, R0,22> 1) 2 host strains, 2 pathogen strains

  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) R0,11 > R0,12 > R0,13 > R0,14, 1234 R0,22 > R0,21 > R0,23 > R0,24, 2134 R0,33 > R0,32 > R0,31 > R0,34, 3214 R0,44 > R0,43 > R0,42 > R0,41. 4321 D0,11 > D0,21 > D0,31 > D0,41, 1234 D0,22 > D0,32 > D0,42 > D0,12, 2341 D0,33 > D0,43 > D0,13 > D0,23, 3412 D0,44 > D0,14 > D0,24 > D0,34. 4123

  14. Cyclic stable(4 host strains with 4 pathogen strains) R0,11 > R0,12 > R0,13 > R0,14, 1234 R0,22 > R0,23 > R0,24 > R0,21, 2341 R0,33 > R0,34 > R0,31 > R0,32, 3412 R0,44 > R0,41 > R0,42 > R0,43. 4123 D0,11 > D0,21 > D0,31> D0,41, 1234 D0,22 > D0,32 > D0,42 > D0,12, 2341 D0,33 > D0,43 > D0,13 > D0,23, 3412 D0,44 > D0,14 > D0,24 > D0,34. 4123

  15. Possible equilibria for n host strains with n pathogen strains Uninfected: H = K , Yhp = 0 for all h and p. Infected Xh = X*hp = HT,hp , (monomorphic): Xk = Ykq = 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 R0in pathogen virulence. • Importance of D0in host resistance. • The interaction of R0 and D0. • 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

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