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

Community dynamics, invasion criteria and the co-evolution of host and pathogen. Rachel Bennett. Contents. Understanding the biology Model Equilibria h host strains with p pathogen strains Coexistence of 2 host strains with 2 pathogen strains Future investigations. Biological Background.

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

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  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 • Communitydynamics • Co-evolution not evolution • R0= . It is known that pathogen virulence evolves to maximise R0 which yields monomorphism. • D0= . It is known thathost resistance evolves to minimise D0 which yields monomorphism. • Do R0 and D0 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 , Yhp = 0 for all h and p. Infected (monomorphic) Xh = X*hp = HT,hp , Xk = Ykq = 0 for all k  h and q  p. Provided that the threshold density HT,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 (HT,hp > K) • endemic infection (HT,hp < K)

  9. 1 host strain, 2 pathogen strains Equilibrium points with conditions: • host and pathogen strain die out (unstable) • pathogen strain dies out (HT,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 X1= K - X2 and X2= X2, or X2= K and X1= 0, or X1= K and X2= 0. (HT,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 : X1= K - X2 and X2= X2 (HT,hp > K) • host strain 1 with pathogen strain 1 (D0,21>D0,11, R0,11>R0,12) • host strain 1 with pathogen strain 2 (D0,22>D0,12, R0,12>R0,11) • host strain 2 with pathogen strain 1 (D0,11>D0,21, R0,21>R0,22) • host strain 2 with pathogen strain 2 (D0,12>D0,22, R0,22>R0,21) • 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 R0 in pathogen virulence • Importance of D0 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

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