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

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

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

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Biological Background

• Strains
• Community dynamics
• Co-evolution not evolution
• R0 . It is known that pathogen virulence
  evolves to maximise R0 which yields monomorphism.
• D0 . It is known that host resistance
  evolves to minimise D0 which yields monomorphism.
• Do R0 and D0 interact to give polymorphism?

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Model

• Ignoring latency, immunity etc.
• Where

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E.g. Of Model 2 host strains, 2 pathogen strains
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Analysis of model

• Find equilibrium points
• Feasibility conditions
• Jacobian
• Stability - determinant gt 0 and
• trace lt 0 (2x2 matrices only)
• - eigenvalues (Re lt 0)
• Stability conditions
• Dynamical illustrations by numerical integration

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Equilibria

• Uninfected
• H K , Yhp 0 for all h and p.
• Infected (monomorphic)
• Xh Xhp HT,hp ,
• Xk Ykq 0 for all k ? h and q ? p.
• Provided that the threshold density HT,hp lt K.
• What about equilibria for polymorphisms in host
  and pathogen strains?

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1 host strain, 1 pathogen strain

• Equilibrium points with conditions
• host and pathogen strain die out (unstable)
• pathogen strain dies out
• (HT,hp gt K)
• endemic infection
• (HT,hp lt K)

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1 host strain, 2 pathogen strains

• Equilibrium points with conditions
• host and pathogen strain die out (unstable)
• pathogen strain dies out (HT,hp gt
  K)
• host strain 1 with pathogen strain 2
• host strain 1 with pathogen strain 1

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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 gt K)
• host strain 2 with pathogen strain 1
• host strain 1 with pathogen strain 1

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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 gt K)
• host strain 1 with pathogen strain 1
  (D0,21gtD0,11, R0,11gtR0,12)
• host strain 1 with pathogen strain 2
  (D0,22gtD0,12, R0,12gtR0,11)
• host strain 2 with pathogen strain 1
  (D0,11gtD0,21, R0,21gtR0,22)
• host strain 2 with pathogen strain 2
  (D0,12gtD0,22, R0,22gtR0,21)
• coexistence/polymorphism

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Coexistence of 2 host strains and 2 pathogen
strains

• At equilibrium we have
• Feasibility conditions ?
• Conjecture Feasibility conditions Stability
  conditions

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Jacobian

• diagonalised Jacobian
• 2 negative eigenvalues so far

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• 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.

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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

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Future Investigation

• Complete 2 host, 2 pathogen strain case

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Future Investigation

• Complete 2 host, 2 pathogen strain case
• n host, n pathogen strain case

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Future Investigation

• Complete 2 host, 2 pathogen strain case
• n host, n pathogen strain case
• Adaptive dynamics

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Future Investigation

• Complete 2 host, 2 pathogen strain case
• n host, n pathogen strain case
• Adaptive dynamics
• Evolution of sex