How do the basic reproduction ratio and the basic depression ratio determine the dynamics of a syste

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
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Contents

• Understanding the biology
• Definitions
• Mathematical Approach
• Examples
• n host strains with n pathogen strains

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Biological Background

• Strains
• Community dynamics
• Co-evolution not evolution

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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.

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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?

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Model

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

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Mathematical approach

• Find equilibrium points
• Feasibility conditions
• Jacobian
• Stability conditions
• Dynamical illustrations by numerical integration

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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
• (R0,11 lt 1)
• endemic infection
• (R0,11 gt 1)

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1 host strain, 2 pathogen strains

• Equilibrium points with conditions
• host and pathogen strain die out
  (unstable)
• pathogen strains die out
  (R0,11 lt 1, R0,12 lt 1)
• host strain 1 with pathogen strain 2
• (R0,12 gt 1, R0,12 gt R0,11)
• host strain 1 with pathogen strain 1
• (R0,11 gt 1, R0,11 gt R0,12)

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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
• X1 X2 K
• (R0,11 lt 1, R0,21 lt 1)
• host strain 2 with pathogen strain 1
• (R0,21 gt 1, D0,11 gt D0,21)
• host strain 1 with pathogen strain 1
• (R0,11 gt 1, D0,21 gt D0,11)

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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 X2 K
• (K gt X1R0,11 X2R0,21, K gt X1R0,12
  X2R0,22 )
• host strain 1 with pathogen strain 1
• (D0,21 gt D0,11, R0,11 gt R0,12, R0,11
  gt 1)
• host strain 1 with pathogen strain 2
• (D0,22 gt D0,12, R0,12 gt R0,11,
  R0,12 gt 1)
• host strain 2 with pathogen strain 1
• (D0,11 gt D0,21, R0,21 gt R0,22, R0,21
  gt 1)
• host strain 2 with pathogen strain 2
• (D0,12 gt D0,22, R0,22 gt R0,21, R0,22
  gt 1)

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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.

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Point Stable (4 host strains with 4 pathogen
strains)

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

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Cyclic stable (4 host strains with 4 pathogen
strains)

• R0,11 gt R0,12 gt R0,13 gt R0,14, 1234
• R0,22 gt R0,23 gt R0,24 gt R0,21, 2341
• R0,33 gt R0,34 gt R0,31 gt R0,32, 3412
• R0,44 gt R0,41 gt R0,42 gt R0,43. 4123
• D0,11 gt D0,21 gt D0,31 gt D0,41, 1234
• D0,22 gt D0,32 gt D0,42 gt D0,12, 2341
• D0,33 gt D0,43 gt D0,13 gt D0,23, 3412
• D0,44 gt D0,14 gt D0,24 gt D0,34. 4123

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Possible equilibria for n host strains with n
pathogen strains

• Uninfected H K , Yhp 0 for all h and p.
• Infected Xh Xhp HT,hp ,
• (monomorphic)
• Xk Ykq 0 for all k ? h and q ? p.
• Coexistence States
• (polymorphic)

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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.

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Summary

• Co-evolution not evolution.
• Importance of R0 in pathogen virulence.
• Importance of D0 in host resistance.
• The interaction of R0 and D0.
• Multi-strain (polymorphism) co-existence is
  possible.
• Stable cycles can occur.

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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