Download Spatial structure and the ecology and evolution of host

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Host population structure and
the evolution of parasites
Mike Boots
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Our
Infectious
Diseases
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Theory on the evolution of
parasites
Evolutionary
game theory
‘Adaptive Dynamics’
Can strains invade when rare?
Generally a simple haploid genetic
assumption
Small mutations
Ecological feedbacks
Theory on the evolution of
parasites
Infectivity
is maximised
Infectious period maximised
Mortality due to infection (virulence)
minimised
Recovery rate minimised
Trade-offs related to exploitation of the host
explain variation
Virulence as a cost to
transmission
Transmission
Virulence
Lattice Models (Spatial structure within populations)
S
I
Natural
Mortality + Virulence
Natural
Mortality
S
S
Transmission
S
I
S
S
I
S
I
Reproduction
Simulation results for the evolution of transmission
with individuals on a lattice where interactions are all local
35
30
Mean

Transmission
25
20
15
10
5
200
400
600
800
1000
t
TIME
Max transmission = 150
No trade-offs between transmission and virulence
Intermediate Levels of Spatial
Structure
I
S
I
Global Infection (L)
S
Local Infection (1-L)
Maximum virulence
5
4
3
Mean Virulence
2
1
Linear
trade-off
with virulence
and transmission
0
0.0
0.2
0.4
0.6
L (Proportion of global infection)
0.8
1.0
Host Parasite models between local and meanfield
Pair-wise Approximation: differential equations for pair densities
eg,
PSI(t) =prob randomly chosen pair is in state SI
event
r(SI  II ) =
transmission
rate

z
PSI 

(z  1)PSI qI /SI
z
# neighbours
(fixed)
conditional prob that
I is a neighbour of an S
site in an SI pair
Host Parasite models between local and meanfield
Pair-wise Approximation: differential equations for pair densities
eg,
PSI(t) =prob randomly chosen pair is in state SI
event


r(SI  II ) =  PSI   (z  1)PSI qI /SI 
z

z
Host Parasite models between local and meanfield
Pair-wise Approximation: differential equations for pair densities
eg, PSI(t) =prob randomly chosen pair is in state SI
LI
event
(1-LI)
prob that a site is
infected


r(SI  II ) =  PSI   (z  1)PSI qI /SI  1  LI   LI  PSI PI
z

z
LI=0 (local), LI=1 (mean-field)
proportion
of global infection
Host Parasite models between local and meanfield
• Derive correlation Eqns:
dPSI
  r(SI    ), for each pair and singleton from
dt
states S, I, R and 0 (empty sites).
events
with params 0<LI,Lr<1 for global proportions of reproduction for
pathogen and host.
• Pair closure: determine qI/SI in terms of qI/S (from Monte Carlo sims).
• Analysis: Stability analysis (long term behaviours)
Bifurcation analysis, continuation (limit cycles)
Invasion Condition
Local density of infecteds
Transmission
Virulence Background
Mortality
1 d J
 (J | I ) 
  J {L ̂S  (1  L)q̂ 0S / J }  ( J  d) > 0
 J dt
Global density of susceptibles
J is a mutant strain
I is the resident strain
Hat notation denotes quasi steady state
Pairwise Invasion Plots
(Linear trade-off between transmission and virulence)
Does the analysis agree with
the simulations?
Yes: There is an ES virulence with spatial
structure and maximization with global
infection
 Yes: The ES virulence increases as the
proportion of global infection increases
 But: The ESS is lost before L=1.0


Weak selection gradients mean this is not
seen when simulation is run for a set time
period
The ESS is lost
Bistability
Bistability
The role of trade-off shape
Standard
assumption
of the evolution
of virulence theory
Transmission
Virulence
Evolution with a saturating trade-off in
a spatial model
Simulation
Approximation
The role of recovery: The Spatial
Susceptible Infected Removed (SIR) Model
S
S
S
I
S
R
I
R
I
The role of recovery
No recovery
=0
The role of recovery
=0.1
Increased ES virulence
Wider region of bistability
The role of recovery
=0.2
Bi-stability region reduces
The role of recovery
=0.3
The role of recovery
=0.4
The role of recovery
Max ES virulence increases
Recovery rate
Conclusions

Spatial structure crucial to evolutionary
outcomes

Bi-stability leading to the possibility of dramatic
shifts in virulence

Shapes of trade-offs are important

Approximate analysis is useful in spatial
evolutionary models
Collaborators

Akira Sasaki (Kyushu University)

Masashi Kamo (Kyushu: Institute for risk
management, Tsukuba)

Steve Webb