host population structure and the evolution of parasites mike boots
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![Page 1: Host population structure and the evolution of parasites Mike Boots](https://reader030.vdocuments.site/reader030/viewer/2022033104/56649d235503460f949f8f84/html5/thumbnails/1.jpg)
Host population structure and the evolution of parasites
Mike Boots
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MALARIA
OurInfectious 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 assumptionSmall mutationsEcological feedbacks
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Theory on the evolution of parasites
Infectivity is maximisedInfectious period maximised
Mortality due to infection (virulence) minimisedRecovery rate minimised
Trade-offs related to exploitation of the host explain variation
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Virulence as a cost to transmission
Transmission
Virulence
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S I S S
S
I S I
Lattice Models (Spatial structure within populations)
S
Transmission
Reproduction
S
Natural Mortality
I
NaturalMortality + Virulence
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200 400 600 800 1000
5
10
15
20
25
30
35
t
MeanTransmission
TIME
No trade-offs between transmission and virulence
Simulation results for the evolution of transmissionwith individuals on a lattice where interactions are all local
Max transmission = 150
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Intermediate Levels of Spatial Structure
I
SI
SGlobal Infection (L)
(1-L)Local Infection
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Mean Virulence
1.00.80.60.40.20.00
1
2
3
4
5
L (Proportion of global infection)
Maximum virulence
Lineartrade-offwith virulenceand transmission
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Host Parasite models between local and mean-field
Pair-wise Approximation: differential equations for pair densities
PSI(t) =prob randomly chosen pair is in state SI
z
(z 1)PSIqI /SI
conditional prob thatI is a neighbour of an Ssite in an SI pair
event
z
PSI =
transmission rate
# neighbours(fixed)
r(SI II )
eg,
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Host Parasite models between local and mean-field
Pair-wise Approximation: differential equations for pair densities
eg, PSI(t) =prob randomly chosen pair is in state SI
z
(z 1)PSIqI /SI
event
z
PSI
=r(SI II )
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Host Parasite models between local and mean-field
Pair-wise Approximation: differential equations for pair densities
eg, PSI(t) =prob randomly chosen pair is in state SI
z
(z 1)PSIqI /SI
event
z
PSI
=r(SI II ) 1 LI LIPSI PI
LI=0 (local), LI=1 (mean-field) proportionof global infection
(1-LI)
LI
prob that a site is infected
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• Derive correlation Eqns:
dPSI
dt r(SI )
events , for each pair and singleton from
states S, I, R and 0 (empty sites).
• 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)
Host Parasite models between local and mean-field
with params 0<LI,Lr<1 for global proportions of reproduction forpathogen and host.
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Invasion Condition
(J | I ) 1
J
dJ
dtJ {L̂S (1 L)q̂0
S / J } ( J d) > 0
J is a mutant strainI is the resident strainHat notation denotes quasi steady state
Transmission Virulence Background Mortality
Global density of susceptibles
Local density of infecteds
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Pairwise Invasion Plots (Linear trade-off between transmission and virulence)
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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
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The ESS is lost
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Bistability
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Bistability
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The role of trade-off shape
Transmission
Virulence
Standardassumptionof the evolution of virulence theory
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Evolution with a saturating trade-off in a spatial model
Approximation
Simulation
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The role of recovery: The Spatial Susceptible Infected Removed (SIR) Model
S I S R
S
I R I
S
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The role of recoveryNo recovery=0
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The role of recovery=0.1
Increased ES virulenceWider region of bistability
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The role of recovery=0.2
Bi-stability region reduces
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The role of recovery=0.3
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The role of recovery=0.4
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The role of recovery
Recovery rate
Max ES virulence increases
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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
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Collaborators
Akira Sasaki (Kyushu University)
Masashi Kamo (Kyushu: Institute for risk management, Tsukuba)
Steve Webb