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

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

Virulence as a cost to transmission

Transmission

Virulence

S I S S

S

I S I

Lattice Models (Spatial structure within populations)

S

Transmission

Reproduction

S

Natural Mortality

I

NaturalMortality + Virulence

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

Intermediate Levels of Spatial Structure

I

SI

SGlobal Infection (L)

(1-L)Local Infection

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

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,

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 )

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

• 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.

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

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

Transmission

Virulence

Standardassumptionof the evolution of virulence theory

Evolution with a saturating trade-off in a spatial model

Approximation

Simulation

The role of recovery: The Spatial Susceptible Infected Removed (SIR) Model

S I S R

S

I R I

S

The role of recoveryNo recovery=0

The role of recovery=0.1

Increased ES virulenceWider 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

Recovery rate

Max ES virulence increases

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