1. Host population structure and the evolution of parasites
Mike Boots
I’m an evolutionary biologist, based in Animal and plant sciences in Sheffield. I’m going to talk about some theory on the general role of spatial structure in the evolution of parasites.
As well as general theory, in Sheffield we work on a number of diseases

2. Our Infectious Diseases
these are our diseases. look at two tropical diseases. Malaria & Dengue, But there are also lots of wildlife diseases. These are the ones that I work on……………. Evolutionary theory of parasites in general is very well developed, because there’s an obvious importance to human health, there is lots of variation in their virulence. So there’s lots of scope to understand what processes select for these different life-histories: solutions to being an infectious organism.

3. 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

4. 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

5. Virulence as a cost to transmission

6. Lattice Models (Spatial structure within populations)
Natural
Mortality I
Natural
Mortality + Virulence S S
Transmission S I S S
OK . Of course the thing that all these models ignore is within population spatial structure. Populations are not completely mixed. You will infect individuals that are near to you. So we wanted to add space simply. So we produce a simple model on a lattice. We track where each individual is….
Reproduction I S I

7. Simulation results for the evolution of transmission
with individuals on a lattice where interactions are all local
Mean
Transmission
Why? Space matters….. You surround yourself. Local availabilty of sussceptibles is important
TIME
Max transmission = 150
No trade-offs between transmission and virulence

8. Intermediate Levels of Spatial Structure
Global Infection (L)
(1-L)
Local Infection
We have built models with an intermediate level of spatial structure in a very simple way. We assume that a proportion of the interactions are global. That is they can be with any individual on the lattice. While the remaining interactions are local with nearest neighbours only. Biologically this is not unreasonable - although simplistic- for a number of organisms and it has been shown to do nearly as good a job as more complex methods of mixing populations. It has the advantage of being simple. This way we can check whether the spatial effects we see are only for extreme spatial structure. For them to be generally relevant I would argue they need to be not just extreme local structure phenomenon.

9. Linear trade-off with virulence and transmission Mean Virulence
Maximum virulence 1 2 3 4 5
Mean Virulence
Linear
trade-off
with virulence
and transmission
1.0
0.8
0.6
0.4
0.2
0.0
L (Proportion of global infection)

10. 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
conditional prob that
I is a neighbour of an S
site in an SI pair
event =
transmission
rate
# neighbours
(fixed)

11. 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
eg,
event =

12. 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
eg, LI
event
prob that a site is
infected
(1-LI) =
proportion
of global infection
LI=0 (local), LI=1 (mean-field)

13. Host Parasite models between local and mean-field
Derive correlation Eqns:
for each pair and singleton from
states S, I, R and 0 (empty sites).
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)

14. Invasion Condition Local density of infecteds > 0
Transmission
Virulence
Background
Mortality
> 0
Global density of susceptibles
In order to obtain the value of , we assume that the conditional densities in the nearest neighbourhood of a rare mutant strain change much faster than the global density of the resident strain, which can be justified during the initial phase of invasion in which the global density of mutant-infected hosts remains small. We can solve the pair equations numerically to obtain the quasi equilibrium value of and then calculate the invasibility of mutant strain from this equation. When we repeat the procedure for a various combination of resident and mutant parameters, we can draw pair wise invasibility plots (PIPs).
J is a mutant strain
I is the resident strain
Hat notation denotes quasi steady state

15. Pairwise Invasion Plots (Linear trade-off between transmission and virulence)
For those that are unfamiliar, let me take you through these.

16. 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

17. The ESS is lost

18. Bistability

19. Bistability

20. The role of trade-off shape
Transmission
Virulence
Standard
assumption
of the evolution
of virulence theory

21. Evolution with a saturating trade-off in a spatial model
Simulation
Approximation
What do we find.

22. The role of recovery: The Spatial Susceptible Infected Removed (SIR) Model
As you know many hosts can acquire immunity to some of their diseases. We felt that the spatial relationships of these immune individuals are likely to be important. They are going to be around infected individuals to some degree. So we built a simple model with recovered individuals. They are removed as we assume for simplicity that immunity doesn’t wain.

23. The role of recovery No recovery =0
So let’s start with recovery at zero. This is what we had before..

24. The role of recovery =0.1 Increased ES virulence
Wider region of bistability

25. The role of recovery
=0.2
Bi-stability region reduces

26. The role of recovery
=0.3

27. The role of recovery
=0.4

28. The role of recovery Max ES virulence increases Recovery rate
This blue line is here. ES virulence - when you have it - goes up with increasing recovery. The critical infection proportion goes up at first - so you have a more chance of an ESS early for low recovery and then falls as
Max ES virulence increases
Recovery rate

29. 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
The simple model shows very different results in space. Wide range of population structures which makes us think it may be important. In particular we get bistability between locally stable points. This can lead to dramatic shifts. Large shifts in virulence have occurred due to recombination between less virulent strains in RHD.

30. Collaborators Akira Sasaki (Kyushu University)
Masashi Kamo (Kyushu: Institute for risk management, Tsukuba)
Steve Webb