1. Persistence and dynamics of disease in a host-pathogen model with seasonality in the host birth rate. Rachel Norman and Jill Ireland

2. Motivation: RHD Rabbit Haemorhaggic disease kills up to 95% of those infected within 48 hours. Rabbit Haemorhaggic disease kills up to 95% of those infected within 48 hours. First recorded in China in 1984 where it killed 140 million farmed rabbits. First recorded in China in 1984 where it killed 140 million farmed rabbits. The disease is now in wild rabbits and is a conservation issue in Europe and a pest control issue in Australia/New Zealand. The disease is now in wild rabbits and is a conservation issue in Europe and a pest control issue in Australia/New Zealand. Arid areas: 65-90% population reduction. Arid areas: 65-90% population reduction. Temperate areas: less impact. Temperate areas: less impact.

3. The original hypothesis Since breeding patterns for rabbits will be different in temperate (longer) than arid (shorter) regions then the number of susceptibles available will vary and so disease persistence is likely to be effected. It might also be the case that the time of year that the disease appears will have an effect on persistence.

4. The basic model (no seasonality):

5. Analysis Three possible biologically relevant equilibria of the form (H,Y,Z) Three possible biologically relevant equilibria of the form (H,Y,Z) (0,0,0), nothing there, only stable if the growth rate is negative. (0,0,0), nothing there, only stable if the growth rate is negative. (K,0,0) host at carrying capacity, no infection, stable iff (K,0,0) host at carrying capacity, no infection, stable iff Coexistence equilibrium Coexistence equilibrium

6. Coexistence equilibrium We can show that there is one root of this cubic lying between 0 and K iff We assume that this equilibrium is stable or there are stable limit cycles if no other equilibrium is stable.

7. Dynamics of the basic model Numerical experiments have not shown stable limit cycles but under some parameter regimes we get oscillations to a stable equilibrium, in other words we have a spiral point in state space: Numerical experiments have not shown stable limit cycles but under some parameter regimes we get oscillations to a stable equilibrium, in other words we have a spiral point in state space:

8. Seasonality The basic model assumes a constant birth rate. In many real biological systems this is not the case e.g rabbits, grouse, foxes, bank voles. The basic model assumes a constant birth rate. In many real biological systems this is not the case e.g rabbits, grouse, foxes, bank voles. We therefore make the birth rate in the model seasonal, in the first instance we use a sine wave We therefore make the birth rate in the model seasonal, in the first instance we use a sine wave

9. The seasonal model:

10. Analysis: disease persistence In this case we cannot carry out equilibrium analysis in the same way. In this case we cannot carry out equilibrium analysis in the same way. We therefore took a different approach and assumed that as a(t) is bounded then we can use those bounds to determine disease persistence: We therefore took a different approach and assumed that as a(t) is bounded then we can use those bounds to determine disease persistence: Assumption: Ro is of the same form as in the basic model i.e. Assumption: Ro is of the same form as in the basic model i.e.

11. Initial Conjectures If R 0 (t)<1 for the whole season, then the disease will not persist. If R 0 (t)<1 for the whole season, then the disease will not persist. If R 0 (t)>1 for the whole season then the disease will persist. If R 0 (t)>1 for the whole season then the disease will persist. When neither of the above cases hold then there is some “average” value of R 0 (t) above which the disease persists. When neither of the above cases hold then there is some “average” value of R 0 (t) above which the disease persists.

12. R 0 <1 for all t.

13. R 0 >1 for all t.

14. In between:

15. Conjecture If then the disease If then the disease will die out. If then the disease If then the disease will persist

16. Dynamics and applications We will now consider two biological applications which will illustrate interesting aspects of the dynamics of this model. We will now consider two biological applications which will illustrate interesting aspects of the dynamics of this model. These systems are cowpox in bank voles and rabbit haemorhaggic disease in rabbits. We have parameterised the model for both of these systems. These systems are cowpox in bank voles and rabbit haemorhaggic disease in rabbits. We have parameterised the model for both of these systems.

17. Cowpox in bank voles This disease is endemic in bank voles and the system is of interest since ecologists have been trying to explain the cycles seen in vole populations. This disease is endemic in bank voles and the system is of interest since ecologists have been trying to explain the cycles seen in vole populations. One interesting aspect of this disease is that it does no kill the host, this simplifies the model. However, the results which follow do not rely, mathematically, on this. One interesting aspect of this disease is that it does no kill the host, this simplifies the model. However, the results which follow do not rely, mathematically, on this.

18. Cowpox without seasonality H time

19. Seasonality If we add seasonality to the model we get R 0,min =16.14 and R 0,max =21.24, clearly both of these values lie above 1 and we would expect the disease to persist. If we add seasonality to the model we get R 0,min =16.14 and R 0,max =21.24, clearly both of these values lie above 1 and we would expect the disease to persist. These simulations are consistent with the data on cowpox in bank voles since they predict an annual cycles in vole dynamics. These simulations are consistent with the data on cowpox in bank voles since they predict an annual cycles in vole dynamics. In addition, the peaks and troughs in the infected section of the population come after those in the susceptible section, again this is consistent with field observations. In addition, the peaks and troughs in the infected section of the population come after those in the susceptible section, again this is consistent with field observations. Infecteds Susceptibles time

20. RHD without seasonality

21. RHD with seasonality In this case, if we add seasonality R 0,min =0, R 0,max =353.2 and R 0,ave =180, again, in this case we predict disease persistence. In this case, if we add seasonality R 0,min =0, R 0,max =353.2 and R 0,ave =180, again, in this case we predict disease persistence. Y time

22. As we can see from the previous simulation, the dynamics of RHD are predicted to be extremely complex, in fact what we have is resonance between the underlying oscillations in the non-seasonal model and the seasonality. This phenomenon has been studied by Greenman et al (2004) for a model with seasonality in the transmission rate. Their model explains many of the characteristics of the dynamics of childhood diseases such as measles and whooping cough. As we can see from the previous simulation, the dynamics of RHD are predicted to be extremely complex, in fact what we have is resonance between the underlying oscillations in the non-seasonal model and the seasonality. This phenomenon has been studied by Greenman et al (2004) for a model with seasonality in the transmission rate. Their model explains many of the characteristics of the dynamics of childhood diseases such as measles and whooping cough.

23. Method of analysis We scale the equations by a factor 1/p and rescale time such that t’=tp We scale the equations by a factor 1/p and rescale time such that t’=tp We now have all of the parameters divided by p and, more importantly We now have all of the parameters divided by p and, more importantly We then vary p We then vary p As p varies we effectively looking at a different member of the family of models defined by our equations. As p varies we effectively looking at a different member of the family of models defined by our equations.

24. Bifurcation diagram for RHD

25. Resonance diagram for RHD

26. When do we get resonance? We get resonance when our non-seasonal model exhibits oscillations towards the equilibrium, which is when the eigenvalues of the Jacobian are complex with negative real parts. We can work out a formula for the line which separates real and complex eigenvalues and plot it in parameter space. We get resonance when our non-seasonal model exhibits oscillations towards the equilibrium, which is when the eigenvalues of the Jacobian are complex with negative real parts. We can work out a formula for the line which separates real and complex eigenvalues and plot it in parameter space.

27. Summary and conclusions Many real wildlife systems have seasonal birth rates. Many real wildlife systems have seasonal birth rates. We cannot analyse the model presented using equilibrium analysis. We cannot analyse the model presented using equilibrium analysis. We can find a simple condition for persistence of the disease. We can find a simple condition for persistence of the disease. However, the dynamics of the disease can be much more complex as we get resonance between the oscillations in the non-seasonal model and the seasonal forcing term. However, the dynamics of the disease can be much more complex as we get resonance between the oscillations in the non-seasonal model and the seasonal forcing term. In real biological terms this means the if a disease enters a seasonally oscillating population then resonance and chaotic dynamics could occur. In real biological terms this means the if a disease enters a seasonally oscillating population then resonance and chaotic dynamics could occur. Adding seasonal birth rates to models of real systems gives us dynamics which are at least qualitatively similar to the observed dynamics in RHD, rabies and cowpox in bank voles. Adding seasonal birth rates to models of real systems gives us dynamics which are at least qualitatively similar to the observed dynamics in RHD, rabies and cowpox in bank voles.