6 Diseases of ecological populations
Introduction
In the models of infectious diseases we have covered so far we have been focussing on human populations. However, infectious diseases are also extremely important to the dynamics of many animal, plant and microbial populations (for example, Foot and Mouth, Bovine TB, Wheat Rust, etc.). We should therefore extend our models to consider these cases. There are quite a few changes to our models that we may wish to make so that our models are more reflective of wider ecological populations. Three particular ones we will consider here are:
- We shall now assume that there is no immunity. Hosts can still recover from infection, but they will simply become susceptible once again. (The degree to which even simple organisms possess adaptive immunity is in fact a fascinating question, but for now we will assume it is absent).
- We can no longer reasonably assume that births and deaths are equal.
- We should include the fact that disease causes significant damage to hosts (something we have so far ignored). In particular, we shall assume that infected hosts suffer an additional mortality, or virulence at rate [latex]\alpha[/latex].
Putting these assumptions together with our previous model, we have a new SIS (Susceptible-Infected-Susceptible) model given by,
[latex]\begin{align} &\frac{dS}{dt} = b N - \beta SI - d S + \gamma I\\ &\frac{dI}{dt} = \beta SI - (d+\alpha+\gamma)I. \end{align}[/latex]
A case study: controlling rabbits in Australia
Prior to European settlement, there were no rabbits in Australia (see chapter references). Initially bred for food, their numbers stayed low until a small number were released for hunting purposes on an estate. Within 10 years rabbit numbers were well into the millions.
How might we model this initial population growth? During this early period, we might actually suppose that our very first population growth model provided a good approximation of rabbit dynamics,
[latex]\begin{equation} \frac{dN}{dt} = bN - dN=rN \end{equation}[/latex]
predicting exponential growth for [latex]b\gt d[/latex] (births greater than deaths). We previously criticised this model for not having a carrying capacity, but for populations of rabbits in the hundreds or thousands with the whole continent of Australia to exploit, we might argue that in fact there wasn’t much limiting their growth.
A number of control strategies were attempted through the late 19th- and early 20th-centuries targeting rabbits. In 1950, the myxoma virus was deliberately released in to the rabbit population. From a modelling perspective, the effect of this is to transform the system to the SIS model we introduced above, that is,
[latex]\begin{align} &\frac{dS}{dt} = b N - \beta SI - d S + \gamma I\\ &\frac{dI}{dt} = \beta SI - (d+\alpha+\gamma)I.\\ \end{align}[/latex]
We want to use this model to answer a simple question: will the introduction of the disease successfully control rabbit numbers? Initially we had exponential