5 The SIR model with demographics
Including Births and Deaths
In our initial epidemic model in the previous chapter we had only two mechanisms involved – infection and recovery. Let us now add some increased detail into the model by including birth and death processes. Given the timescales of epidemic processes, this likely means looking at dynamics over a much longer time period, at least in human populations. Let’s begin by drawing a schematic of the system again:
We will make the simplifying assumption that the birth rate and death rate are equal, so that the population would be at equilibrium in the absence of disease (this is a questionable assumption for many ecological populations, but perhaps not too unreasonable for modern human populations). If we assume all individuals produce offspring at rate [latex]\mu[/latex], that everyone is born susceptible, and that every individual has the same death rate, also [latex]\mu[/latex], this leads us to the equations,
[latex]\begin{align} &\frac{dS}{dt} = \mu N - \beta SI - \mu S\\ &\frac{dI}{dt} = \beta SI - (\gamma +\mu)I\\ &\frac{dR}{dt} = \gamma I-\mu R \end{align}[/latex]
Once more, [latex]dN/dt=0[/latex] so we can eliminate [latex]R=N-(S+I)[/latex] (again, this is somewhat by design – if the rates of birth and death were not equal this simplification could not be made). One thing to note here is that the infectious period has changed to be [latex]1/(\gamma+\mu)[/latex]. When we define [latex]R_0[/latex], therefore, in our updated model we will have [latex]R_0=\beta N/(\gamma+\mu)[/latex].
Endemic disease
At this point we could non-dimensionalise our system as we have seen previously. This would allow us to reduce the number of parameters in our model to make life easier, as well as revealing potentially useful information about the scales involved. You will see some studies do this and others don’t. For now, let us continue with our analysis without doing so, since it means we retain the clear biological meanings of all of our parameters and variables.
First we should find the equilibria of our system, where [latex]dS/dt=0[/latex] and [latex]dI/dt=0[/latex] simultaneously. Recall in the previous model this only happened when [latex]I=0[/latex], with this condition satisfying both ODEs and leaving us with a line of equilibria. Now, we find more ‘standard’ unique equilibria. There are two cases where [latex]dI/dt=0[/latex]:
- [latex]I^*=0[/latex], giving [latex]dS/dt=0\implies S^*=N[/latex];
This means the population is disease-free.
- [latex]S^*=\dfrac{\gamma+\mu}{\beta}[/latex], giving [latex]dS/dt=0\implies I^*=\dfrac{\mu(N-S^*)}{\beta S^*}=\dfrac{\mu}{\beta}\left(\dfrac{N}{S^*}-1\right)[/latex];
This means the disease is endemic.
We can do a bit more manipulation of that last equilibrium,
[latex]\begin{equation} I^*=\frac{\mu}{\beta}\left(\dfrac{N}{S^*}-1\right)=\frac{\mu}{\beta}\left(\frac{\beta N}{\gamma+\mu}-1\right)=\frac{\mu}{\beta}\left(R_0-1\right). \end{equation}[/latex]
While both e