Modelling epidemics with real-world data
Problem: You are an epidemiologist or public health worker. You have obtained data on \(\mathcal{R}\) and other parameters and want to use epydemic to model the disease – but then discover that it needs things like infection probability rather than the data you have available.
Solution: The relationship between the \(\mathcal{R}\) number and the probabilities involved in simulation can be modelled in several ways, depending on the degree of accuracy you want.
\(\mathcal{R}\) is defined as the average number of secondary infections that arise from every new infection. It’s probably the easiest parameter to understand in the disease dynamics. It actually forms a critical point – what a mathematician would called a separatrix – between two possible regimes of epidemic evolution. If \(\mathcal{R} < 1\) then the current “generation” of infected individuals gives rise to fewer infected individuals in the next generation, and so (if this is sustained for enough generations) the epidemic will die out; conversely, if \(\mathcal{R} > 1\) the next generation will be larger, and the disease will become epidemic.
\(\mathcal{R}\) will change over time both as a result of the population becoming gradually immune (if this happens with the disease model in question), and as a result of countermeasures that seek to` reduce infection. (Both of these effects are explored experimentally by Dobson [Dob20].) At the very start of an epidemic we have the disease spreading without immunity (in what epidemiologists refer to as a naive population), and we speak of \(\mathcal{R}\) at this point as \(\mathcal{R}_0\) – \(\mathcal{R}\) at time \(t = 0\).
A compartmented model of disease is described by dynamical parameters concerning the way the disease spreads at each contact. For SIR these parameters are the probability that a contact leads to infection (\(\beta\)) and the probability that an infected individual recovers (\(\alpha\)). In disease models that use differential equations as their basis, we can define
So \(\mathcal{R}\) is simply the ratio of the rate of infection to the rate of recovery. This means the actual choices of numbers for \(\beta\) and \(\alpha\) aren’t important in themselves, only their relationship to each other, so we could set \(\alpha = 1\), use this to set \(\beta = \mathcal{R}\), and get a simulation of the measured disease.
However, this takes no account of the network over which infections are spreading, and we would expect, in a realistic model, that (for example) infected individuals with a large number of contacts spread the disease more effectively than those with only a small number. This means we should expect the degree distribution (and possibly other topological parameters) to appear in our calculations.
Let’s change notation slightly and define \(T\) to be the transmissibility of the disease, defined by \(\frac{\beta}{\alpha}\). This is purely a disease-level value, taking no account of the network. We should be able then to express \(\mathcal{R}\) in terms of \(T\) and the network’s topology.
Choose a node at random and infect it. If we take the view that we will choose an “average” node, we expect it to have the average number of neighbours (which we denote \(\langle k \rangle\)), and this node will infect a fraction \(\beta\) of those neighbours in the time until it recovers. The expected average recovery time is \(\frac{1}{\alpha}\), and so we have that
This is known as the mean field solution of the epidemic equations: it assumes that every node behaves like the average node. Again it gives us a route to define \(\beta\) and \(\alpha\) given that we know both \(\mathcal{R}\) (from measurements of the disease) and \(\langle k \rangle\) (from measurements of the network).
The mean field is clearly a strong assumption to make in networks where the degree distribution is not normal. We can account for these effects by taking account of higher moments of the degree distribution, for example:
where \(\langle k^2 \rangle\) is the variance of the degree distribution. Again, we measure the topological parameters of the network and use them to compute the probabilities.
There are a few observations we can make here. The first is that the maximum number of people that an infected node can infect is bounded by its degree, \(\langle k \rangle\) on average. Secondly, the higher moments of the distribution limit this still further. In order for there to be an epidemic we need \(\mathcal{R} > 1\). To get this from the last equation implies a threshold
which is the epidemic threshold for the network, often denotes \(\phi_c\): below this no epidemic will take hold. This is known as the Molloy-Reed criterion [MR96] for epidemics to spread on networks.
The third observation pertains to the second. If we have
then if \(\langle k^2 \rangle\) is very large then \(\phi_c\) will be very small: there will be an epidemic even for very small transmissibilities. Indeed, in the limit of \(\langle k^2 \rangle \rightarrow \infty\) it will be that \(\phi_c \rightarrow 0\). It so happens that networks with powerlaw degree distributions do indeed have such infinite variance, and powerlaw networks with a cutoff (which are used as models of human populations) often have very high variances, which implies that such networks will be very susceptible to epidemic outbreaks. (The network science community captures this with the saying that “powerlaw networks always percolate”.)
This has been a long answer to what seems like a straightforward question about the relationship between measurements collected in the field for “real” epidemics and their simulation on networks. In general we can say that we need to account for both disease and network characteristics when building a simulation intended to model a real disease. But the disease we’ve measured is itself spreading over a network of individuals in the real world – which is a network we can’t directly observe, measure, or control very precisely. In other words we have a lot of work to do to determine what disease parameters are actually in play.