4 Epidemics in human populations
the Susceptible-Infected-Recovered model
At the time of writing, it seems reasonable to say that mathematical modelling of disease spread has never been as topical or well-known amongst the general public. Many modellers and scientists have been contributing to our understanding of the Covid-19 pandemic, from research being conducted and published, to directly advising government policy, to public engagement and information. In the next few chapters we will cover the fundamentals of the mathematical models that are the basis of most of this modelling work. The models that have been used to advise governments are of course rather more detailed than those you will see here (and we’ll discuss some extensions as we go along), but there is a common engine behind them all.
Our first few models focussed on the dynamics of ecological populations. As part of this we have assumed that every individual within each population is identical. Such a generalisation has the benefit of reducing our model to a low number of variables. But often certain individuals within a population are fundamentally distinct from one another. In this case, we might want to divide our population up in to different compartments. The choice of how to divide our population is often dictated by the biological question we are seeking to address. For example we may wish to have an age-structured model, with adults and juveniles, or perhaps individuals with different roles in a population such as foragers or workers.
For the next few chapters we will focus on epidemic models, where we will need to use such a compartment model. Broadly, we will think about disease dynamics within populations of humans, animals or plants that act as hosts for diseases such as viruses or bacteria, often termed parasites or pathogens. We will initially assume we are interested in infectious diseases of human populations, and this will guide our model design. When thinking about the spread of disease through a population, while we might have some interest in the total population at any time, it is more likely that we want to know about individuals’ infection status – are they currently infected, for example. This will be the basis of our compartment model. Once exposed to the disease, individuals may then become infected (and also infectious). Finally, individuals will eventually fight off the disease to become recovered and immune.
This is the ‘SIR’ model (i.e. Susceptible, Infected, Recovered), and has been central to our understanding of disease dynamics in human populations since it was first proposed in the 1920s. Disease is transmitted by ‘direct contact’ between infected and susceptible hosts and is a ‘mass action’ process, meaning we do not take any account of contact networks and the risk of infection is proportional to the total density of infection in the population. When hosts have recovered, they gain immunity and cannot be infected again (this is a key aspect of