Modelling an epidemic on a network

Introduction

In MAS371 Applied Probability we have seen a simple model for an SIR (Susceptible, Infected, Removed) epidemic. This model is a continuous time Markov chain where each individual in the population is classified as susceptible, infected or removed. A currently susceptible individual becomes infected at rate λ times the number of infected individuals, and a currently infected individual becomes removed at rate μ. We can think of the process as starting at time zero with a single infected individual and all others being susceptible.

The aim of this page is to consider how this model might be made a little more realistic, and to use this to consider some features of epidemics and their control. However, it should of course be noted that this is still a toy model; it is certainly not a fully realistic model of the spread of COVID-19.

Epidemic on a network

One disadvantage of the model seen in MAS371 is that an individual is equally likely to get the disease from anyone else in the population: the rate of becoming infected is λ if there is one currently infected individual regardless of who that individual is. However, in a real population individuals are in contact with only certain other individuals, and are only at risk of getting the disease from them. We can model this by considering the population as a network, where vertices represent individuals and edges represent contacts between individuals.

We will again consider the epidemic as a continuous time Markov chain. However, the rate at which a susceptible individual becomes infected will now be λ times the number of infected neighbours they have in the network (alternatively, λ times the number of edges from that individual to an infected individual). The rate at which an infected individual becomes removed remains μ. The state space for the Markov chain is now the set of all possible settings of states for each individual; note that this is a very large state space.

Various network models are available. On this page we will work with a variant of a model known as the Watts-Strogatz small world graph, which includes an element of local connectivity (representing the idea that people are likely to have contacts with their physical neighbours) but also some random connections (representing the idea that people will have some long distance contacts, which in this context would be due to travel).

In a more sophisticated model, the graph could be modelled as changing over time. With one exception, we will not consider this on this page.

Simulation

The video linked below shows a simulation of this model with a particular graph where each individual has about 12 contacts, λ=0.4 and μ=1. Turquoise vertices are susceptible, red vertices are infected and blue vertices are removed.

As can be seen the epidemic grows slowly at first, then more rapidly as more individuals are infected. Once the number of susceptible individuals falls low enough, the epidemic peaks and starts to die out. However, in this run almost all individuals get infected eventually.

With these parameters the initial reproduction number, R0, is about 3.4. This is the average number of individuals who catch the disease from a single infected individual near the start of the epidemic.

Vaccination

One way to control an epidemic is by vaccination. Typically a large proportion of the population (but usually not the whole population) is vaccinated, meaning that they cannot (or are much less likely to) catch the disease.

The video below shows two versions of the model running side by side (and using the same random variables, to enhance the comparison). On the left no vertices are vaccinated, while on the right 75% of the vertices are initially vaccinated (coloured pink) and are immune to the disease.

The vaccination almost entirely prevents the spread of the epidemic. Note that even individuals who have not been vaccinated are protected, because the vaccination of others prevents the disease from getting to them. This is the concept of "herd immunity": if enough individuals are vaccinated the whole population is protected.

A lower proportion of individuals being vaccinated may not be enough to prevent the spread of the epidemic. The video below and the plot below that show a similar simulation but with the initial vaccination proportion only 40%. The epidemic is reduced compared to the scenario with no vaccination, but still spreads.

Social distancing

During the COVID-19 pandemic, social distancing has been widely used to combat the spread of the disease. Social distancing can be thought of as an effort to reduce the number of contacts, so in the graph model it can be thought of as the removal of a proportion of the edges in the graph.

he following simulation runs versions of the model with and without social distancing in parallel. The version on the left runs with no control measures; the version on the right imposes social distancing when the infection rate exceeds 15% of the population, and does so by deleting each edge with probability 0.75. As can be seen the imposition of social distancing here gradually suppresses the epidemic.

More realistically, social distancing might be relaxed when the epidemic eases off. The simulation below imposes distancing, as above, when the infection rate exceeds 15%. However, it removes it again, restoring the original network, when the infection rate falls below 5%. As can be seen, this has the potential to allow the epidemic to grow again, causing a "second wave", requiring distancing to be re-imposed. (In fact, in this simulation three waves can be seen.)

Other control methods

Some other methods which have been suggested for controlling epidemics can also be interpreted in terms of this type of model. Wearing masks aims to reduce the chance of a contact actually spreading the infection, so can be seen as trying to reduce the value of λ. "Test and trace" methods aim to isolate known cases and their immediate contacts, so can be seen as removing edges, in a similar way to social distancing, but only in the vicinity of cases.