Contagion spread

All models are wrong, but some are useful

Agent-based models have the potential to inform policy decisions, for example in response to an outbreak of a unknown contagion. Rather than experimenting with different policies, policy-makers can predict the effects of different scenarios and interventions through agent-based simulation. Of course, the usefulness of such predictions relies on the accuracy of the model and the estimations of its parameter. Agent-based models need to carefully select a level of detail. Too little detail, and the model is unable to reliably predict the effect of policies. Too much detail, and accurately estimating all necessary parameters in reasonable time become prohibitively difficult.

On this page, we take a look at the SIR model, one of the simplest models of the spread of contagion, and consider different ways of increasing the level of detail (and thereby the complexity) of this model. The result is a simplified model of the spread of COVID-19, shown in the script below (open script in a new tab), which extends the SIR model with small-world networks and contagion modelling.
The scripts on this page make use of HTML5.

SIR model

The SIR model (Kermack & McKendrick, 1927) provides a convenient starting point for constructing an agent-based model of the spread of contagion. This model separates the population into three classes: susceptible individuals who can be infected, infected individuals who spread the contagion to susceptible individuals, and recovered individuals who have become immune to the contagion.

The model is controlled by up to three parameters. The advantage of having very few parameters is that the model is more easy to evaluate and understand, and that it is relatively easy to find suitable values for these parameters. The downside is that the conclusions of the model for the purpose of formulating policy decisions are rather limited. To slow down or even stop the spread of the contagion, the SIR model suggests that policies should aim to reduce the infection rate or increase the recovery rate. However, how policies can achieve these effects is beyond the scope of the SIR model.

Small world network

To draw more fine-grained conclusions, the SIR model needs to be extended. For example, one of the more restrictive assumptions in the SIR model is that the population is well-mixed. This means that every interaction between agents is equally likely. While this may be a reasonable assumption for a small community, it is difficult to defend that people that live in the same street are equally likely to interact as people that live at opposite ends of the country.

In real life, social networks are highly clustered. For example, many people have friends in common with their friends. In addition, social networks tend to have the small world property, so that it takes relatively few steps to move from one individual to another in the network (cf. six degrees of separation.) While it’s not entirely certain that physical interactions can also be modelled as such a small-world network as well, it seems more reasonable than assuming the population is well-mixed.

In the script above, interactions are modelled through a small-world network, where agents are arranged on a circle. When the clustering coefficient is maximal, agents are only connected to neighbours on this circle. The lower the clustering coefficient, the more random the connections between agents become.

In addition to the small world network, the daily infection rate has been separated into the infection probability per interaction and the average number of interactions per agent per day. By making these two changes, the conclusions drawn from the model can be more fine-grained than for the SIR model. This extended model suggests that to slow down or even stop the spread of the contagion, policies should aim to limit the connectivity of people, reduce the number of interactions per person per day, reduce the infection rate per interaction, or increase the recovery rate.

Contagion modelling

Until now, the contagion has been rather abstract. The progression from susceptible to infected to recovered assumes that individuals are contagious directly after exposure to the contagion. Also, the contagion model ignores any symptoms altogether. In particular, individuals may behave differently depending on symptoms, and policy may be based on these observable symptoms.

We extend the model by representing the contagion as a Markov process. That is, the contagion is divided into discrete states. Once an individual has been exposed, the path of progression through the states of the infection is determined by a transition table. This transition table should then reflect the contagion being modelled. On this page, the default characteristics and transition rates have been set to COVID-19 characteristics (see Section 2.2 of Kerr et al., 2020).

Beyond the scope of this page

Of course, the model can be extended in many different ways. For example, CovaSim (Kerr et al., 2020) takes into account that COVID-19 has a different effect on different age groups. In addition, SynthPops takes a multi-layer approach to interactions, where interactions of households, schools, and workplaces are modelled separately and combined to create a more accurate model of interactions across different age groups. While these extension have the potential for making more accurate predictions of the spread of the contagion and its effects on society, they are beyond the scope of the scripts on this page.