There is widespread anxiety about the threat of the Covid-19 virus. Mathematics now plays a vital role in combating the spread of epidemics, and will help us to bring this outbreak under control. For centuries, mathematics has been used to solve problems in astronomy, physics and engineering. But now biology and medicine have become topics of mathematical investigation, and applications in these areas are certain to expand in the future [TM183 or search for “thatsmaths” at irishtimes.com].

How rapidly will the viral infection spread? How long will it remain a problem? When will it reach a peak and how quickly will it die out? Most important, what effective steps can we can take to control the outbreak and to minimize the damage caused? When vaccines become available, what is the optimal strategy for their use? Models provide valuable evidence for decision makers.

A model is a set of mathematical equations or a computer program that can simulate a physical, social or biological system. Models allow us to understand systems, to discover and explain patterns and to predict the effects of planned changes. For example, weather forecasts are made using models that simulate the atmospheric flow over the coming days. The model predictions are very valuable but are not infallible. All models involve simplifications and expertise is crucial in deciding what to include and what to omit. Validation is vital to ensure realistic outputs that can be reliably used for guidance.

**Equation-based Models**

For several decades, mathematical models have been applied to epidemiological problems. One of the earliest was the SIR model, with three variables that described how an outbreak evolves. The model divided the population into three categories: susceptible, infected and recovered people, denoted respectively by S, I and R, with each changing over time. It was successful in predicting the behaviour of some epidemics. However, many critical factors were omitted from the model.

**Agent-Based Models**

We now understand that classical equation-based models are too simple to answer most of our questions. Agent-based models consist of autonomous “agents” that interact with each other and have varied characteristics. For a model of a contagious outbreak, each agent represents an individual or a group of people who share certain characteristics, such as age, economic status or geographical location. This reflects the non-homogeneous nature of society and the wide range of behaviour that may be important in determining how infections spread.

Specific agents can interact with some other agents and not with others. The agents may change over time, and may have different life cycles. Since they account for specific characteristics of an outbreak, and for variations in individual behaviour, agent-based models can find new and better solutions to problems and provide more detailed and reliable guidance for policymakers. The rapid growth in computer power has enabled simulation using models with thousands or even millions of agents.

**Stochasticity**

The spread of an infection results from numerous chance events. Transmission occurs when infected people come into contact with others susceptible to the disease. Deterministic models like the SIR model make definite predictions, but they may not be accurate. “Stochastic” models allow for uncertainties by including criteria based on probabilities. Agent-based models incorporate random effects to allow for uncertainty and to provide confidence intervals for predictions.

Mathematical models have proved useful in responding to several outbreaks: Sars in 2003, Swine flu in 2009 and Ebola in 2014. The models are becoming more sophisticated and powerful and will play an indispensable role in controlling the Covid-19 pandemic.

**Sources**

Steven F. Railsback and Volker Grimm, 2019: *Agent-Based and Individual-Based Modeling: A Practical Introduction,* Second Edn., Princeton University Press, 340pp.

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