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  • Review Article
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Mathematical models of infectious disease transmission

Nature Reviews Microbiology volume 6, pages 477–487 (2008)Cite this article

Key Points

  • Mathematical analysis and modelling is an important part of infectious disease epidemiology. Application of mathematical models to disease surveillance data can be used to address both scientific hypotheses and disease-control policy questions.

  • The link between the biology of an infectious disease, the process of transmission and the mathematics that are used to describe them is not always clear in published research. An understanding of this link is needed to critically interpret these publications and the policy recommendations and scientific conclusions that are contained within them.

  • This Review describes the biology of the transmission process and how it can be represented mathematically. It shows how this representation leads to a mathematical model of infectious disease epidemics as a function of underlying disease natural history and ecology. The mathematical description of disease epidemics immediately leads to several useful results, including the expected size of an epidemic and the critical level that is needed for an intervention to achieve effective disease control.

  • Statistical methods to fit mathematical models of disease surveillance data are outlined and the fundamental importance of the concept of likelihood is highlighted. The fit of mathematical models to surveillance data can provide estimates of key model parameters that determine a disease's natural history or the impact of an intervention, and are crucially dependent on the appropriate choice of mathematical model.

  • The Review ends with four outstanding challenges in mathematical infectious disease epidemiology that are essential for progress in our understanding of the ecology and evolution of infectious diseases. This understanding could lead to improvements in disease control.

Abstract

Mathematical analysis and modelling is central to infectious disease epidemiology. Here, we provide an intuitive introduction to the process of disease transmission, how this stochastic process can be represented mathematically and how this mathematical representation can be used to analyse the emergent dynamics of observed epidemics. Progress in mathematical analysis and modelling is of fundamental importance to our growing understanding of pathogen evolution and ecology. The fit of mathematical models to surveillance data has informed both scientific research and health policy. This Review is illustrated throughout by such applications and ends with suggestions of open challenges in mathematical epidemiology.

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Figure 1: Biological infectiousness over time after infection for three different human pathogens.
Figure 2: Offspring and generation-time distribution for an epidemic.
Figure 3: The emergent dynamics of infectious diseases.
Figure 4: Estimating R0 for influenza during the 1918 pandemic.

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Acknowledgements

The authors thank the Royal Society for funding.

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  1. Department of Infectious Disease Epidemiology, Medical Research Council Centre for Outbreak Analysis and Modelling, Imperial College London, London, W2 1PG, UK

    Nicholas C. Grassly & Christophe Fraser

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Glossary

Infectiousness

A characteristic of the infected individual that determines the rate of infection of susceptible members of the population and can be broken down into biological, behavioural and environmental components.

Superspreading

An individual who infects an 'unusually large' number of secondary individuals. The definition of unusually large can be subjective or be more formally defined with respect to the expectation under a random (Poisson) process.

Index case

The earliest infected individual who goes on to infect other individuals in the sample of cases that are being examined.

R0

The basic reproduction number, which is typically defined as the expected number of secondary infections that result from a single infected individual in an entirely susceptible (non-immune) population. The key property of R0 is its use as a threshold parameter, such that a major epidemic can only occur if R0 is more than one. In demography, ecology and the epidemiology of macroparasites (which typically do not multiply within the host), R0 has the analogous interpretation of the expected number of female offspring that result from a single female during her entire life in the absence of density-dependent constraints.

Stochastic

Involves random processes; the opposite of deterministic.

Mass action

The law of mass action states that the rate at which individuals of two types contact one another in a population is proportional to the product of their densities. Thus, the rate of increase in infected individuals accelerates early in an epidemic as the number of infected individuals increases and then declines as the number of susceptibles decreases, which often leads to a bell-shaped epidemic curve.

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Grassly, N., Fraser, C. Mathematical models of infectious disease transmission. Nat Rev Microbiol 6, 477–487 (2008). https://doi.org/10.1038/nrmicro1845

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