Mathematical models of infectious disease transmission

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.
The dynamics of infectious diseases are complex, so developing models that can capture key features of the spread of infection is important. Grassly and Fraser provide an introduction to the mathematical analysis and modelling of disease transmission, which, in addition to informing public health disease control measures, is also important for understanding pathogen evolution and ecology.
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.