Modeling Epidemics With Compartmental Models

INTRODUCTION

This JAMA Guide to Statistics and Methods reviews the susceptible-infected-recovered (SIR) model for predicting the course of infectious disease outbreaks, which describes the transition of individuals from susceptible to infected and from infected to recovered, and discusses the model’s limitations, including oversimplification of complex disease processes.

During epidemics, there is a critical need to understand both the likely number of infections and their time course to inform both public health and health care system responses. Approaches to forecasting the course of an epidemic vary and can include simulating the dynamics of disease transmission and recovery^1,2 or empirical fitting of data trends.³ A common approach is to use epidemic compartmental models, such as the susceptible-infected-recovered (SIR) model.^1,2

WHY IS A SIR MODEL USED?

The SIR model aims to predict the number of individuals who are susceptible to infection, are actively infected, or have recovered from infection at any given time. This model was introduced in 1927, less than a decade after the 1918 influenza pandemic,⁴ and its popularity may be due in part to its simplicity, which allows modelers to approximate disease behavior by estimating a small number of parameters.

DESCRIPTION OF THE SIR MODEL

In compartmental models, individuals within a closed population are separated into mutually exclusive groups, or compartments, based on their disease status. Each individual is considered to be in 1 compartment at a given time, but can move from one compartment to another based on the parameters of the model. The SIR model is one of the most basic compartmental models, named for its 3 compartments (susceptible, infected, and recovered). In this model, the assumed progression is for a susceptible individual to become infected through contact with another infected individual. Following a period as an infected individual, during which that person is assumed to be contagious, the individual advances to a noncontagious state, termed recovery, although that stage may include death or effective isolation.

In most modeled epidemics, all of a population begins in the susceptible compartment (Figure 1), which contains individuals who might become infected if exposed to the pathogen. This implies that no one has immunity to the disease at the beginning of the outbreak. The infected compartment is defined as individuals who have the ability to infect individuals in the susceptible compartment. As such, this compartment includes asymptomatic transmitters of the pathogen as well as hospitalized patients who require intensive levels of care. One simplification in the SIR model is that it does not consider the latent period following exposure, rather it assumes that newly infected individuals are immediately contagious.