SIR models of epidemics

Level 1 module

Basic problem

Infectious diseases are a major cause of death worldwide, and have in the past killed many more people than all the wars (think, for instance, of the Spanish flu). Mathematical modelling of infectious diseases was initiated by Bernoulli in 1760. The work of Kermack and McKendrick, published in 1927, had a major influence on the modelling framework. Their SIR model is still used to model epidemics of infectious diseases. We will study this basic model, and some of its extensions.

General approach

The SIR model tracks the numbers of susceptible, infected and recovered individuals during an epidemic with the help of ordinary differential equations (ODE). The model can be coded in a few lines in R. We will learn how to simulate the model and how to plot and interpret the results. We will use simulation to verify some analytical results. We will plot time courses, phase diagrams and contour plots.

What can be learned?

Concepts

Basic reproductive ratio (R0)
Herd immunity

Methods

Numerical simulation of ordinary differential equations
Graphical tools (phase portrait, contour plot)

Starting point

Download reader and script describing the basic SIR model.

Interesting questions that you can investigate

• What conditions are necessary for the outbreak of an epidemic?
• What fraction of a population is going to be infected in a transient epidemic?
• Can partial vaccination in a population protect against the outbreak of an epidemic?
  

Advanced questions

Modify the model to
o allow for the loss of immunity
o model treatment of the disease
o model the emergence of drug resistance and find the optimal rate of treatment
o model longer time scales that allow for the birth and death of individuals

Glossary

• Compartment models: models where the population is divided between several spatial compartments or classes, which are connected by the flow (migration or transformation) of individuals.
• SIR models: models where the population is divided into 3 classes - susceptible individuals are uninfected and susceptible to the disease; infected individuals are infected and can infect susceptibles; recovered individuals have recovered from the infection and are immune to re-infection.
• Basic reproductive ratio: the key parameter of epidemiology. It represents the number of secondary cases initiated by the introduction of a single infected individual into a 'naive' population.
• Phase portrait: also called a phase diagram, it shows the temporal evolution of two or three variables of a system. It consists of trajectories plotted in a coordinate system that has axes corresponding to the variables of the system.
• Herd immunity: immunity of a population to the outbreak of an epidemic, provided by the immunity of only a fraction of the individuals.

Literature and weblinks

Kermack, W. and McKendrick, A., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. London A 115, 700-721. Proc. R. Soc. London A 115, 700-721.

Anderson, R. M. and May, R. M. 1991. Infectious Diseases of Humans. Oxford. Oxford University Press

Earn DJ et al (2000). A Simple Model for Complex Dynamical Transitions in Epidemics. Science 287, 667-670.

Matt Keeling's article in Plus: The mathematics of diseases (open access)

Compartment models of epidemiology in Wikipedia