Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Wednesday, October 9, 2013

The SIR model

What does it mean?

This is the standard model for how an infectious disease is spread in a closed population without births, deaths or migration. Individuals are in three categories, the ones not yet infected but susceptible (S), the infected ones (I) and the ones that have recovered from the disease and have become immune (R). This system describes the rate of change of the proportions of the three categories and only two parameters dictate the dynamics - β is the rate by which susceptible individuals become infected, and γ, the recovery rate (1/γ is then the average infectious period). Note that S, I and R are proportions, not the number of individuals in the three categories.

When a pathogen invades the population everybody is susceptible (= 1). This proportion decreases as more and more individuals become infected (I increases). But the infected ones recover with rate γ and eventually no-one is infected and everybody has recovered (= 1).

From the second equation we can can infer one of the most important quantities in epidemiology. If this derivative (dI/dt) is less than zero the infection dies out. This happens if the initial fraction of susceptibles is less than γ/β. It is customary to use the inverse of this ratio to indicate whether an infection is going to spread in a population of suceptible individuals. This inverse ratio is called R0 - the basic reproductive ratio and is the average number of secondary cases arising from an average primary case in an entirely susceptible population. We see that if R> 1, the infection can spread and if R< 1 it can not. Human influenza has an estimated R0 around 3-5, measles around 16-18. It follows from this that (in a closed population) an infectious disease can only spread if there is an intial fraction of susceptibles greater than 1/R0.

Here is where vaccination comes in. The initial proportion of susceptibles can be reduced and to what extent that is necessary depends on R0 of the disease. The critical fraction, pc, of vaccinated individuals can be shown to be pc = 1 − 1/R0. Thus, the more infectious the disease (high R0), the larger the fraction of the population has to be vaccinated in order to stop the disease from spreading.

Where does it come from?

The simple SIR model (Susceptible, Infected, Recovered) was first formulated by W.O. Kermack and A.G. McKendrick in 1927. This was the first mathematical theory of the spread of diseases and they identified the critical threshold for a pandemic (an infection that is spread in a very large proportion of the population over large areas).


Despite its simplicity, the SIR model captures many important and useful aspects of the spread of infections. The model is easily expanded to include births and deaths in the population. In that case R0 = β/(γ + μ), where μ is the natural death rate in the population (not the mortality caused by the disease).

Many modifications have been made to the SIR model to better understand (and prevent) the spread of diseases, both in humans and in wild animal populations. The SIRS models allow the recovered individuals to be susceptible again, the SEIR model includes a class of individuals that are infected but not infectious (the "E"), and the MSEIR model accounts for immunity passed on from the mother to newborn, to mention a few of the many original SIR modifications. An important extension is to model the spread of the diseas in space (e.g., Finkenstädt & Grenfell 1998).

It is of utmost importance to be able to model the epidemiology of a disease in order to prevent pandemics, to design efficient vaccination programs and to understand the role of diseases in ecological system. The SIR model is the backbone of all such efforts.

Per Lundberg


Finkenstädt, B. & Grenfell, B. 1998. Empirical determinants of measels metapopulation dynamics in England and Wales. Proc. Roy. Soc. B. 265: 211-220.

Keeling, M. J. & Rohani, P. 2008. Modeling infectious diseases. Princeton UP.

Kermack, W. O. & McKendrick, A. G. 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A 115: 700-721.

Ostfeld, R. S., Keesing, F. & Eviner, V. T. (eds) 2008. Infectious disease ecology. Princeton UP.

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