A Theory of How Disease Spreads
The mathematical theory of disease epidemics was first proposed in 1927 by
William Ogilvy Kermack, a Scottish biochemist, and Anderson Gray
McKendrick, a Scottish physician. It marked a revolution in medical thinking by
providing a realistic framework for understanding the dynamics of infectious
diseases.
Their simplest model divided the population into three compartments:
susceptible, infective, and recovered. It is therefore called an SIR model or
compartmental model. S is the percentage of the population who are susceptible,
people who have not had the disease and are vulnerable to getting it. I is the
percentage of the population who have caught the disease and are infective, who
are actively spreading it. R is the percentage of the population who are
recovered, who have had the disease and gotten over it, who have acquired
immunity, and who are no longer capable of catching the disease again or
spreading it. Nobody dies in this original model. The sum of the percentages is
100%, 100% = S + I + R, and the population is assumed constant.
According to the Kermack-McKendrick mathematical theory of disease
epidemics, in a thoroughly mixing constant population the rate of increase of
infectives in a disease epidemic is equal to a constant contagion parameter c
times the product of the fraction of the total population who are susceptible S
and the fraction infective I, minus a constant recovery rate r times the fraction of
infectives I. Each time a susceptible person meets an infective person, there is a
chance of infection. In a large population, the chance averages out to a certainty.
The number of such meetings per unit of time depends on the number of
susceptible-infective pairs in the population, hence the product SI.1 The three-
equation Kermack-McKendrick SIR model is:
There is no algebraic solution to this model, only approximations.2 Similar

equations also appear in chemistry, where they are called rate equations or
consecutive chemical reactions.3
In the model used in this book, the contagion rate is cS, the product of a
constant contagion parameter c and the time-varying fraction of susceptible
people S. The recovery rate is constant, r. If we divide both sides of the second
equation by the fraction of infective people I, we can see that the second
equation is nothing more than a statement that the growth rate of the fraction of
the population who are infectives is equal to the contagion rate cS minus the
recovery (or forgetting) rate r. This conclusion makes sense: if it is to grow, the
epidemic has to be spreading faster than people are recovering, and it is common
sense that the contagion rate should depend on the fraction of the population
susceptible to infection.
The first and third equations are very simple. The first equation says that the
number of susceptibles falls by one with every new infection, because a
susceptible turns into an infective. The third equation says that the number of
recovereds rises by one with every new recovery, because when a person
recovers from the illness (or in our context forgets a narrative) an infective turns
into a recovered. We will see below that this elementary model, which carries an
essential insight about the path of epidemics, can be modified to include a
growing population and many other factors specific to a particular epidemic.

FIGURE A.1. Theoretical Epidemic Paths
Solution to Kermack-McKendrick SIR model for I0 = .0001%, c = .5, r = .05. The heavy bars show the
percentage of the population who are sick and spreading the disease. The model assumes no medical
intervention; the epidemic ends on its own, even though there are still susceptibles in the population, and
not everyone was ever infected. Source: Author’s calculations.
Figure A.1 shows an example, implied by the above three equations, where
one person in a million is initially exposed, I0 = 0.0001%, and parameters c = .5,
r = .05. In this case, almost 100% of the population eventually gets infected.
During a disease epidemic, the public tends to focus on the infectives, the bell-
shaped curve in the figure. Attention also focuses on the number of newly
reported cases, the speed of transition from susceptible to infective, which
follows a similarly shaped bell-shaped curve if r is not too far below c. For
narratives, we compare plots of counts of words and publications to the infective
curve as in the figure.
The SIR model implies that from a small number of initial infectives, the
number of infectives follows much the same hump pattern, from epidemic to
epidemic, rising at first, then falling. A mutation in an old, much-reduced disease
may produce a single individual who is infective with the new strain. Then there
will be a lag, possibly a long lag if c is small, before the disease has infected
enough people to be noticed in public. The epidemic will then rise to a peak.