### This story is over 5 years old. # The Equations Behind Epidemics

Understanding the SIR model, which describes how an outbreak can catch fire from humble beginnings.
October 5, 2014, 12:30pm

Within epidemiology, the SIR model of is a way of predicting the spread of an infectious disease through a closed population using three broad, easily defined categories of individuals: susceptible (S), infectious (I), recovered (R). These three population segments are related by a series of relatively unmessy differential equations that demonstrate some striking things about how a disease can be spread so quickly from so small of a start.

The SIR model was first published in 1927 by W. O. Kermack and A. G. McKendrick, epidemiologists who were looking to explain the observation that epidemics can get very large with extremely humble beginnings. Epidemiologists of the time were at a loss, lacking a single causal factor sufficient to account for the frequent outbreaks tearing through society on a regular basis. Epidemics often just made no sense.

What the duo found in their equations were two fundamental principles: an epidemic may exhaust itself before the susceptible population reaches zero, and that an initial threshold value exists of susceptible members in a population below which an epidemic will not form. The SIR model was accurate enough that it persists today.

Its description below comes with help from course materials provided by the Mathematical Association of America. The equations are included not to induce pain so much as to illustrate the relative simplicity of the basic relationships.

#### S-I-R

In the SIR model, people are never added or subtracted from the total population. That is, dying is considered recovery here and no one is ever added to the susceptible group via things like immigration or births. As a member of a population, you either have, had (whether fully recovered, dead, or otherwise no longer infectious), or have not had the disease in question. Immune members can be said to occupy the recovered group.

So, we have three different equations showing the rates of change for each population segment. The rate of change in the susceptible group is modeled as such: The B coefficient in the above equation corresponds to average number of contacts an average member of population connects with per day. The next piece, the S, is the total number of susceptible members at a given time (as a fraction of the total population) and the last one, I, is what gives us the total number of infected members at a given time (as a bare number, not a fraction).

This tells us the rate of change in susceptible members of a population. If the present fraction of susceptible members is relatively smaller, we can expect a slower rate of change in the susceptible population, as the larger numbers of infected or recovered patients limits the number of people in the population that can become infected (because they can't get sick again and, thus, can't be infectious again).

As the relationship between the susceptible fraction and the infected population changes, the number of daily contacts then takes on less and less weight. At some critical point, however, the rate of change in the susceptible population is basically just the number of infected people times the average number of contacts per day with an ugly negative sign. That's how things get out of control—when there's no buffer of infected or recovered members to absorb some of those daily B contacts.

The "recovered" equation is easier: The γ here is the fraction of infected members expected to recover per day for a given illness. That recovery fraction just gets multiplied by the fraction of infected members, leaving those that are recovered, e.g. once but no longer sick. Again, this doesn't mean that these members are healthy and walking around, just that they're no longer infectious (and so don't factor into the first equation anymore). What γ actually is depends on the specific disease and its specific mortality rate.

So, when you look at the following graph, remember that r(t) might be 90 percent dead people. Again, these are the stakes in catching an outbreak early. The whole equation, with all three population segments accounted for, is this, which gives us the rate of change in the fraction of infected individuals. This one is basically the change in recovery rates subtracted by the change in the susceptible population. Eventually these two terms find some sort of equilibrium, and, as you can see from the graph, the rate of change in the infected population levels off.

The graph above is specifically for the spread of Hong Kong Flu through New York City in the 1960s. In that situation, the number of immune or recovered members of the population was as low as 10 at the start of the outbreak, leaving an enormous susceptible population, and an enormous potential for the disease to spread.