## The SIR model of epidemics

The mathematical study of disease was put on firm footing in the early 1930s in a series of papers by Kermack and McKendrick (1927, 1932, 1933); a discussion of these papers and their intellectual history, c. 1990, is found in R. M. Anderson (1991). When Kermack and McKendrick did their work, computing was difficult, so that good thinking (analytic ability, finding closed forms of solutions and their approximations) was even more important than now (of course, one might argue that since these days it is so easy to blindly solve a set of equations on the computer, it is even more important now to be able to think about them carefully).

We consider a closed population in which individuals are either susceptible to disease (S), infected (I) or recovered or removed by death (R). Since the population is closed, at any time t we have S(t) + I(t) + R(t) = N. If we assume mass action transmission of the disease and that removal occurs at rate v, the dynamics of the disease become d t dI

and in general, the initial conditions would be S(0) = S0,I(0) = I0 and R(0) = N — S0 —10 (since the population may already contain individuals who have experienced and recovered from the disease).

Let us begin with the special case of S(0) = N — 1 and I(0) = 1. As in the model of hepatitis, we can ask the following question: how many new cases of the disease are caused directly by this one infected individual entering a population in which everyone else is susceptible. We proceed in very much the same way as we did with hepatitis. If we set I = 1 in the first line of Eq. (5.6), the solution is S(t) = (N — 1)exp(—bt). The one infected individual is infectious for a period of time approximately equal to 1/v, at which time the number of susceptibles is (N — 1)exp(—b/v). The number of new cases caused by this one infected individual is then N — 1 — [(N — 1)exp(—b/v)] = (N — 1)(1 — exp(—b/v)) and if we Taylor expand the exponential, keeping only the linear term, and assume that the population is large so that N — 1« N we conclude that R0 « bN/v, just as with the model for hepatitis C.

Now let us think about Eq. (5.6) in general. The only steady state for the number of infected individuals is I = 0, but there are two choices for the steady states of S: either S = 0 (in which case an epidemic has run through the entire population) or S = v/b (in which case an epidemic has run its course, but not every individual became sick). We would like to know which is which, and how we determine that. The phase plane for Eq. (5.6) is shown in Figure 5.2, and it is an exceptionally simple phase plane. Indeed, from this phase plane we conclude the following remarkable fact: if S(0) > v/b then there will be a wave of epidemic in the population in the sense that I(t) will first increase and then decrease. Note that this condition, S(0) > v/b, is the same as the condition that

Figure 5.2. The phase plane for the SIR model. This is an exceptionally simple phase plane: since dS/dt is always negative, points in the phase plane can move only to the left. If S(0) > v/b, then I(t) will increase, until the line S = v/b is crossed. If S(0) < v/b, then I(t) only declines.

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