The SIR model of an epidemic

The SIR model is a three-compartment model of the time development of an epidemic. After normalizing the dependent variables, the model is a system of two non-linear differential equations for the susceptible proportion S and the infected proportion I. After normalizing the time variable there is only one remaining parameter. This largely expository article is mainly about aspects of this model that can be understood with calculus. It also discusses an alternative exactly solvable model that appeared in early work of Kermack and McKendrick. This model may be obtained by replacing SI factors by √ 2S − 1I factors. For a mild epidemic, where S is decreasing from 1 but remains fairly close to 1, this is a reasonable approximation. 1 The simplest model of an epidemic The SIR model is the simplest differential equation model that describes how an epidemic begins and ends. It depends on only two parameters: One governs the timing, the other determines everything else. It gives a glimpse into the world of more complicated epidemic models. The SIR model is standard in the literature of epidemiology [2, 3], and it even shows up in textbooks on calculus and differential equation. There is no exact formula for how the infection level depends on time. This account reviews what can be done in spite of this limitation. There is a modification of the SIR model for which there is an exact formula for the time dependence. This account also presents a way of understanding this modified model . 1 ar X iv :2 10 4. 12 02 9v 1 [ m at h. D S] 2 4 A pr 2 02 1 For a fixed population S is the proportion of susceptibles, I is the proportion of infected, and R is the proportion of removed. Removal is supposed to be permanent. The mechanism of removal is not specified. Some individuals might not survive; others might recover with full immunity. Every member of the population is supposed to belong to one of these three compartments, so S + I +R = 1. While S, I, and R change with time, the parameters that define the model do not change with time. This is a closed system, with no reintroduction from the outside. There are two parameters in the model. One is a rate b that governs how fast the susceptible-infected pairs are changing into infected-infected pairs. The other is a rate a that governs how fast infected individuals are being removed. The ratio r0 = b a (1) is the basic reproduction number that determines the scope of the epidemic. This represents the number of new infections per removal at the beginning of the epidemic. When r0 < 1 the epidemic immediately begins to die out. The more interesting situation is when r0 > 1. In this model there are no interventions; the rate constants b and a never change, and so also the reproductive number r0 is constant. The goal is to see how the course of the epidemic depends on the parameters. Here is an outline of what happens. During the course of the epidemic S decreases and R increases. What happens to I is governed by the effective reproduction number r0S. This represents the average number of new infections per removal. As the pool of susceptibles is depleted, this number decreases with time. As long as r0S is greater than one, the proportion I of infected increases. It reaches a peak when r0S = 1. Afterward r0S is less than one, and I decreases to zero. Meanwhile, R increases to a final value representing the proportion of the population eventually infected and removed. It is sometimes useful to take the fundamental parameters as r0 and a. In that case, the role of r0 is to describe the size of the epidemic, while the role of a is to describe how fast the epidemic takes place as a function of time. Figures 1, 2, and 3 show the behavior of the SIR model for r0 values 2, 3, and 6. The following discussion will clarify what these pictures mean. Even when r0 is 2, the final value of R is 80 percent. Also, the r0 equal to 6 picture shows that the infection proportion curve I is not symmetric about its peak. This is because at this level of r0 infection is fast and removal is