The SIR model of an epidemic

TL;DR

This paper explains the SIR epidemic model using calculus, discusses its properties, and introduces an alternative solvable model that approximates mild epidemics.

Contribution

It provides an expository analysis of the SIR model and presents an alternative exactly solvable model for mild epidemics.

Findings

The SIR model reduces to two non-linear differential equations.

An alternative model with a different interaction term is introduced.

The alternative model approximates mild epidemic dynamics effectively.

Abstract

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 $\sqrt{2S-1}I$ factors. For a mild epidemic, where $S$ is decreasing from 1 but remains fairly close to 1, this is a reasonable approximation.