An analytic approximate solution of the SIR model

TL;DR

This paper introduces explicit approximate solutions to the complex SIR model equations using generalized logistic functions, providing practical, accurate, and insightful tools applicable across various epidemic scenarios.

Contribution

The paper presents a novel method to derive explicit approximate solutions to the SIR model using simple geometric considerations, improving practicality and understanding.

Findings

Approximate solutions match asymptotic values accurately.

Method is robust across different epidemic parameters.

Solutions reveal features similar to numerical solutions of exact equations.

Abstract

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure {\em exact} (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the {\em exact} equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work's method and result prove to be robust over a range of values of the well known non-dimensional parameter called {\em basic reproduction ratio}, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn't appear much discussed in analogous works.