Exact and approximate analytic solutions in the SIR epidemic model

TL;DR

This paper derives new exact and approximate analytical solutions for the SIR epidemic model, relating susceptible and infective populations explicitly and modeling epidemic curves accurately for various R0 values.

Contribution

It introduces novel explicit solutions for the SIR model using Lambert W functions and simple functions based on the R0 limit, enhancing analytical understanding.

Findings

Explicit relation between S and I using Lambert W function

Accurate modeling of epidemic curves for any R0 value

Effect of initial ratio I0/S0 is a simple time shift

Abstract

In this work, some new exact and approximate analytical solutions are obtained for the SIR epidemic model, which is formulated in terms of dimensionless variables and parameters. The susceptibles population (S) is in this way explicitly related to the infectives population (I) using the Lambert W function (both the principal and the secondary branches). A simple and accurate relation for the fraction of the population that does not catch the disease is also obtained. The explicit time dependences of the susceptibles, infectives and removed populations, as well as that of the epidemic curve are also modelled with good accuracy for any value of R0 (basic multiplication number) using simple functions that are modified solutions of the R0 -> infinity limiting case (logistic curve). It is also shown that for I0 << S0 the effect of a change in the ratio I0/S0 on the population evolution curves amounts to a time shift, their shape and relative position being unaffected.