Two critical times for the SIR model

TL;DR

This paper analyzes key transition times in the SIR epidemic model, deriving integral formulas and PDE characterizations to estimate when infections peak and decline in large populations.

Contribution

It introduces a PDE-based framework to precisely estimate critical times in the SIR model, enhancing understanding of epidemic dynamics.

Findings

Derived integral representations for critical times.

Characterized transition times as solutions to PDEs.

Provided estimates for large population scenarios.

Abstract

We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.