Epidemics: towards understanding undulation and decay

TL;DR

This paper introduces a mathematical model that explains infection waves and decay in epidemics, based on a reinterpretation of the SIR model with delay differential equations, capturing periodic outbreaks and potential limit cycles.

Contribution

It presents a novel delay differential equation model derived from the SIR framework that naturally exhibits epidemic undulation and decay behaviors.

Findings

Model exhibits periodic solutions and limit cycles.

Linearization supports heuristic and numerical analysis.

Captures epidemic waves and decay dynamics.

Abstract

Undulation of infection levels, usually called waves, are not well understood. In this paper we propose a mathematical model that exhibits undulation and decay towards a stable state. The model is a re-interpretation of the original SIR-model obtained by postulating different constitutive relations whereby classical logistic growth with recovery is obtained. The recovery relation is based on the premise that infectiousness only lasts for some time. This leads to a differential-difference (delay) equation which intrinsically exhibits periodicity in its solutions but not necessarily decay to asymptotically stable equilibrium. Limit cycles can indeed occur. An appropriate linearization of the governing equation provides a firm basis for heuristic reasoning as well as confidence in numerical calculations.