Recurrent patterns of disease spread post the acute phase of a pandemic: Insights from a coupled system of a differential equation for disease transmission and a delayed algebraic equation for behavioral adaptation

Abstract

We introduce a coupled system of a disease transmission differential equation and a behavioral adaptation algebraic renewal equation to understand the mechanisms of nonlinear oscillations post-acute phase of a pandemic. This extends the Zhang–Scarabel–Murty–Wu model, which was formulated and analyzed to describe multi-wave patterns observed at the early stage during the acute phase of the COVID-19 pandemic. Our extension involves the depletion of susceptible population due to infection and contains a nonlinear disease transmission term to reflect the recovery and temporal immunity in the infected population past the acute phase of the pandemic. Examining whether and how incorporating this depletion of susceptible population impacts interwoven disease transmission dynamics and behavioral adaptation is the objective of our current research. We introduce some prototypical risk aversion functions to characterize behavioral responses to perceived risks and show how the risk aversion behaviors and the logistic delay in implementation of behavioral adaptation combined contribute to a dynamic equilibrium state described by a periodic oscillatory wave. We also link the period between two consecutive peaks to basic epidemic parameters, the community flexibility to behavioral change, and the population’s tolerance to perceived risks.