Memory Effects in Disease Modelling Through Kernel Estimates with Oscillatory Time History

TL;DR

This paper introduces a linear chain trick algorithm to model memory effects in disease dynamics using oscillatory time histories, revealing how adaptive behavior influences stability and reduces attack rates in epidemic models.

Contribution

It develops a Markovian reformulation of disease models with memory effects via kernel estimates, enabling stability analysis and insights into adaptive behavior impacts.

Findings

Adaptive behavior can stabilize disease dynamics or induce limit cycles.

Memory effects lower the epidemic attack rate.

The model's dampening effect diminishes as the epidemic subsides.

Abstract

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR - model and on a susceptible-exposed-infected-recovered SEIR - model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e.\ locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.