Oscillating behavior of a compartmental model with retarded noisy dynamic infection rate

TL;DR

This paper extends the SIRS epidemiological model by incorporating delayed feedback control and environmental noise, revealing oscillatory and irregular outbreak patterns through analytical and numerical methods.

Contribution

It introduces a delay-based feedback mechanism into the SIRS model and analyzes its impact on disease dynamics, including oscillations and stochastic outbreak patterns.

Findings

Delayed feedback causes endemic equilibrium to become unstable.

Nonlinear solutions exhibit persistent oscillations in infection levels.

Environmental noise leads to irregular infection waves with diverse frequencies.

Abstract

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected(I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) into an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.