Alternative stable states and disease induced extinction

¹ Department of Mathematics & Philosophy, Western Illinois University, 1 University Circle, Macomb, IL 61455, USA
² The Preston M. Green Department of Electrical & Systems Engineering, Washington University in St. Louis, 1 Brookings Drive, St. Louis, MO 63130, USA
³ The Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 102G Crowley Hall, Notre Dame, IN 46556, USA

^* Corresponding author: db-ekanayake@wiu.edu

Received: 20 June 2023
Accepted: 24 June 2024

Abstract

In this research, we study a generalization of the SIR epidemic model to observe diseaseinduced extinction in the presence of alternative stable states. We model per capita reproduction and mortality rates as functions of both population density and external indicators reflecting temporal resource variations that impact stable states. We then obtain conditions to guarantee a unique global solution for the SIR model when the rate functions are discontinuous. We further obtain conditions for the stability of two states when the external indicators are assumed to be constant. We use both deterministic and stochastic epidemic simulations to analyze models with alternative stable states for real mammalian populations. Through numerical examples, we show that changes in external indicators can, in fact, lead to the collapse of a population subject to an epidemic. However, we find a high probability of species survival in the presence of environmental stochasticity, even when the corresponding deterministic models predict extinction of the host population.