A look at endemic equilibria of compartmental epidemiological models and model control via vaccination and mitigation

TL;DR

This paper introduces two new epidemiological models, analyzes their endemic equilibria, and explores control strategies like vaccination and mitigation, providing theoretical insights and simulations relevant to disease management.

Contribution

The paper proposes two novel compartmental models, derives an endemic threshold theorem, and demonstrates that one model is static feedback linearizable for control purposes.

Findings

Endemic threshold theorem established for the first model

Model control system is static feedback linearizable

Simulations illustrate control strategies effectiveness

Abstract

Compartmental models have long served as important tools in mathematical epidemiology, with their usefulness highlighted by the recent COVID-19 pandemic. However, most of the classical models fail to account for certain features of this disease and others like it, such as the ability of exposed individuals to recover without becoming infectious, or the possibility that asymptomatic individuals can indeed transmit the disease but at a lesser rate than the symptomatic. In the first part of this paper we propose two new compartmental epidemiological models and study their equilibria, obtaining an endemic threshold theorem for the first model. In the second part of the paper, we treat the second model as an affine control system with two controls: vaccination and mitigation. We show that this system is static feedback linearizable, present some simulations, and investigate of an optimal control version of the problem. We conclude with some open problems and ideas for future research.