An epidemic model for cholera with optimal control treatment

TL;DR

This paper develops a mathematical model for cholera transmission incorporating quarantine treatment, analyzes its stability and equilibria, and determines optimal quarantine strategies to minimize infections, supported by a real outbreak simulation.

Contribution

It introduces a novel cholera model with quarantine control, proves its mathematical properties, and derives optimal quarantine strategies based on real outbreak data.

Findings

Model is epidemiologically well-posed and solutions are positive and bounded.

Existence and stability of disease-free and endemic equilibria are established.

Optimal quarantine strategies effectively reduce infections and bacteria concentration.

Abstract

We propose a mathematical model for cholera with treatment through quarantine. The model is shown to be both epidemiologically and mathematically well posed. In particular, we prove that all solutions of the model are positive and bounded; and that every solution with initial conditions in a certain meaningful set remains in that set for all time. The existence of unique disease-free and endemic equilibrium points is proved and the basic reproduction number is computed. Then, we study the local asymptotic stability of these equilibrium points. An optimal control problem is proposed and analyzed, whose goal is to obtain a successful treatment through quarantine. We provide the optimal quarantine strategy for the minimization of the number of infectious individuals and bacteria concentration, as well as the costs associated with the quarantine. Finally, a numerical simulation of the cholera outbreak in the Department of Artibonite (Haiti), in 2010, is carried out, illustrating the usefulness of the model and its analysis.