A Mathematical Model to Control the Prevalence of a Directly and Indirectly Transmitted Disease

Begoña Cantó,Maria Jesús Pagán,Elena Sánchez,Carmen Coll,Joan Poveda

doi:10.3390/math9202562

Abstract

In this paper, a mathematical model to describe the spread of an infectious disease on a farm is developed. To analyze the evolution of the infection, the direct transmission from infected individuals and the indirect transmission from the bacteria accumulated in the enclosure are considered. A threshold value of population is obtained to assure the extinction of the disease. When this size of population is exceeded, two control procedures to apply at each time are proposed. For each of them, a maximum number of steps without control and reducing the prevalence of disease is obtained. In addition, a criterion to choose between both procedures is established. Finally, the results are numerically simulated for a hypothetical outbreak on a farm.

Highlights

Mathematical models are frequently used to analyze the behavior of an infectious disease on a group of individuals
The infectious agent that causes the disease can be transferred between an infected and a susceptible individual, from a reservoir where susceptible individuals are exposed to the pathogen, or even through both types of transmission
We suggest a direct and indirect transmission model where the pathogen cannot be reproduced in the environment

Summary

Introduction

Mathematical models are frequently used to analyze the behavior of an infectious disease on a group of individuals. The case in which the transmission is direct and indirect has not been studied This contrasts markedly the behavior of most diseases such as salmonella, typhoid fever or COVID-19. Unlike other papers that appear in the literature on models of direct or/and indirect disease transmission, the novelty of this work consists of the method followed to control the disease. The main result is obtaining a threshold for each of these strategies This provides us with the optimal time interval in which we can be without acting on the process and with the assurance that the disease tends to disappear.

Mathematical Model and Problem Statement

Periodic Control Leading the Infection to Extinction

Control Model Simulation

Conclusions