Optimal Control Strategies for Epidemic Dynamics: Integrating SIR-SI and Lotka--Volterra Models

TL;DR

This paper develops a mathematical framework combining epidemiological and ecological models to optimize predator-based control strategies for vector-borne diseases, demonstrating how natural predators and interventions can reduce infection spread.

Contribution

It introduces the ecological reproduction number and formulates an optimal control problem for predator release to manage disease transmission effectively.

Findings

Natural predator control depends on predator-prey density ratios.

Optimal predator release reduces epidemic peaks.

Interventions stabilize ecological and epidemiological dynamics.

Abstract

In this work we present a mathematical model that integrates the epidemiological dynamics of a vector-borne disease (SIR-SI) with Lotka Volterra predator prey ecological interactions. The study analyzes how the presence of natural predators acts as a biological control mechanism to regulate the vector population and, consequently, disease transmission in host. We introduce the concept of the ecological reproduction number, a threshold that links the amplitude of predator prey cycles to disease persistence, showing that natural control depends critically on the ratio between the maximum vector density and the minimum predator density. In scenarios where natural control is insufficient, we formulate an optimal control problem based on the release of predators. Using the Pontryagin Maximum Principle, we characterize the optimal strategy that minimizes the cumulative number of infected individuals and intervention costs, while simultaneously maximizing the susceptible host population at the end of the time horizon. Numerical simulations validate the effectiveness of the model, showing that external intervention mitigates the epidemic peak and stabilizes the system against the natural oscillations of biological populations.