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

TL;DR

This paper addresses epidemic control for vector-borne diseases by integrating SIR–SI host–vector dynamics with Lotka–Volterra predator–prey interactions. It introduces an ecological reproduction number $R_0(k_0)$ that links predator–prey oscillation amplitude to disease persistence and develops an optimal-control framework where captive predator releases, governed by $u(t)$, minimize infections and intervention costs while maximizing final susceptibility $S_h(T)$; the Pontryagin Maximum Principle yields explicit structures for the optimal control. Theoretical results establish existence and characterize the optimality conditions, and numerical simulations demonstrate that predator releases can sharply reduce the epidemic peak and stabilize dynamics, with horizon length and cost structure shaping the intervention. Overall, the work provides a mechanistic, ecologically grounded approach to vector control that integrates biological and economic considerations for managing vector-borne disease outbreaks.

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.

Figures (3)

• Figure 1: Dynamics of the controlled and uncontrolled system for $c=1$ and a short time horizon ($T=30$). The optimal release of predators substantially mitigates the epidemic by reducing the initial peak and the cumulative number of infected humans. The control acts mainly during the early phase of the outbreak, weakening transmission through a rapid reduction of the vector population. However, due to the limited control horizon, the underlying vector--predator oscillations are not fully suppressed, which may allow for a resurgence of the epidemic outside the controlled time window.
• Figure 2: Dynamics of the controlled and uncontrolled system for $c=1$ and a long time horizon ($T=120$). Using the same parameters and cost weights as in Figure 1, the optimal control maintains the number of infected humans at low levels throughout the entire epidemic period. The extended intervention prevents secondary growth of the vector population during the epidemic phase. After the epidemic season, seasonal effects naturally drive the mosquito population to extinction, so that no further control is required.
• Figure 3: Controlled and uncontrolled dynamics for $c=0$, showing a bang--bang optimal control, in agreement with the theoretical predictions of the Pontryagin Maximum Principle (\ref{['eq: u*_c_igual_0']}). The final time is $T=120$ days.

Theorems & Definitions (13)

• Proposition 3.1
• proof
• Remark 3.2: Simplifying the equations
• Remark 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Remark 4.1
• Proposition 4.2