Analysis of the dynamics and optimal control of cutaneous Leishmania during human immigration

Leishmania is an infectious disease that is difficult to control and has an impact on morbidity and mortality around the world. This study investi-gates the dynamics of cutaneous Leishmania and optimal control measures, particularly in regards to human immigration. Applying a mathematical model to evaluate the dynamics of human immigration and sand flies pop-ulation. The human population is classified into four compartments: sus-ceptible, exposed, infectious, and recovered. The sand fly population is divided into three categories: susceptible, exposed, and infectious. The mathematical analysis involves positivity, existence and the uniqueness of the solution. We analyzed the global stability of the system around the endemic equilibrium point by contracting the Lyapunov function. Optimal control measures are used to reduce the number of infected and exposed individuals among humans, sand flies, and migrants. These techniques are described using Pontryagin’s Maximum Principle to derive necessary conditions for optimal control. The numerical simulations confirm the the-oretical results by showing that following these controls effectively reduces the spread of the disease, and immigration has a major impact on the spread of human-borne Leishmania.