Analisis Kestabilan dan Kontrol Optimal Model Matematika Penyebaran Leptospirosis dengan Saturated Incidence Rate

Abstract

Leptospirosis is a disease caused by the bacteria Leptospira inchterohemorrhagiaea. Leptospirosis can attack humans and other animals, through rodents, especially rats. This research aims to analyze the stability of the equilibrium point in the mathematical model of the spread of Leptospirosis and apply optimal control variables in the form of prevention and treatment efforts. Based on the results of the mathematical model analysis of the spread of Leptospirosis, two equilibrium points were obtained, there are the non-endemic equilibrium point and the endemic equilibrium point. Local stability and the existence of an equilibrium point depend on the basic reproduction number 𝑅0. The non-endemic equilibrium point is local asymptotically stable if 𝑅0 < 1, while the endemic equilibrium point tends to be asymptotically stable if 𝑅0 > 1. Next, the problem of control variables in the model is determined using Pontryagin's Maximum Principle. Numerical simulation results show that providing control in the form of prevention efforts and treatment efforts simultaneously provides effective results in minimizing the population of individuals exposed to and infected by Leptospirosis at the cost of providing optimal control.