Kontrol Optimum Vaksinasi dan Pengobatan pada Penyakit Menular Tipe SIR

Abstract

Nowadays, there are many infectious diseases that cause epidemic in the population. Therefore, an effective program is required to eliminate the infectious diseases, such as vaccination and treatment. The spread of infectious diseases can be modeled by SIR model, in which it enables us to predict the outbreak and the control rate. In this research, the SIR model is applied to analyze the population dynamic and to optimize the control of the outbreak with vaccination and treatment. As the result, the model provides two equilibrium points, i.e. dieasesfree and diseases-endemic equilibrium points. Moreover, the analysis show that if the reproduction number or ℛ0 is less than or equal 1, then the disease-free equilibrium point is globally asymptotically stable. While if ℛ0 is more than 1, then the disease-endemic equilibrium point is globally asymptotically stable. In this research the optimal control uses Pontryagin’s minimum principle with vaccination and treatment as control measures. The control objective is to find the optimal combination of vaccination and treatment that minimizes the cost of the two control measures as well as the number of infected individuals at the end of control period.