Abstract
Measles is an acute highly contagious disease caused by Paramyxovirus. Measles is considered as a dangerous disease because it cause complications, brain and other organs damage, lifelong disability, paralysis and even death. In the previous studies, it was known that the spread of measles highly dependent on number of infected individuals so it is necessary to control measles through treatment. In this paper, we study about the application of the optimal control theory on the system of differential equations of the SIR endemic model. Determination of the optimal control is obtained through the application of the Pontryagin minimum principle. The main target in this paper is to find a unique optimal control where the optimal control can be described as an efficiency rate of treatment in individuals infected with measles to decrease the number of infected individuals.
Highlights
Perubahan lingkungan hidup dan perkembangan ilmu pengetahuan serta teknologi dapat mempengaruhi berbagai macam penyakit yang dapat menimbulkan endemi
Measles is an acute highly contagious disease caused by Paramyxovirus
Measles is considered as a dangerous disease because it cause complications, brain and other organs damage, lifelong disability, paralysis and even death
Summary
Perubahan lingkungan hidup dan perkembangan ilmu pengetahuan serta teknologi dapat mempengaruhi berbagai macam penyakit yang dapat menimbulkan endemi. Salah satu penyakit yang menyebabkan endemi dan membahayakan manusia adalah penyakit campak (Measles). Kejadian penularan wabah penyakit campak yang terjadi pada suatu populasi dapat dimodelkan ke dalam model matematika salah satunya adalah model SIR (Susceptible, Infected,Recovered). Pada penelitian (Kurniawan,dkk,2012) telah dibahas analisis kestabilan penyebaran penyakit campak (measles) menggunakan model endemi SIR untuk mencari titik kesetimbangan model yaitu titik kesetimbangan bebas penyakit dan titik kesetimbangan endemi. Model endemi SIR menjelaskan pengobatan penyakit campak dapat direpresentasikan sebagai sistem persamaan diferensial biasa nonlinear dengan tiga variabel tak bebas yang meliputi Susceptible (S), Infected (I) dan Recovered (R) (Kurniawan,dkk,2012). Penelitian ini akan mengaplikasikan teori kontrol optimal pada sistem persamaan diferensial dari model matematika SIR yang terbentuk, untuk menentukan bentuk kontrol optimal yang tepat yang dapat digunakan untuk menekan laju pertambahan individu yang terinfeksi penyakit campak
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