Stability analysis of SIRS epidemic model on measles disease spreading with vaccination and migration

Measles or also called Rubeola 9 days measles is an infectious disease caused by the Paramyxovirus virus and is transmitted through coughing and sneezing characterized by fever, cough, runny nose, and reddish rashes on the skin. Measles is included in the endemic category in developing countries, one of which is Indonesia. In this paper, we analyze the stability of the SIRS epidemic model with vaccination and migration and simulate the model based on data obtained from the Indonesian Ministry of Health in 2015 and migration data from the SUPAS (Survei Penduduk Antar Sensus) in 2015. Based on the results of the analysis that has been done, we obtained two equilibrium points, namely disease-free equilibrium point and endemic equilibrium point. If the value of R₀ <1, then the disease-free equilibrium point is said to be locally asymptotically stable, and if the value of R₀ > 1, then the endemic equilibrium point is locally asymptotically stable. In the case of the spread of measles based on data in East Java Province in 2015, a basic reproduction number is obtained R₀ = 4,48609.10⁻⁵ <1 which means that measles is no longer epidemic in a certain time provided that vaccination must be carried out continously and evenly.