Analisis Kestabilan Model Matematika Penyebaran Penyakit Campak dengan Dua Vaksinasi dan Karantina

Abstract

Measles is a contagious disease that can spread rapidly and has the potential to cause outbreaks if not properly controlled. One control measure that can be implemented is through vaccination programs and the quarantine of infected individuals. This study aims to develop an 𝑆𝑉1𝑉2𝐸𝐼𝑄𝑅 mathematical model to describe the spread of measles with two-dose vaccination and quarantine in Jember Regency. The model consists of seven compartments: Susceptible (S), first-dose vaccinated (𝑉1), second-dose vaccinated (𝑉2), exposed (E), infected (I), quarantined (Q), and recovered (R). Model analysis was conducted by determining the disease-free equilibrium point, performing stability analysis using the Jacobian matrix and Routh-Hurwitz criteria, and determining the basic reproduction number using the Next Generation Matrix (NGM) method. Numerical simulations were performed using the fourth-order Runge-Kutta method with the aid of python. The results of the study indicate that the model has a disease-free equilibrium point. The basic reproduction number obtained is 𝑅0 ≈ 0,00000026 < 1, so the disease-free equilibrium point is locally asymptotically stable. The results of the numerical simulations show that the infected population gradually decreases, while the recovered population increases over time. This indicates that vaccination and quarantine play a role in curbing the spread of measles in Jember Regency.