Mathematical Model of Preventing The Spread of Dengue Disease in Yogyakarta Using Wolbachia Mosquitoes

Abstract

The purpose of this study is to explain the mathematical model of preventing the spread of dengue fever in the Special Region of Yogyakarta by involving Wolbachia mosquitoes, explain the equilibrium point and stability analysis and conduct numerical simulations. The model used in this study is SIS-SI model, using population data of Yogyakarta Special Region Province in 2023. The steps involved include forming the SIS-SI mathematical model, determining the disease-free and endemic equilibrium points, calculating the basic reproduction number, analyzing the stability of the equilibrium point, and conducting numerical simulation. The results showed that the disease-free equilibrium point is asymptotically stable when . The endemic equilibrium point is asymptotically stable when . Research has shown that  is related to the inhibitory effect of Wolbachia bacteria ( ). When , , the threshold for disease spread. If , the disease will not spread in the population, conversly if , the disease will spread. Based on the numerical simulations conducted, it was observed that the smaller the inhibitory effect of Wolbachia bacteria on dengue replication in mosquitoes, the more effective Wolbachia bacteria are in suppressing the spread of dengue virus. As a result, this leads to the disappearance of dengue disease from the population.