Model Matematika pada Penyebaran Penyakit Covid-19 dengan Pengaruh Vaksinasi di DKI Jakarta

Abstract

Coronavirus Disease 2019 or Covid-19 is a disease caused by a coronavirus that attacks the respiratory tract causing high fever, cough, flu, shortness of breath, and sore throat. To see the spread of Covid-19 with the effect of vaccination in DKI Jakarta, this study developed the SIR epidemic model into SVIR by adding a population of vaccinated individuals to prevent the spread of Covid-19. This model assumes that individuals are given the vaccine until the second dose. Individuals vaccinated for two doses can still be infected with Covid-19 if they interact with individuals infected with Covid-19. The population is divided into four classes: the vulnerable individual class, the vaccinated individual class, the Covid-19 infected individual class, and the recovered individual class. Construction of the models starts by making a flow chart of the spread of Covid-19 with the effect of vaccination. This model obtains two equilibrium points, namely the disease-free equilibrium  point and the endemic equilibrium point . Analysis of the system's stability around the equilibrium point gives the primary reproduction number . From the analysis results, the system around the disease-free equilibrium point is  locally asymptotically stable when . Then a numerical simulation is carried out to provide a geometric picture related to the results that have been analyzed. The simulation results show that when conditions  occur, the disease will disappear, and under conditions , the disease will become epidemic. To prevent the spread of Covid-19, efforts can be made to reduce direct contact with infected individuals, implement health protocols and increase the proportion of vaccinated individuals.