On The Study of Covid-19 Transmission Using Deterministic and Stochastic Models with Vaccination Treatment and Quarantine

In this study, we propose deterministic and stochastic models of the spread of Covid-19 with vaccination and quarantine programs. The model considers the facts that vaccines do not provide full protection, the efficacy of current vaccines only lasts for a limited time, and recovered people could be reinfected. The routine analysis was carried out for the deterministic model, including calculating an expression for the basic reproduction number. The stochastic formulation makes use of a Continuous-Time Markov Chain (CTMC) model. The basic reproduction number from the deterministic model relates to the stochastic model's analysis in producing a formula for the probability of extinction of Covid-19. Furthermore, numerical simulations are carried out to analyze the sensitivity of the dynamical states and the basic reproduction number to the model parameters. An expression for the probability of disease extinction in terms of the model parameters and initial conditions is given. The results of this study suggest that current conditions in Indonesia will lead to a longterm Covid-19 epidemic. One of the efforts to overcome the Covid-19 epidemic is by increasing the provision of vaccines to the susceptible population. However, the number of vaccinated people in the population is not always an ideal control for dealing with the spread of the disease. The vaccine efficacy is also important to reduce the infection. As long as the efficacy is not sufficient to give a good protection to the human population and it lasts only for a short period of time, quarantine is still needed.