Lyapunov Stability Analysis of Covid-19 SIRV Model

Until now, it is not known when the COVID-19 pandemic in Indonesia will end. As COVID-19 cases continue to increase, predicting the number of cases infected with COVID-19 is very important to design a control strategy to reduce the disease spread. Towards the end of 2020, several manufacturers announced high efficacy rates of COVID-19 vaccine candidates. Vaccines have been believed to be an important tool for improving the health of the population so that the disease spread can be controlled without hindering economic growth. A Mathematical model of infectious diseases is an important tool that has been focused on predicting the dynamics of the disease spread. It can be used to predict the future situation of a potential outbreak and evaluate the best strategy to reduce the spread of the outbreak. There are many types of mathematical models to predict the behavior of an infectious disease that is transmitted from human to human. One of the commonly used is called the compartment model. In this paper, we use a modified SIR model with vaccination to predict the behavior of the disease spread after vaccination. Theoretically, a successful vaccination program should slow down the rate of the virus spread. The modified SIR model with vaccination is adopted to predict the spread of coronavirus. Here, we proof that the model has a unique equilibrium point that is globally asymptotically stable by using Lyapunov function if the vaccination rate is greater than zero. Otherwise, if there is no vaccination is done, the equilibrium points only stable if reproduction number of infection is less than one. Further, the model will be implemented to Indonesia data to predict the behavior of the spread of the disease after the vaccination program.