Study of Stochastic and Fractional Order Deterministic Type Epidemic Model

Viral diseases have been remained a great threat for human on this globe from ancient time. These viruses enter to human body from various sources and causes infectious diseases. For instance, it is possible for viruses to enter ground and surface waterways from septic tanks, sewage sludges, wastewater, and other sources. Microorganisms known as viruses are made of RNA or DNA and encased in a capsid protein. This protein coating has a major role in determining the fate and mode of movement of viruses in the environment. Researchers have conducted various studies which indicate that due to their relative small size, they were considered to offer the most potential for transportation. There is proof that some viruses can travel over saturated zones and vadose zones over signicant distances. Recently, the COVID-19 disease has been transmitted all over the globe from person to person due to local and international travel of people from one place to other. There are other sources also were responsible in transmitting the said disease. Researchers have studied the mentioned disease through mathematical models very well. Also, the tools of fractional calculus were used to investigate various models of COVID-19 from different perspectives. In this paper, a mathematical model of COVID-19 disease is considered with the vaccination class. For the desired study, we use the Caputo-Fabrizio derivative (CFD) and stochastic differential equations (SDEs). A compartmental nonlinear model for COVID-19 with different classes uninfected, vaccinated and infected has been considered under CFD. Exis-tence theory is established by using fixed point theory. Also, the fundamental results related to the model under consideration such as feasible region, boundedness, and trivial and non-trivial equilibrium points are deduced. Basic reproduction number is computed and local and global stability analysis are derived. In addition, the numerical solution to the model is given via using Adam Bashforth numerical method. Graphical presentations are also given for various fractional orders values to show that CFD is also a realistic operator which can be used to analyze various infectious disease dynamical system. Additionally, various graphical presentations corresponding to stochastic type derivative have also given. In addition, comparison with real data has presented also.