Construction of a stochastic SVIS model for prevention of influenza spread with vaccines

A lot of researches related to influenza vaccination have been done in epidemiology. However, additional researches in mathematics are also needed to understand the potential biases in estimating the benefits of vaccination, since mathematical analysis and modelling is central to infectious disease epidemiology. Here, we construct a system of stochastic differential equations model for preventing the spread of influenza with vaccines. Three subpopulations are considered as compartments of the model, they are susceptible, vaccinated, and infected individuals. The model assumes there are demographic variabilities in the number of births, deaths and transitions rates for each compartment. The model is constructed based on the continuous time Markov chain model by studying changes in the components over small time interval. We perform some numerical simulations to give an interpretation and understanding about the model. For a better simulation result, we estimate values of the epidemiological parameters of the model based on weekly influenza surveillance and influenza vaccine distribution report in the United States.