Transition Probability Density Function for Number of Infections in a Population Satisfying a Stochastic SIS-Epidemic Model

By assuming a certain population is randomly fluctuating and subject to a continuous spectrum of disturbances at a given time, we derive and analyze the time-dependent probability density function (PDF) for the number of susceptible and infected individuals at a given time in a stochastic Susceptible-Infective-Susceptible (SIS) epidemic model with vital dynamics. The PDF is obtained for the case where fluctuation is present in the transmission and recovery rates of the disease. With the PDF, the closed-form expression for the mean number of infections at each given time is obtained. The effect of noise, together with the effect of changes in epidemiological parameters on the distribution are investigated. Properties of the distribution, namely, the time-dependent mean, variance, skewness, and kurtosis are obtained and analyzed. The correctness of the work done is verified and validated using population and published parameters.