Some properties of an extended simple stochastic epidemic model involving two additional parameters

In a fixed population of size N, we assume that the probability for a new infective in the time interval (t, t + dt) is βr^1-b(N - r)^a dt, where r is the number of susceptibles at time t, N - r is the number of infectives at time t, β is the infection rate, a is the infection power, and b is the safety-in-numbers power. If a = 1 and b = 0, then this probability would determine the familiar simple stochastic epidemic model. This article investigates the effect of the parameters a and b on the state probabilities and other derived quantities of the model.