Four versions of an SIRS model with demography are described in a paper with the title "The influence of immunity loss on persistence and recurrence of endemic infections", by Ingemar Nåsell, written in 2012. We give here Maple worksheets for each of the four models that contain derivations of the stationary distributions of their diffusion approximations. All of the models are Markov population processes, and all of them are closely related. One of the models is basic, since the other models can be established by successive approximations of the basic model. Three of the models are trivariate with the state space {S,I,R}, where S stands for the number of susceptible individuals, I for the number of infected and infective individuals, and R for the number or recovered and immune individuals, while one model is bivariate with the state space {S,I}. The basic model is a trivariate model with proper infection rate. This means that the denominator of the infection rate equals S+I+R-1. This model is denoted sirs3pr. The first approximation of the basic model is a trivariate model with a modified proper infection rate, with its denominator equal to S+I+R. This model is called sirs3prm. The second approximation of the basic model is found from the first approximation by replacing the stochastically varying denominator S+I+R in the denominator of the infection rate by the constant N. The resulting infection rate is called conventional, and the resulting model is denoted sirs3c. The third and last approximation of the basic model is found from the second approximation by replacing the state variable R by R=N-S-I. The resulting model is therefore bivariate, with conventional infection rate. It is referred to as sirs2c.
Maple worksheets that show the derivation of the stationary distribution of the diffusion approximation of each of the four SIRS models are found below:
A Maple worksheet for the basic trivariate model with proper infection rate.
A Maple worksheet for the trivariate model with modified proper infection rate.
A Maple worksheet for the trivariate model with conventional infection rate.
A Maple worksheet for the bivariate model with conventional infection rate.