Given initial parameters $T \ge 0 \in \mathbb{R}$, $S, I, R \ge 0 \in \mathbb{N} $, $ N = S+I+R$ the constants $\beta, \gamma$ and a uniform distribution $\mathcal{U}_{[0,1] in \mathbb{R}}$. I wish to evaluate the number of iterations of given series:
$$ T_n = T_{n-1} - \frac{\ln(\mathcal{U})}{\frac{\beta}{N}SI + \gamma I }$$
In this model $S, I, R$ are three variables of a system of differential equations given by:
\begin{equation} Pr\{\Delta S(t) = i, \Delta I(t) = j | (S(t), I(t))\} = \left\{ \begin{array}{cc} \dfrac{\beta}{N}S(t)I(t)\Delta t + o(\Delta t), & (i, j) = (-1, 1)\\ \gamma I(t)\Delta t + o(\Delta (t), & (i, j) = (0, -1)\\ 1 - \left[\dfrac{\beta}{N}S(t)I(t) + \gamma I(t)\right] \Delta t, & (i, j) = (0, 0)\\ o(\Delta t), & otherwise \end{array} \right. \nonumber \end{equation}
I can pick any initial values for $S, I$ and $R$, but to make it simpler pick just a initial $I_0$ and $N$ value and find $S$ from it, $R$ is always initialized as 0. This gives only 2 free variables to handle.
An algorithmic approach can be written as:
$t, S, I, R \gets 0, N - I_0, I_0, 0$
WHILE $t < T$ and $I > 0$
$t \gets t - \frac{\ln{\left(\mathcal{U}_{[0,1] \in \mathbb{R}}\right)}}{\frac{\beta}{N} S I + \gamma I}$
IF
$Prob \left( X \le \frac{\frac{\beta S}{N}}{\left(\frac{\beta S}{N} + \gamma\right)}\right)$, $X \in \mathcal{U}$
THEN
$S, I, R \gets S - 1, I + 1, R$
ELSE
$S, I, R \gets S , I - 1, R + 1$
ENDWHILE
There is two stochastic events, one chooses the step in time , the second ones chooses if we will Increment $I$, or decrement $I$.
A graph of evolution of this system is shown below
Any directions are very welcome.