Stochastic SIS project
Let \(I\) be the number infectious in a population \(N\), so \(S=N-I\). Use infection and recovery rates
\[b(i)=\beta\frac{i(N-i)}N,\qquad d(i)=\gamma i.\]Unlike the deterministic SIS equation, the stochastic model generates different epidemic histories from the same initial state.
Simulation experiment
Run many independent trajectories from \(I(0)=i_0\). Record whether infection reaches zero before a chosen time and estimate
\[P(\text{extinction by }T)\approx\frac{\text{number extinct by }T}{\text{number simulated}}.\]Compare outcomes
Plot several individual trajectories together with their empirical mean. The mean summarises the simulations but does not show the probability of extinction or the spread of possible paths.
Project outcome. Demonstrate why a distribution of stochastic outcomes contains information that a single deterministic trajectory cannot provide.