Though population and epidemic dynamics are by nature discrete and stochastic, they are usually modelized as systems of ODE's. Indeed, in large population size discreteness can be neglected and size can be considered as continuous (or treated as population densities); likewise randomness is "averaged" and may be neglected. 

From this perspective, epidemic models can be revisited, as their central problem is the spread of a disease starting from a few infected individuals. The present work is adopting this approach mainly from the numerical viewpoint. 

The divergence of the stochastic model from its corresponding deterministic ODE limit in large population can be evaluated from a large deviation principle. The ODE can be simulated using a specific non-standard finite difference scheme that respects the qualitative properties of the ODE. 

The discrete stochastic model can be exactly simulated using the stochastic simulation algorithm (SSA). 

As this last approach can be time consuming, it is possible to make use approximated schemes such as the $\tau$-leaping algorithms. These numerical methods are applied to two examples. 

Fabien Campillo (INRIA Montpellier)