## Section: Application Domains

Keywords : Biomathematics, population dynamics, host-microparasite system, host-macroparasite system, deterministic model, individual based model, spatially explicit model, parasite aggregation, health.

### Population Dynamics

In population dynamics, systems can present very complex behaviors and can be difficult to analyse from a purely mathematical point of view. The aim of this interdisciplinary project was to develop numerical tools for population dynamics models arising in modelling complex heterogeneous host-parasite systems. Some typical heterogeneities we consider are spatial locations, age or ability to recruit macroparasites for hosts, age of macroparasites. Our main goals are: understanding the impact of a host population structure on a parasite population dynamics, developing accurate numerical simulations using parallelization, designing prophylactic methods. For many host-parasite systems different time scales between the host population (e.g. a one year period) and the virus (e.g. an infected host dies with a few weeks) require a small time step. Numerical schemes of the resulting nonlinear epidemiological model in spatially heterogeneous environment are complex to perform and reliable numerical results become difficult to get when the size of the spatial domain is increasing. In addition, many input parameters (biological and environmental factors) are taken into account to compare results of simulations and observations from field studies. Therefore, a realistic simulator has a significant computation cost and parallelization is required.

Individual-Based Models (IBM) are becoming more and more useful to describe biological systems. Interactions between individuals are simple and local, yet can lead to complex patterns at a global scale. The principle is to replicate several times the simulation program to obtain statistically meaningful results. The Individual-Based Model approach contrasts with a more aggregate population modeling approach in providing low level mechanisms to manage the population interactions. Stochastic simulations reproduce elementary processes and often lead to prohibitive computations; thus we need parallel algorithmic.

In our developments of both stochastic and deterministic models, biological processes are combined to reach a good level of realism. For host-parasite systems, it make a big difference with purely mathematical models, for which numerical results could hardly be compared to observations. Parallel numerical simulations mimic some of the dynamics observed in the fields, and supply a usable tool to validate the models. This work is a collaborative effort in an interdisciplinary approach between population dynamics, mathematics and computer science.