## Section: New Results

Keywords : coupling model, deterministic simulator, stochastic simulator.

### Numerical schemes and algorithmic for population dynamics

Participants : Guillaume Latu, Jean Roman, Gaël Tessier.

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 modeling complex heterogeneous host-parasite systems. Some typical heterogeneities we consider are spatial locations, age or ability to recruit macroparasites for hosts. 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 environnmental 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.

A cooperation involving a biologist (Agnès Calonnec - INRA UMR Santé
végétale 1065 - Villenave d'Ornon) and a thesis student in computer
science (Gaël Tessier) began since october 2003. Using numerical methods and
parallel technics, we are interested in modeling the spread of
*powdery mildew* , a disease of vineyard. Correct prediction of
this type of parasite epidemics needs an realistic simulator, and
could have an industrial impact.

An architectural model of vine stocks is used for two purposes: the study of the growth of stocks and the influence of its structure on the dispersal of powdery mildew. In this model, we consider a large number of infectious elements and several spatially hetereogeneous environmental parameters. Indeed, the dispersal of powdery is a multiscale mechanism that takes place within vine stocks, and along and across the rows of the vineyard. An initial version of a parallel simulator using MPI communications has been developed. A characterization of the implemented algorithms is presented in [82] ; we evaluate particularly the communication costs and the load imbalance. First results indicated a good scalability up to 24 processors. Further experiments were carried out on clusters of SMP nodes, up to 128 processors [73] . This revealed that the part of time spent for communications and synchronizations highly increases for simulations that uses 64 processors and more. Relative efficiency drops to 63 % with 128 processors.

An hybrid approach mixing processes and threads has been considered: the idea is to benefit from the high speed of shared memory accesses by replacing n monothreaded processes in the previous parallel simulator by n/p processes, each one containing one master thread responsible for inter-process MPI communications and p simulation threads running inside the same SMP node. Simulation threads compute the growth of vinestocks and colonies of powdery mildew, and the dispersal of aerial spores. Threads in a same process can exchange data via the shared memory, avoiding MPI communications. Furthermore, communications between nodes can be aggregated for all the threads of the nodes, and load-balancing can be improved by exchanging vinestocks between threads of a process. The implementation and the performances of this hybrid simulator were presented in [10] . A partial dynamic load-balancing turned out to be necessary to reduce the cost of synchronizations between threads, and permit to improve the scalability of the simulator.