Resistance to antibiotics: limit theorems for a stochastic SIS model structured by level of resistance.

The rise of bacterial resistance to antibiotics is a major Public Health concern. It is the result of two interacting processes: the selection of resistant bacterial strains under exposure to antibiotics and the dissemination of bacterial strains throughout the population by contact between colonized and uncolonized individuals. To investigate the resulting time evolution of bacterial resistance, Temime et al. (Emerg Infect Dis 9:411-417, 2003) developed a stochastic SIS model, which was structured by the level of resistance of bacterial strains. Here we study the asymptotic properties of this model when the population size is large. To this end, we cast the model within the framework of measure valued processes, using point measures to represent the pattern of bacterial resistance in the compartments of colonized individuals. We first show that the suitably normalized model tends in probability to the solution of a deterministic differential system. Then we prove that the process of fluctuations around this limit tends in law to a Gaussian process in a space of distributions. These results, which generalize those of Kurtz (CBMS-NSF regional conference series in applied mathematics, vol 36. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981, chap. 8) on SIR models, support the validity of the deterministic approximation and quantify the rate of convergence.