  The aim of this paper is to study the asymptotic behavior of a system of
birth and death processes in mean field type interaction in discrete space. We
first establish the exponential convergence of the particle system to
equilibrium for a suitable Wasserstein coupling distance. The approach provides
an explicit quantitative estimate on the rate of convergence. We prove next a
uniform propagation of chaos property. As a consequence, we show that the limit
of the associated empirical distribution, which is the solution of a nonlinear
differential equation, converges exponentially fast to equilibrium. This paper
can be seen as a discrete version of the particle approximation of the
McKean-Vlasov equations and is inspired from previous works of Malrieu and al
and Caputo, Dai Pra and Posta.
