  In this study we extend the results of Kurtz (1970,1971) to show the weak
convergence of epidemic processes that include explicit time dependence,
specifically where the transmission parameter,$\beta(t)$, carries a time
dependency. We first show that when population size goes to infinity, the time
inhomogeneous process converges weakly to the solution of the mean-field ODE.
Our second result is that, under proper scaling, the central limit type
fluctuations converge to a diffusion process.

Probability