  The study of epidemic models plays an important role in mathematical
epidemiology. There are many researches on epidemic models using ordinary
differential equations, partial differential equations or stochastic
differential equations. In contrast to these researches, our work analyzes the
SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) model using
stochastic partial differential equations. Specifically, we consider the
effects of spatial variables and space-time white noise on the epidemic model.
In the means of mild solution, we construct a contraction mapping and prove the
well-posedness for the stochastic partial differential equation using the fixed
point argument and the factorization method. We also find sufficient conditions
for permanence and extinction to determine the longtime behavior of the
solution. We hope that this work can provide some new ideas for the study of
increasingly complex epidemic models and related numerical simulations.
