<p>This paper generalizes and enhances an epidemic model with recurrence and latency from two aspects. On the one hand, models with an incidence rate limited to a unary function of the infected class have been comprehensively studied in the former work. To deal with the situation where the transmission pattern is unclear, a general binary function with respect to both susceptible and infected classes that removes the constraint of monotonicity and concavity is considered. On the other hand, to formulate widespread human mobility and random noises, each compartment is assumed to have a diffusion term and the Black–Karasinski process is adopted as the parameter perturbation method. After establishing the stochastic reaction-diffusion system and the well-posedness of the solution, we construct the suitable integral functions to investigate under what conditions the time average of the number of patients within a bounded region would have a positive infimum, and the criterion that populations with disease tend to zero exponentially is also established. Then, we prove that the solution of the corresponding linearization system asymptotically follows the normal distribution and further calculate the local density function by decomposing the solution into general and particular solutions. From the conducted results, which are also strengthened by simulation examples, two findings are necessary to be emphasized. Firstly, even if the disease dies out in the deterministic setting, it still has the potential to spread in the population when reversion speed is small and the perturbation intensity is large, implying that the stochastic noise significantly changes dynamic properties. Secondly, the obtained conditions for the stochastic model are equal to the basic reproduction number when the noise disappears, meaning that the proposed model not only extends the deterministic model during modeling but also generalizes the threshold.</p>