Threshold behavior and exponential ergodicity of an sir epidemic model: the impact of random jamming and hospital capacity.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number [Formula: see text]. If [Formula: see text], the disease is eradicated with a probability of 1, while [Formula: see text] results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if [Formula: see text], the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.