Coexistence and extinction for a stochastic sheep brucellosis model motivated by Black–Karasinski process

To capture the underlying realistic dynamics of brucellosis infection, we propose a stochastic SEIVB-type model, where the concentration of brucella in the environment is incorporated. This paper is the first mathematical attempt to consider the Black–Karasinski process as the random effect in the modeling of epidemic transmission. It turns out that Black–Karasinski process is a both biologically and mathematically reasonable assumption compared with existing stochastic modeling approaches. We derive two critical values R0S and R0E to classify the long-term properties of the model. It is shown that (i) if R0E<1, the brucellosis will die out exponentially; (ii) if R0S>1, the stochastic model has a stationary distribution ϖ(·), which means the brucellosis prevalence; (iii) R0E=R0S=R0 if there are no random fluctuations in brucellosis transmission, where R0 is the basic reproduction number of its deterministic system. Finally, some numerical examples are provided to support our findings. It should be highlighted that our theoretical methods and techniques used can be applied to other complex high-dimensional epidemic models perturbed by Black–Karasinski process.