Random environment and human activities have important effects on the survival of listeria. In this paper, treating infected people and removing bacteria from the environment as control strategies, we developed a listeriosis model that considers random noise and spatial diffusion. By constructing a Lyapunov function, we demonstrated the existence and uniqueness of the global positive solution of the model. However, it was a challenging task to realize the optimal control of the model by solving the Pontryagin random maximum principle with the lowest control cost. Therefore, our study on near-optimal controls is of great significance for controlling the spread of listeriosis. Initially, we gave some adjoint equations and a priori estimates. Subsequently, the Pontryagin random maximum principle was utilized to establish the sufficient and necessary conditions for achieving near-optimal controls. Ultimately, the theoretical findings are corroborated through numerical analysis.

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô's lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value ${\mathcal{R}}_0^e$ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when ${\mathcal{R}}_0^e>1$, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.

In response to the pressing need to understand anthrax biology, this paper focused on the dynamical behavior of the anthrax model under environmental influence. We defined the threshold parameter $R^s$, when $R^s>1$; the disease was almost certainly present and the model exists a unique ergodic stationary distribution. Subsequently, statistical features were employed to analyze the dynamic behavior of the disease. The exact representation of the probability density function in the vicinity of the quasi-equilibrium point was determined by the Fokker-Planck equation. Finally, some numerical simulations validated our theoretical results.