In this study, we investigated the threshold dynamics of a spatially heterogeneous nonlocal diffusion West Nile virus model. By employing semigroup theory and continuous Fréchet-differentiable, we established the well-posedness of the solution. The expression for the basic reproduction number derived using the next-generation matrix method. The authors demonstrated the threshold dynamics of the system by constructing a Lyapunov function and applying the comparison principle. Finally, numerical simulations were used to validate the theorem results. It can be suggested that to control disease development rapidly, measures should be taken to reduce the spread of mosquitoes and birds.

In this study, we aimed to determine the stationary distribution of a Susceptible-Vaccine-Infected-Bacteria (SVIB) Cholera model that incorporates environmental noise and reaction-diffusion. First, we demonstrated the model's invariant set. Subsequently, a Lyapunov function was constructed to prove the existence and uniqueness of the solutions, and the model's finite-time stability was demonstrated. Furthermore, we derived the stationary distribution of the stochastic cholera model with reaction-diffusion. Finally, the theorem's results were verified through numerical simulation. Notably, the noise intensity could impact the model's stationary distribution. When the number of infected individuals and cholera bacteria decreases with reduced noise intensity, the system is characterized by a normal distribution. Therefore, appropriate measures should be taken to reduce the interference of external factors when a disease outbreak occurs.