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In this paper, the global exponential stability and stabilization problems are investigated for memristive neural networks (MNNs) with stochastic disturbances, spatial diffusions and distributed delays. The spatial diffusions are not assumed to be symmetric and distributed delays are relaxed to be unbounded. Then, the presented MNNs are modeled as a class of stochastic partial systems with hybrid delays. Based on nonsmooth analysis, the Lyapunov–Krasovskii functional and inequality approach, some simple algebraic criteria are derived for the global exponential stability as well as the stabilization via two kinds of designed feedback controllers. The derived criteria are easily verified and the obtained results are available for other delayed systems with or without stochastic disturbances and spatial diffusions. Finally, the stability and stabilization results are validated by numerical simulations. • The stochastic model with spatial diffusions and distributed delays is discussed. • The derived results are generalized and they contain previous ones as special cases. • The method is available for other stochastic delayed partial differential systems. • The space set of the spatial diffusions in our model is not symmetrical. • The distributed delays are unbounded. [ABSTRACT FROM AUTHOR]

This paper aims to incorporate a high order diffusion term into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and we find a condition ensuring the extinction of an infectious disease. The existence of a stationary distribution for the stochastic epidemic model is investigated as well. Numerical simulations are conducted to support our theoretical results and an example of application with COVID-19 data from Canada is used to estimate the transmission rate and reproduction number associated with the stochastic model, while constructing a model fitting the data. • A stochastic SIQR epidemic model with higher-order perturbation is investigated. • The existence and uniqueness of the global positive solution are studied. • A stochastic threshold is established for extinction. • COVID-19 data from Canada are used to estimate the transmission rate. [ABSTRACT FROM AUTHOR]

As the spread of diseases among species becomes more and more common, our society pays increasingly more attention to the harm caused by the disease. Research on eco-epidemiological models will not only promote the development of biology and ecology, but also have important impacts on the economy and health of the society. This paper deals with a stochastic eco-epidemiological model which contains three species and is governed by Ornstein–Uhlenbeck (OU) process. Two common approaches to incorporate the effects of environmental perturbations in stochastic systems are first discussed and compared analytically and computationally. Based on the existence and uniqueness of the global solution to the model, several sharp conditions for the persistence and extinction of the species are established. Then, a criteria for the existence of the stationary distribution to the system is established by constructing suitable Lyapunov functions. Furthermore, the analytical expression of the probability density function of the model around the quasi-equilibrium is obtained. Finally, the impact of the main parameters on the dynamical behaviors and the noise perturbation effect are analysed in the numerical simulations section to illustrate the theoretical results. • A stochastic eco-epidemiological model with Ornstein–Uhlenbeck process is studied. • Some conditions for the persistence and extinction of species are established. • A criteria for the existence of stationary distribution is given. • The probability density function near the quasi-equilibrium is obtained. [ABSTRACT FROM AUTHOR]