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In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number $ \Re _0^{\rho } $ ℜ 0 ρ . The results show that (i) if $ \Re _0^{\rho } \gt 1 $ ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when $ \Re _0^{\rho } \lt 1 $ ℜ 0 ρ < 1. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure. [ABSTRACT FROM AUTHOR]

In order to complement some clues for analytical pursuits and potential applications in disease control and prevention, we propose a diffusive susceptible-infective-susceptible (SIS) epidemic model incorporating a logistic growth source and a vector-mediated infection mechanism. Under the Neumann boundary condition and environmental heterogeneity, we carry out the qualitative analysis to achieve the basic reproduction number served as the threshold-type value of the model. Specifically, without environmental heterogeneity, the global attractivity of the unique constant endemic equilibrium is proved by Lyapunov function method. We further explore the asymptotic profiles of endemic equilibrium as the diffusion rates of humans approaching to zero or infinity. We find that the strategies that limiting and releasing the diffusion rates do not help eliminating disease in our current model since the steady state can stay above a positive level, which brings a useful insight in investigating the effects of the environmental heterogeneity and spatial diffusion on disease spread and control. [ABSTRACT FROM AUTHOR]

The shunting inhibitory cellular neural networks with continuously distributed delays and pseudo almost periodic coefficients are considered. First, we make a generalization of the Halanay inequality, and then establish some sufficient conditions for the existence and asymptotical stability of pseudo almost periodic solutions. Finally, a numerical simulation is presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]