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Complex dynamics of a reaction-diffusion epidemic model. (English) Zbl 1327.92069

Summary: In this paper, we investigate the complex dynamics of a reaction-diffusion \(S-I\) model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes-stripes, stripes, spots-stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with \(R_{0}\) increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatio-temporal dynamics in the epidemic model.

MSC:

92D30 Epidemiology
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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