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Numerical modelling of an SIR epidemic model with diffusion. (English) Zbl 1184.92028

Summary: A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functionals. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.

MSC:

92C60 Medical epidemiology
35K57 Reaction-diffusion equations
65N06 Finite difference methods for boundary value problems involving PDEs
92D30 Epidemiology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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