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A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

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    * Corresponding author

J. Kou and X. Wang were supported by the Scientific and Technical Research Project of Hubei Provincial Department of Education (No.D20192703) and the Technology Creative Project of Excellent Middle & Young Team of Hubei Province (No.T201920). H. Chen was supported by the NSF of China (Grant No.11771363) and the Fundamental Research Funds for the Central Universities (Grant No.20720180003). S. Sun was supported by the grants of King Abdullah University of Science and Technology(KAUST)(No.BAS/1/1351-01, URF/1/4074-01and URF/1/3769-01)

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  • In this paper, we propose an efficient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicit-implicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme.

    Mathematics Subject Classification: Primary: 35K57, 65M06; Secondary: 92D30.


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  • Figure 1.  Initial conditions of Example 1 for susceptible $ (S) $ and infected $ (I) $ fractions in space

    Figure 2.  Susceptible fraction profiles at different times in Example 1

    Figure 3.  Infected fraction profiles at different times in Example 1

    Figure 4.  Recovered fraction profiles at different times in Example 1

    Figure 5.  Temporal evolution of susceptible, infected and recovered fractions locating at the center of the domain

    Figure 6.  Traveling waves of (a) susceptible, (b) infected and (c) recovered fractions along the central horizontal line of the domain at different times

    Figure 7.  Comparison with the real infection data

    Figure 8.  Preservation of the positivity and summation constraint in Example 1

    Figure 9.  Initial conditions of Example 2 for susceptible $ (S) $ and infected $ (I) $ fractions in space

    Figure 10.  Susceptible fraction profiles at different times in Example 2

    Figure 11.  Infected fraction profiles at different times in Example 2

    Figure 12.  Recovered fraction profiles at different times in Example 2

    Figure 13.  Preservation of the positivity and summation constraint of the proposed scheme (2.8) in Example 2

    Figure 14.  Positivity and summation constraint of the scheme (4.1) in Example 2

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