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doi: 10.3934/cpaa.2021094

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

 1 School of Civil Engineering, Shaoxing University, Shaoxing, Zhejiang 312000, China 2 School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China 3 School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432000, China 4 Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia

* Corresponding author

Received  February 2021 Revised  April 2021 Published  June 2021

Fund Project: 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)

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.

Citation: Jisheng Kou, Huangxin Chen, Xiuhua Wang, Shuyu Sun. A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021094
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##### References:
Initial conditions of Example 1 for susceptible $(S)$ and infected $(I)$ fractions in space
Susceptible fraction profiles at different times in Example 1
Infected fraction profiles at different times in Example 1
Recovered fraction profiles at different times in Example 1
Temporal evolution of susceptible, infected and recovered fractions locating at the center of the domain
Traveling waves of (a) susceptible, (b) infected and (c) recovered fractions along the central horizontal line of the domain at different times
Comparison with the real infection data
Preservation of the positivity and summation constraint in Example 1
Initial conditions of Example 2 for susceptible $(S)$ and infected $(I)$ fractions in space
Susceptible fraction profiles at different times in Example 2
Infected fraction profiles at different times in Example 2
Recovered fraction profiles at different times in Example 2
Preservation of the positivity and summation constraint of the proposed scheme (2.8) in Example 2
Positivity and summation constraint of the scheme (4.1) in Example 2