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doi: 10.3934/cpaa.2021094
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## 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 Early access 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 and Applied Analysis, doi: 10.3934/cpaa.2021094
##### References:
 [1] T. Arbogast, M. F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), 828-852.  doi: 10.1137/S0036142994262585. [2] F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [3] H. Chen, J. Kou, S. Sun and T. Zhang, Fully mass-conservative IMPES schemes for incompressible two-phase flow in porous media, Comput. Methods Appl. Mech. Engrg., 350 (2019), 641-663.  doi: 10.1016/j.cma.2019.03.023. [4] W. Chen, C. Wang, X. Wang and S. M. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, Journal of Computational Physics: X, 3 (2019), 100031. doi: 10.1016/j.jcpx.2019.100031. [5] Y. Chen and J. Shen, Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models, J. Comput. Phys., 308 (2016), 40-56.  doi: 10.1016/j.jcp.2015.12.006. [6] Y. Cheng, D. Lu, J. Zhou and J. Wei, Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model, Adv. Differ. Equ., 2019 (2019), 494. doi: 10.1186/s13662-019-2432-6. [7] D. Ding, Q. Ma and X. Ding, A non-standard finite difference scheme for an epidemic model with vaccination, J. Differ. Equ. Appl., 19 (2013), 179-190.  doi: 10.1080/10236198.2011.614606. [8] D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001), 675-699.  doi: 10.1007/PL00005429. [9] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [10] Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020. [11] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., B., 115 (1927), 700-721. [12] N. Khiari, T. Achouri, M. Ben Mohamed and K. Omrani, Finite difference approximate solutions for the Cahn-Hilliard equation, Numer. Methods Partial Differ. Equ., 23 (2007), 437-455.  doi: 10.1002/num.20189. [13] J. Kou, S. Sun and X. Wang, Linearly decoupled energy-stable numerical methods for multicomponent two-phase compressible flow, SIAM J. Numer. Anal., 56 (2018), 3219-3248.  doi: 10.1137/17M1162287. [14] J. Kou, S. Sun and X. Wang, A novel energy factorization approach for the diffuse-interface model with Peng-Robinson equation of state, SIAM J. Sci. Comput., 42 (2020), B30-B56. doi: 10.1137/19M1251230. [15] K. Y. Lam, X. Wang and T. Zhang, Traveling Waves for a Class of Diffusive Disease-Transmission Models with Network Structures, SIAM J. Math. Anal., 50 (2018), 5719-5748.  doi: 10.1137/17M1144258. [16] R. E. Mickens, Dynamic Consistency: a Fundamental Principle for Constructing Nonstandard Finite Difference Schemes for Differential Equations, J. Differ. Equ. Appl., 11 (2005), 645-653.  doi: 10.1080/10236190412331334527. [17] R. E. Mickens, Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition, Numer. Methods Partial Differ. Equ., 23 (2007), 672-691.  doi: 10.1002/num.20198. [18] H. Ramaswamy, A. A. Oberai and Y. C. Yortsos, A comprehensive spatial-temporal infection model, Chem. Eng. Sci., 233 (2021), 116347. [19] H. Rui, D. Zhao and H. Pan., A block-centered finite difference method for Darcy-Forchheimer model with variable Forchheimer number, Numer. Methods Partial Differ. Equ., 31 (2015), 1603-1622.  doi: 10.1002/num.21963. [20] T. Tang and J. Yang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math., 34 (2016), 451-461.  doi: 10.4208/jcm.1603-m2014-0017. [21] G. Tryggvason, R. Scardovelli and S. Zaleski, Direct Numerical Simulations of Gas-Liquid Multiphase Flows,, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511975264. [22] X. Wang, J. Kou and J. Cai, Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential, J. Sci. Comput., 82 (2020), 25. doi: 10.1007/s10915-020-01127-x. [23] X. Wang, J. Kou and H. Gao, Linear energy stable and maximum principle preserving semi-implicit scheme for Allen-Cahn equation with double well potential, Commun. Nonlinear Sci. Numer. Simul., 98 (2021), 105766. doi: 10.1016/j.cnsns.2021.105766. [24] J. Wei, J. Zhou, W. Chen, Z. Zhen and L. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pure Appl. Anal., 19 (2020), 2853-2886.  doi: 10.3934/cpaa.2020125. [25] S. M. Wise., Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations, J. Sci. Comput., 44 (2010), 38-68.  doi: 10.1007/s10915-010-9363-4. [26] S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288.  doi: 10.1137/080738143. [27] J. Xu, J. Hou, Y. Geng and S. Zhang, Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence, Adv. Differ. Equ., 2018 (2018), 108. doi: 10.1186/s13662-018-1560-8. [28] T. Zhang and W. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068. [29] L. Zhao and Z. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033. [30] L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9. [31] J. Zhou, L. Song, J. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.

show all references

##### References:
 [1] T. Arbogast, M. F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), 828-852.  doi: 10.1137/S0036142994262585. [2] F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [3] H. Chen, J. Kou, S. Sun and T. Zhang, Fully mass-conservative IMPES schemes for incompressible two-phase flow in porous media, Comput. Methods Appl. Mech. Engrg., 350 (2019), 641-663.  doi: 10.1016/j.cma.2019.03.023. [4] W. Chen, C. Wang, X. Wang and S. M. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, Journal of Computational Physics: X, 3 (2019), 100031. doi: 10.1016/j.jcpx.2019.100031. [5] Y. Chen and J. Shen, Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models, J. Comput. Phys., 308 (2016), 40-56.  doi: 10.1016/j.jcp.2015.12.006. [6] Y. Cheng, D. Lu, J. Zhou and J. Wei, Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model, Adv. Differ. Equ., 2019 (2019), 494. doi: 10.1186/s13662-019-2432-6. [7] D. Ding, Q. Ma and X. Ding, A non-standard finite difference scheme for an epidemic model with vaccination, J. Differ. Equ. Appl., 19 (2013), 179-190.  doi: 10.1080/10236198.2011.614606. [8] D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001), 675-699.  doi: 10.1007/PL00005429. [9] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [10] Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020. [11] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., B., 115 (1927), 700-721. [12] N. Khiari, T. Achouri, M. Ben Mohamed and K. Omrani, Finite difference approximate solutions for the Cahn-Hilliard equation, Numer. Methods Partial Differ. Equ., 23 (2007), 437-455.  doi: 10.1002/num.20189. [13] J. Kou, S. Sun and X. Wang, Linearly decoupled energy-stable numerical methods for multicomponent two-phase compressible flow, SIAM J. Numer. Anal., 56 (2018), 3219-3248.  doi: 10.1137/17M1162287. [14] J. Kou, S. Sun and X. Wang, A novel energy factorization approach for the diffuse-interface model with Peng-Robinson equation of state, SIAM J. Sci. Comput., 42 (2020), B30-B56. doi: 10.1137/19M1251230. [15] K. Y. Lam, X. Wang and T. Zhang, Traveling Waves for a Class of Diffusive Disease-Transmission Models with Network Structures, SIAM J. Math. Anal., 50 (2018), 5719-5748.  doi: 10.1137/17M1144258. [16] R. E. Mickens, Dynamic Consistency: a Fundamental Principle for Constructing Nonstandard Finite Difference Schemes for Differential Equations, J. Differ. Equ. Appl., 11 (2005), 645-653.  doi: 10.1080/10236190412331334527. [17] R. E. Mickens, Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition, Numer. Methods Partial Differ. Equ., 23 (2007), 672-691.  doi: 10.1002/num.20198. [18] H. Ramaswamy, A. A. Oberai and Y. C. Yortsos, A comprehensive spatial-temporal infection model, Chem. Eng. Sci., 233 (2021), 116347. [19] H. Rui, D. Zhao and H. Pan., A block-centered finite difference method for Darcy-Forchheimer model with variable Forchheimer number, Numer. Methods Partial Differ. Equ., 31 (2015), 1603-1622.  doi: 10.1002/num.21963. [20] T. Tang and J. Yang, Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J. Comput. Math., 34 (2016), 451-461.  doi: 10.4208/jcm.1603-m2014-0017. [21] G. Tryggvason, R. Scardovelli and S. Zaleski, Direct Numerical Simulations of Gas-Liquid Multiphase Flows,, Cambridge University Press, New York, 2011.  doi: 10.1017/CBO9780511975264. [22] X. Wang, J. Kou and J. Cai, Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential, J. Sci. Comput., 82 (2020), 25. doi: 10.1007/s10915-020-01127-x. [23] X. Wang, J. Kou and H. Gao, Linear energy stable and maximum principle preserving semi-implicit scheme for Allen-Cahn equation with double well potential, Commun. Nonlinear Sci. Numer. Simul., 98 (2021), 105766. doi: 10.1016/j.cnsns.2021.105766. [24] J. Wei, J. Zhou, W. Chen, Z. Zhen and L. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pure Appl. Anal., 19 (2020), 2853-2886.  doi: 10.3934/cpaa.2020125. [25] S. M. Wise., Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations, J. Sci. Comput., 44 (2010), 38-68.  doi: 10.1007/s10915-010-9363-4. [26] S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288.  doi: 10.1137/080738143. [27] J. Xu, J. Hou, Y. Geng and S. Zhang, Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence, Adv. Differ. Equ., 2018 (2018), 108. doi: 10.1186/s13662-018-1560-8. [28] T. Zhang and W. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495.  doi: 10.1016/j.jmaa.2014.04.068. [29] L. Zhao and Z. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033. [30] L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.  doi: 10.1007/s00285-018-1227-9. [31] J. Zhou, L. Song, J. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.
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