# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022094
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## Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread

 1 Department of Computer Engineering, "Gheorghe Asachi" Technical University of Iaşi, Strada prof. dr. doc. Dimitrie Mangeron 27, 700050, Iaşi, Romania 2 Department of Computer Science, University "Al. I. Cuza" of Iaşi, Strada General Berthelot 16, 700483, Iaşi, Romania 3 Department of Mathematics, University "Al. I. Cuza" of Iaşi, Bd. Carol I 11, 700506, Iaşi, Romania

Received  March 2022 Revised  March 2022 Early access April 2022

The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: $S_0(x)$, $E_0(x)$, $I_0(x)$ $R_0(x)$, $D_0(x)$, $f(t,x)$ and $w_{_i}(t,x), i = 1,2,3,4,5$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.

The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.

Citation: Silviu Dumitru Pavăl, Alex Vasilică, Alin Adochiei. Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022094
##### References:
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Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. Optimiz., 30 (2009), 199-213.  doi: 10.1080/01630560902841120. [7] M. Bogoya and J. A. Gómez S., On a nonlocal diffusion model with Neumann boundary conditions, Nonlinear Analysis, 75 (2012), 3198-3209.  doi: 10.1016/j.na.2011.12.019. [8] F. Brauer, C. Castillo-Chavez and Z. Feng, Mathematical Models in Epidemiology, 2019, Springer, New York. doi: 10.1007/978-1-4939-9828-9. [9] J. M. Carcione, J. E. Santos, C. Bagaini and J. Ba, A simulation of a COVID-19 epidemic based on a deterministic SEIR model, Front Public Health, 8 (2020), 230.  doi: 10.3389/fpubh.2020.00230. [10] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Analysis. TMA, 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003. [11] M. M. Choban and C. N. Moroşanu, Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions, Carpathian J. Math., 38 (2022), 95-116.  doi: 10.37193/cjm.2022.01.08. [12] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to aproximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [13] M. Craus and S.-D. Pavăl, An accelerating numerical computation of the diffusion term in a nonlocal reaction-diffusion equation, Mathematics, 8 (2020), 2111.  doi: 10.3390/math8122111. [14] A. Croitoru, C. Moroşanu and G. Tănase, Well-posedness and numerical simulations of an anisotropic reaction-diffusion model in case 2D, Journal of Applied Analysis and Computation (JAAC), 11 (2021), 2258-2278.  doi: 10.11948/20200359. [15] A. Gavriluţ and C. Moroşanu, Well-posedness for a nonlinear reaction-diffusion equation endowed with nonhomogeneous Cauchy-Neumann boundary conditions and degenerate mobility, ROMAI J., 14 (2018), 129–141. https://rj.romai.ro/arhiva/2018/1/Gavrilut-Morosanufinal.pdf [16] C. I. Gheorghiu and C. Moroşanu, Accurate spectral solutions to a phase-field transition system, ROMAI J., 10 (2014), 89–99. https://rj.romai.ro/arhiva/2014/2/GheorghiuMorosanu.pdf [17] M. Grave and A. L. G. A. Coutinho, Adaptive mesh refinement and coarsening for diffusion-reaction epidemiological models, Comput. Mech., 67 (2021), 1177-1199.  doi: 10.1007/s00466-021-01986-7. [18] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [19] G. Marinoschi, Parameter estimation of an epidemic model with state constraints, Appl. Math. Optim., 84 (2021), 1903-1923.  doi: 10.1007/s00245-021-09815-2. [20] A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207.  doi: 10.1016/j.apm.2015.04.039. [21] A. Miranville and C. Moroşanu, Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 537-556.  doi: 10.3934/dcdss.2016011. [22] A. Miranville and C. Moroşanu, Qualitative and quantitative analysis for the mathematical models of phase separation and transition. Aplications, Differential Equations & Dynamical Systems, 7 (2020). www.aimsciences.org/fileAIMS/cms/news/info/28df2b3d-ffac-4598-a89b-9494392d1394.pdf, [23] A. Miranville and C. Moroşanu, A qualitative analysis of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Stefan-Boltzmann boundary conditions, Appl. Math. Optim., 84 (2021), 227-244.  doi: 10.1007/s00245-019-09643-5. [24] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. Optimiz., 18 (1997), 623-648.  doi: 10.1080/01630569708816782. [25] C. Moroşanu, On the numerical stability of the cubic splines approximation to solution of phase-field transition system, PanAmer. Math. J., 12 (2002), 31-46. [26] C. Moroşanu, Cubic spline method and fractional steps schemes to approximate the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions, ROMAI J., 8 (2012), 73–91. https://profs.info.uaic.ro/jromai/romaijournal/arhiva/2012/1/Morosanu.pdf [27] C. Moroşanu, Well-posedness for a phase-field transition system endowed with a polynomial nonlinearity and a general class of nonlinear dynamic boundary conditions, J. Fixed Point Theory Appl., 18 (2016), 225-250.  doi: 10.1007/s11784-015-0274-8. [28] C. Moroşanu, Modeling of the continuous casting process of steel via phase-field transition system. Fractional steps method, AIMS Math., 4 (2019), 648-662.  doi: 10.3934/math.2019.3.648. [29] C. Moroşanu, Numerical approximation for a nonlocal Allen-Cahn equation supplied with non-homogeneous Neumann boundary conditions, Proceedings of the Fifth Conference of Mathematical Society of Moldova, IMCS-55, Sept. 28 - Oct. 1,115–118, 2019, Chisinau, Republic of Moldova. [30] C. Moroşanu, Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1567-1587.  doi: 10.3934/dcdss.2020089. [31] C. Moroşanu and A. Croitoru, Analysis of an iterative scheme of fractional steps type associated to the phase-field equation endowed with a general nonlinearity and Cauchy-Neumann boundary conditions, J. Math. Anal. Appl., 425 (2015), 1225-1239.  doi: 10.1016/j.jmaa.2015.01.033. [32] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91-110. [33] C. Moroşanu and S. Pavăl, On the numerical approximation of a nonlinear reaction-diffusion equation with non-homogeneous Neumann boundary conditions. Case 1D, ROMAI J., 15 (2019), 43–60. https://rj.romai.ro/arhiva/2019/2/Morosanu-Paval.pdf [34] C. Moroşanu and S. Pavăl, Rigorous Mathematical investigation of a nonlocal and nonlinear second-order anisotropic reaction-diffusion model: Applications on image segmentation, Mathematics, 9 (2021), 91.  doi: 10.3390/math9010091. [35] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, J. Appl. Anal. Comput., 7 (2017), 1-19.  doi: 10.11948/2017001. [36] M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076. [37] A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli, T. J. Hughes, A. Patton, A. Reali, T. E. Yankeelov and A. Veneziani, Simulating the spread of COVID-19 via spatially-resolved susceptible-exposed- infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 106617, 9 pp. doi: 10.1016/j.aml.2020.106617. [38] A. Viguerie, A. Veneziani, G. Lorenzo, D. Baroli, N. Aretz-Nellesen, A. Patton, T. E. Yankeelov, A. Reali, T. J. R. Hughes and F. Auricchio, Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study, Comput Mech., 66 (2020), 1131-1152.  doi: 10.1007/s00466-020-01888-0. [39] T. I. Zohdi, An agent-based computational framework for simulation of global pandemic and social response on planet $X$, Comput. Mech., 66 (2020), 1195-1209.  doi: 10.1007/s00466-020-01886-2.

show all references

##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal diffusion problems, in: Mathematical Surveys and Monographs, 165 (2010). doi: 10.1090/surv/165. [2] J. Arino and S. Portet, A simple model for COVID-19, Infect. Dis. Model, 5 (2020), 309-315.  doi: 10.1016/j.idm.2020.04.002. [3] V. Arnăutu and C. Moroşanu, Numerical approximation for the phase-field transition system, Intern. J. Com. Math., 62 (1996), 209-221.  doi: 10.1080/00207169608804538. [4] T. Barbu, A. Miranville and C. Moroşanu, A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Neumann boundary conditions, Appl. Math. Comput., 350 (2019), 170-180.  doi: 10.1016/j.amc.2019.01.004. [5] P. W. Bates, S. Brown and J. Han, Numerical analysis for a nonlocal Allen-Cahn equation, Int. J. Numer. Anal. Model., 6 (2009), 33-49. [6] T. Benincasa and C. Moroşanu, Fractional steps scheme to approximate the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. Optimiz., 30 (2009), 199-213.  doi: 10.1080/01630560902841120. [7] M. Bogoya and J. A. Gómez S., On a nonlocal diffusion model with Neumann boundary conditions, Nonlinear Analysis, 75 (2012), 3198-3209.  doi: 10.1016/j.na.2011.12.019. [8] F. Brauer, C. Castillo-Chavez and Z. Feng, Mathematical Models in Epidemiology, 2019, Springer, New York. doi: 10.1007/978-1-4939-9828-9. [9] J. M. Carcione, J. E. Santos, C. Bagaini and J. Ba, A simulation of a COVID-19 epidemic based on a deterministic SEIR model, Front Public Health, 8 (2020), 230.  doi: 10.3389/fpubh.2020.00230. [10] O. Cârjă, A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Analysis. TMA, 113 (2015), 190-208.  doi: 10.1016/j.na.2014.10.003. [11] M. M. Choban and C. N. Moroşanu, Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions, Carpathian J. Math., 38 (2022), 95-116.  doi: 10.37193/cjm.2022.01.08. [12] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to aproximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [13] M. Craus and S.-D. Pavăl, An accelerating numerical computation of the diffusion term in a nonlocal reaction-diffusion equation, Mathematics, 8 (2020), 2111.  doi: 10.3390/math8122111. [14] A. Croitoru, C. Moroşanu and G. Tănase, Well-posedness and numerical simulations of an anisotropic reaction-diffusion model in case 2D, Journal of Applied Analysis and Computation (JAAC), 11 (2021), 2258-2278.  doi: 10.11948/20200359. [15] A. Gavriluţ and C. Moroşanu, Well-posedness for a nonlinear reaction-diffusion equation endowed with nonhomogeneous Cauchy-Neumann boundary conditions and degenerate mobility, ROMAI J., 14 (2018), 129–141. https://rj.romai.ro/arhiva/2018/1/Gavrilut-Morosanufinal.pdf [16] C. I. Gheorghiu and C. Moroşanu, Accurate spectral solutions to a phase-field transition system, ROMAI J., 10 (2014), 89–99. https://rj.romai.ro/arhiva/2014/2/GheorghiuMorosanu.pdf [17] M. Grave and A. L. G. A. Coutinho, Adaptive mesh refinement and coarsening for diffusion-reaction epidemiological models, Comput. Mech., 67 (2021), 1177-1199.  doi: 10.1007/s00466-021-01986-7. [18] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [19] G. Marinoschi, Parameter estimation of an epidemic model with state constraints, Appl. Math. Optim., 84 (2021), 1903-1923.  doi: 10.1007/s00245-021-09815-2. [20] A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207.  doi: 10.1016/j.apm.2015.04.039. [21] A. Miranville and C. Moroşanu, Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 537-556.  doi: 10.3934/dcdss.2016011. [22] A. Miranville and C. Moroşanu, Qualitative and quantitative analysis for the mathematical models of phase separation and transition. Aplications, Differential Equations & Dynamical Systems, 7 (2020). www.aimsciences.org/fileAIMS/cms/news/info/28df2b3d-ffac-4598-a89b-9494392d1394.pdf, [23] A. Miranville and C. Moroşanu, A qualitative analysis of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Stefan-Boltzmann boundary conditions, Appl. Math. Optim., 84 (2021), 227-244.  doi: 10.1007/s00245-019-09643-5. [24] C. Moroşanu, Approximation of the phase-field transition system via fractional steps method, Numer. Funct. Anal. Optimiz., 18 (1997), 623-648.  doi: 10.1080/01630569708816782. [25] C. Moroşanu, On the numerical stability of the cubic splines approximation to solution of phase-field transition system, PanAmer. Math. J., 12 (2002), 31-46. [26] C. Moroşanu, Cubic spline method and fractional steps schemes to approximate the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions, ROMAI J., 8 (2012), 73–91. https://profs.info.uaic.ro/jromai/romaijournal/arhiva/2012/1/Morosanu.pdf [27] C. Moroşanu, Well-posedness for a phase-field transition system endowed with a polynomial nonlinearity and a general class of nonlinear dynamic boundary conditions, J. Fixed Point Theory Appl., 18 (2016), 225-250.  doi: 10.1007/s11784-015-0274-8. [28] C. Moroşanu, Modeling of the continuous casting process of steel via phase-field transition system. Fractional steps method, AIMS Math., 4 (2019), 648-662.  doi: 10.3934/math.2019.3.648. [29] C. Moroşanu, Numerical approximation for a nonlocal Allen-Cahn equation supplied with non-homogeneous Neumann boundary conditions, Proceedings of the Fifth Conference of Mathematical Society of Moldova, IMCS-55, Sept. 28 - Oct. 1,115–118, 2019, Chisinau, Republic of Moldova. [30] C. Moroşanu, Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1567-1587.  doi: 10.3934/dcdss.2020089. [31] C. Moroşanu and A. Croitoru, Analysis of an iterative scheme of fractional steps type associated to the phase-field equation endowed with a general nonlinearity and Cauchy-Neumann boundary conditions, J. Math. Anal. Appl., 425 (2015), 1225-1239.  doi: 10.1016/j.jmaa.2015.01.033. [32] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91-110. [33] C. Moroşanu and S. Pavăl, On the numerical approximation of a nonlinear reaction-diffusion equation with non-homogeneous Neumann boundary conditions. Case 1D, ROMAI J., 15 (2019), 43–60. https://rj.romai.ro/arhiva/2019/2/Morosanu-Paval.pdf [34] C. Moroşanu and S. Pavăl, Rigorous Mathematical investigation of a nonlocal and nonlinear second-order anisotropic reaction-diffusion model: Applications on image segmentation, Mathematics, 9 (2021), 91.  doi: 10.3390/math9010091. [35] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, J. Appl. Anal. Comput., 7 (2017), 1-19.  doi: 10.11948/2017001. [36] M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076. [37] A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli, T. J. Hughes, A. Patton, A. Reali, T. E. Yankeelov and A. Veneziani, Simulating the spread of COVID-19 via spatially-resolved susceptible-exposed- infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 106617, 9 pp. doi: 10.1016/j.aml.2020.106617. [38] A. Viguerie, A. Veneziani, G. Lorenzo, D. Baroli, N. Aretz-Nellesen, A. Patton, T. E. Yankeelov, A. Reali, T. J. R. Hughes and F. Auricchio, Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study, Comput Mech., 66 (2020), 1131-1152.  doi: 10.1007/s00466-020-01888-0. [39] T. I. Zohdi, An agent-based computational framework for simulation of global pandemic and social response on planet $X$, Comput. Mech., 66 (2020), 1195-1209.  doi: 10.1007/s00466-020-01886-2.
The boundary $\Gamma = \cup_{\ell = 1}^4\Gamma_\ell$ of the rectangle $\Omega = [0,a]\times [0,b]$
Initial data
Epidemic evolution showing the affected areas for the respective populations: S - susceptible, E - exposed, I - infected, R - recovered, D - deceased
The maximum values evolution for all population categories
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