# American Institute of Mathematical Sciences

October  2018, 23(8): 3213-3235. doi: 10.3934/dcdsb.2018242

## Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections

 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

wanbiao_ma@ustb.edu.cn(Corresponding author)

Received  December 2017 Revised  March 2018 Published  October 2018 Early access  August 2018

Fund Project: The research is partly supported by the China Scholarship Council for W. Wang, the National Natural Science Foundation of China (11471034) and National Key R-D Program of China (2017YFF0207400-0207403) for W. Ma.

We consider a class of non-cooperative reaction-diffusion system, which includes different types of incidence rates for virus dynamical models with nonlocal infections. Threshold dynamics are expressed by basic reproduction number $R_0$ in the following sense, if $R_0 < 1$, the infection-free steady state is globally attractive, implying infection becomes extinct; while if $R_0 > 1$, virus will persist. To study the invasion speed of virus, the existence of travelling wave solutions is studied by employing Schauder's fixed point theorem. The method of constructing super-solutions and sub-solutions is very technical. The mathematical difficulty is the problem constructing a bounded cone to apply the Schauder's fixed point theorem. As compared to previous mathematical studies for diffusive virus dynamical models, the novelty here is that we successfully establish the general existence result of travelling wave solutions for a class of virus dynamical models with complex nonlinear transmissions and nonlocal infections.

Citation: Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
##### References:
 [1] A. L. Cox and R. F. Siliciano, HIV: Not-so-innocent bystanders, Nature, 505 (2014), 492-493.  doi: 10.1038/505492a. [2] N. W. Cummins and A. D. Badley, Mechanisms of HIV-associated lymphocyte apoptosis, Cell Death and Disease, 1 (2010), 1-9. [3] Y.-Y. Chen, J.-S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predatorprey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071. [4] G. Doitsh, et al., Pyroptosis drives CD4 T-cell depletion in HIV-1 infection, Nature, 505 (2014), 509-514. [5] O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [6] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112. [7] A. Ducrot and M. Langlais, Traveling waves in invasion processes with pathogens, Math. Models Methods Appl. Sci., 18 (2008), 325-349.  doi: 10.1142/S021820250800270X. [8] G. A. Doitsh, et al., Abortive HIV infection mediates CD4 T cell depletion and inflammation in human lymphoid tissue, Cell, 143 (2010), 789-801. [9] V. Doceul, M. Hollinshead, L. van der Linden and G. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173. [10] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. [11] Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y. [12] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417.  doi: 10.1093/imamat/hxq009. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [14] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504. [15] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925. [16] C. Hsu, C. Yang, T. Yang and T. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differ. Equ., 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008. [17] W. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644.  doi: 10.1007/s10884-012-9255-4. [18] W. Huang, A geometric approach in the study of traveling waves for some classes of nonmonotone reaction-diffusion systems, J. Differ. Equ., 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060. [19] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008. [20] X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [21] Y. Lou and X. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [22] X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Diff. Equat., 23 (2011), 903-921.  doi: 10.1007/s10884-011-9220-7. [23] W. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion models with applications to diffusion-competition models, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003. [24] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z. [25] G. Lin, W. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.  doi: 10.3934/dcdsb.2010.13.393. [26] X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [27] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [28] S. Ma, Traveling wavefronts for delayed reaction-diffusion models via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846. [29] C. A. Muro-Cacho, G. Pantaleo and A. S. Fauci, Analysis of apoptosis in lymph nodes of HIV-infected persons. Intensity of apoptosis correlates with the general state of activation of the lymphoid tissue and not with stage of disease or viral burden, J. Immunol., 154 (1995), 5555-5566. [30] X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-tocell transmission, J. Math. Biol., 76 (2018), 1831-1872, https://doi.org/10.1007/s00285-017-1202-x. doi: 10.1007/s00285-017-1202-x. [31] H. L. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [32] H. L. Smith, Monotone dynamic systems: an introduction to the theory of competitive and Cooperative Systems, Math Surveys Monogr, vol 41. American Mathematical Society, Providence, RI, 1995. [33] H. R. Thieme and X. Zhao, Convergence results and Poincar'e-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [34] H. R. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real Word Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7. [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [36] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Equat., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [37] W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic model, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942. [38] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [39] Z. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread, Proc. Amer. Math. Soc., 149 (2012), 3931-3946. [40] X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303. [41] W. Wang and W. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, J. Math. Anal. Appl., 457 (2018), 868-889.  doi: 10.1016/j.jmaa.2017.08.024. [42] H. Wang and X. Wang, Traveling wave phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., 28 (2016), 143-166.  doi: 10.1007/s10884-015-9506-2. [43] P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ., 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020. [44] F. Wang, Y. Huang and X. Zou, Global dynamics of a PDE in-host viral model, Appl. Anal., 93 (2014), 2312-2329.  doi: 10.1080/00036811.2014.955797. [45] J. Wang, J. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027. [46] W. Wang, W. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. Real World Appl., 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013. [47] W. Wang and X. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890. [48] W. Wang, W. Ma and X. Lai, A diffusive virus infection dynamic model with nonlinear functional response, absorption effect and chemotaxis, Commun. Nonlinear Sci. Numer. Simulat., 42 (2017), 585-606.  doi: 10.1016/j.cnsns.2016.05.010. [49] W. Wang and W. Ma, Block effect on HCV infection by HMGB1 released from virus-infected cells: An insight from mathematical modeling, Commun. Nonlinear Sci. Numer. Simulat., 59 (2018), 488-514.  doi: 10.1016/j.cnsns.2017.11.024. [50] W. Wang and W. Ma, Hepatitis C virus infection is blocked by HMGB1: A new nonlocal and time-delayed reaction-diffusion model, Appl. Math. Comput., 320 (2018), 633-653.  doi: 10.1016/j.amc.2017.09.046. [51] W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4$^{+}$ T cells death: A nonlocal spatial mathematical model, Bull. Math. Biol., 80 (2018), 540-582.  doi: 10.1007/s11538-017-0389-8. [52] J. Wu, Theory and Applications of Partial Functional Differential Equations, (Springer-Verlag, New York, 1996). doi: 10.1007/978-1-4612-4050-1. [53] R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001. [54] Z. Yu and R. 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show all references

##### References:
 [1] A. L. Cox and R. F. Siliciano, HIV: Not-so-innocent bystanders, Nature, 505 (2014), 492-493.  doi: 10.1038/505492a. [2] N. W. Cummins and A. D. Badley, Mechanisms of HIV-associated lymphocyte apoptosis, Cell Death and Disease, 1 (2010), 1-9. [3] Y.-Y. Chen, J.-S. Guo and C.-H. Yao, Traveling wave solutions for a continuous and discrete diffusive predatorprey model, J. Math. Anal. Appl., 445 (2017), 212-239.  doi: 10.1016/j.jmaa.2016.07.071. [4] G. Doitsh, et al., Pyroptosis drives CD4 T-cell depletion in HIV-1 infection, Nature, 505 (2014), 509-514. [5] O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [6] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112. [7] A. Ducrot and M. Langlais, Traveling waves in invasion processes with pathogens, Math. Models Methods Appl. Sci., 18 (2008), 325-349.  doi: 10.1142/S021820250800270X. [8] G. A. Doitsh, et al., Abortive HIV infection mediates CD4 T cell depletion and inflammation in human lymphoid tissue, Cell, 143 (2010), 789-801. [9] V. Doceul, M. Hollinshead, L. van der Linden and G. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173. [10] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. [11] Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y. [12] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417.  doi: 10.1093/imamat/hxq009. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [14] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504. [15] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925. [16] C. Hsu, C. Yang, T. Yang and T. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differ. Equ., 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008. [17] W. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Diff. Equat., 24 (2012), 633-644.  doi: 10.1007/s10884-012-9255-4. [18] W. Huang, A geometric approach in the study of traveling waves for some classes of nonmonotone reaction-diffusion systems, J. Differ. Equ., 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060. [19] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008. [20] X. Liang and X. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [21] Y. Lou and X. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [22] X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Diff. Equat., 23 (2011), 903-921.  doi: 10.1007/s10884-011-9220-7. [23] W. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion models with applications to diffusion-competition models, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003. [24] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z. [25] G. Lin, W. Li and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.  doi: 10.3934/dcdsb.2010.13.393. [26] X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [27] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590. [28] S. Ma, Traveling wavefronts for delayed reaction-diffusion models via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846. [29] C. A. Muro-Cacho, G. Pantaleo and A. S. Fauci, Analysis of apoptosis in lymph nodes of HIV-infected persons. Intensity of apoptosis correlates with the general state of activation of the lymphoid tissue and not with stage of disease or viral burden, J. Immunol., 154 (1995), 5555-5566. [30] X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-tocell transmission, J. Math. Biol., 76 (2018), 1831-1872, https://doi.org/10.1007/s00285-017-1202-x. doi: 10.1007/s00285-017-1202-x. [31] H. L. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [32] H. L. Smith, Monotone dynamic systems: an introduction to the theory of competitive and Cooperative Systems, Math Surveys Monogr, vol 41. American Mathematical Society, Providence, RI, 1995. [33] H. R. Thieme and X. Zhao, Convergence results and Poincar'e-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [34] H. R. Thieme and X. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real Word Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7. [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [36] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Equat., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [37] W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic model, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942. [38] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [39] Z. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread, Proc. Amer. Math. Soc., 149 (2012), 3931-3946. [40] X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303. [41] W. Wang and W. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, J. Math. Anal. Appl., 457 (2018), 868-889.  doi: 10.1016/j.jmaa.2017.08.024. [42] H. Wang and X. Wang, Traveling wave phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., 28 (2016), 143-166.  doi: 10.1007/s10884-015-9506-2. [43] P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ., 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020. [44] F. Wang, Y. Huang and X. Zou, Global dynamics of a PDE in-host viral model, Appl. Anal., 93 (2014), 2312-2329.  doi: 10.1080/00036811.2014.955797. [45] J. Wang, J. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564.  doi: 10.1016/j.jmaa.2016.07.027. [46] W. Wang, W. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. Real World Appl., 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013. [47] W. Wang and X. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890. [48] W. Wang, W. Ma and X. Lai, A diffusive virus infection dynamic model with nonlinear functional response, absorption effect and chemotaxis, Commun. Nonlinear Sci. Numer. Simulat., 42 (2017), 585-606.  doi: 10.1016/j.cnsns.2016.05.010. [49] W. Wang and W. Ma, Block effect on HCV infection by HMGB1 released from virus-infected cells: An insight from mathematical modeling, Commun. Nonlinear Sci. Numer. Simulat., 59 (2018), 488-514.  doi: 10.1016/j.cnsns.2017.11.024. [50] W. Wang and W. Ma, Hepatitis C virus infection is blocked by HMGB1: A new nonlocal and time-delayed reaction-diffusion model, Appl. Math. Comput., 320 (2018), 633-653.  doi: 10.1016/j.amc.2017.09.046. [51] W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4$^{+}$ T cells death: A nonlocal spatial mathematical model, Bull. Math. Biol., 80 (2018), 540-582.  doi: 10.1007/s11538-017-0389-8. [52] J. Wu, Theory and Applications of Partial Functional Differential Equations, (Springer-Verlag, New York, 1996). doi: 10.1007/978-1-4612-4050-1. [53] R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001. [54] Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitivecooperative systems, IMA J. Appl. Math., 76 (2011), 493-513.  doi: 10.1093/imamat/hxq048. [55] T. Zhang, X. Meng and T. Zhang, Global dynamics of a virus dynamical model with cell-to-cell transmission and cure rate, Comput. Math. Methods Med., 2015 (2015), Article ID 758362, 8pp. doi: 10.1155/2015/758362. [56] 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. [57] T. Zhang, W. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusionreaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017. [58] T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Differ. Equ., 262 (2017), 4724-4770.  doi: 10.1016/j.jde.2016.12.017. [59] G. Zhang, W. Li and Z. 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