# 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  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 & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
##### References:
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##### References:
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Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [39] Z. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread, Proc. Amer. Math. Soc., 149 (2012), 3931-3946.   Google Scholar [40] X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. 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