# American Institute of Mathematical Sciences

September  2019, 24(9): 4783-4797. doi: 10.3934/dcdsb.2019030

## Global dynamics of a virus infection model with repulsive effect

 Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China

Received  October 2017 Revised  April 2018 Published  September 2019 Early access  February 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11671359, No. 11271342), the provincial Natural Science Foundation of Zhejiang (No. LY19A010027, No. LY18A010013) and the Science Foundation of Zhejiang Sci-Tech University under Grant No. 15062173-Y.

This paper is devoted to investigate a virus infection model with a spatially heterogeneous structure and nonlinear diffusion. First we establish the properties of the basic reproduction number $R_0$ for infected cells and free virus particles. Then we prove that the comparison principle can be applied to an auxiliary system with quasilinear diffusion under appropriate conditions. Then the sufficient conditions for the globally asymptotical stability of infection-free steady state are obtained, which indicates that $R_0<1$ is necessary for infected cells and free virus particles to be extinct. Next we prove the existence of positive non-constant steady states and the persistence of infected cells and free virion where $R_0>1$ is required. Finally, it is shown that, for the spatially homogeneous case when the infected cells rate of change of the repulsive effect is small enough, $R_0$ is the only determinant of the global dynamics of the underlying virus infection system. The obtained results give an insight into the optimal control of the virion.

Citation: Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030
##### References:
 [1] H. Amann, Synamical theory of quasilinear parabolic equations Ⅲ: Global existence, Math. Z, 202 (1989), 219-250.  doi: 10.1007/BF01215256. [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser HJ, Triebel H (eds) Function spaces, differential operators and nonlinear analysis (Friedrichroda), Teubner-Texte zur Mathematik. Teubner, Stuttgart, 133 (1992), 9-126.  doi: 10.1007/978-3-663-11336-2_1. [3] M. Doceul, V. Hollinshead, L van der Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173. [4] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, J. Appl. Math., 75 (2010), 392-417.  doi: 10.1093/imamat/hxq009. [5] S. B. Hsu, F. B. Wang and X. Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006. [6] X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cells, Bulletin of Mathematical Biology, 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [7] Y. J. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, Journal of Mathematical Biology, 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [8] P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173. [9] M. A. Nowak, S. Bonhoeffer, A. M. Hill and et al., Viral dynamics in hepatitis B virus infection, Proceedings of the National Academy of Sciences of the United States of America, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [10] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. [11] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995. [12] H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [13] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosciences, 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3. [14] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Mathematical Biosciences, 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [15] W. D. Wang and X. Q. Zhao, Spatial invasion threshold of lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769. [16] W. D. Wang and X. Q. Zhao, Basic Reproduction Numbers for Reaction-Diffusion Epidemic Models, SIAM J. Appl. Dynamical Systems, 11 (2012), 1562-1763.  doi: 10.1137/120872942. [17] 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. [18] X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, Second Edition, 2017. doi: 10.1007/978-3-319-56433-3.

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##### References:
 [1] H. Amann, Synamical theory of quasilinear parabolic equations Ⅲ: Global existence, Math. Z, 202 (1989), 219-250.  doi: 10.1007/BF01215256. [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser HJ, Triebel H (eds) Function spaces, differential operators and nonlinear analysis (Friedrichroda), Teubner-Texte zur Mathematik. Teubner, Stuttgart, 133 (1992), 9-126.  doi: 10.1007/978-3-663-11336-2_1. [3] M. Doceul, V. Hollinshead, L van der Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010), 873-876.  doi: 10.1126/science.1183173. [4] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, J. Appl. Math., 75 (2010), 392-417.  doi: 10.1093/imamat/hxq009. [5] S. B. Hsu, F. B. Wang and X. Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006. [6] X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cells, Bulletin of Mathematical Biology, 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9. [7] Y. J. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, Journal of Mathematical Biology, 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8. [8] P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173. [9] M. A. Nowak, S. Bonhoeffer, A. M. Hill and et al., Viral dynamics in hepatitis B virus infection, Proceedings of the National Academy of Sciences of the United States of America, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [10] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. [11] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995. [12] H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2. [13] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosciences, 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3. [14] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Mathematical Biosciences, 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [15] W. D. Wang and X. Q. Zhao, Spatial invasion threshold of lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769. [16] W. D. Wang and X. Q. Zhao, Basic Reproduction Numbers for Reaction-Diffusion Epidemic Models, SIAM J. Appl. Dynamical Systems, 11 (2012), 1562-1763.  doi: 10.1137/120872942. [17] 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. [18] X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, Second Edition, 2017. doi: 10.1007/978-3-319-56433-3.
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