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

Global dynamics of a virus infection model with repulsive effect

Hui li and  Manjun Ma

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

Received  October 2017 Revised  April 2018 Published  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.

Keywords: Quasilinearly parabolic system, globally asymptotical stability, uniform persistence, Basic reproduction number, spatially heterogeneous structure.
Mathematics Subject Classification: Primary: 35B40, 5K55, 35k57; Secondary: 37N25, 92D25.
Citation: Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

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H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, Second Edition, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

References:
[1]

H. Amann, Synamical theory of quasilinear parabolic equations Ⅲ: Global existence, Math. Z, 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, Second Edition, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

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