-
Previous Article
An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries
- DCDS-B Home
- This Issue
-
Next Article
Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models
Global dynamics of a virus infection model with repulsive effect
Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China |
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.
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. |
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. |
[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. |
[1] |
Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 |
[2] |
Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 |
[3] |
Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 |
[4] |
Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 |
[5] |
Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5519-5549. doi: 10.3934/dcdsb.2020357 |
[6] |
Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 |
[7] |
Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 |
[8] |
Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002 |
[9] |
Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033 |
[10] |
Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29 (4) : 2599-2618. doi: 10.3934/era.2021003 |
[11] |
Wenxian Shen, Shuwen Xue. Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2893-2925. doi: 10.3934/dcds.2022003 |
[12] |
Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 |
[13] |
Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170 |
[14] |
Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure and Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 |
[15] |
Zhilan Feng, Wenzhang Huang, Donald L. DeAngelis. Spatially heterogeneous invasion of toxic plant mediated by herbivory. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1519-1538. doi: 10.3934/mbe.2013.10.1519 |
[16] |
Naveen K. Vaidya, Xianping Li, Feng-Bin Wang. Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 321-349. doi: 10.3934/dcdsb.2018099 |
[17] |
Julián López-Gómez, Eduardo Muñoz-Hernández. A spatially heterogeneous predator-prey model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2085-2113. doi: 10.3934/dcdsb.2020081 |
[18] |
Svend Christensen, Preben Klarskov Hansen, Guozheng Qi, Jihuai Wang. The mathematical method of studying the reproduction structure of weeds and its application to Bromus sterilis. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 777-788. doi: 10.3934/dcdsb.2004.4.777 |
[19] |
Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 |
[20] |
George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]