January  2020, 19(1): 85-102. doi: 10.3934/cpaa.2020005

Dynamics of spatially heterogeneous viral model with time delay

1. 

School of Science, Jiangsu Ocean University, Lianyungang, Jiangsu, 222005, China

2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China

3. 

School of Science, Jimei University, Xiamen, Fujian, 361021, China

* Corresponding author

Received  August 2018 Revised  January 2019 Published  July 2019

Fund Project: Hong Yang is supported by China-DSTF grant zd-2016-091, Junjie Wei is supported by China-NNSF grant 11771109.

A delayed reaction-diffusion virus model with a general incidence function and spatially dependent parameters is investigated. The basic reproduction number for the model is derived, and the uniform persistence of solutions and global attractively of the equilibria are proved. We also show the global attractivity of the positive equilibria via constructing Lyapunov functional, in case that all the parameters are spatially independent. Numerical simulations are finally conducted to illustrate these analytical results.

Citation: 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
References:
[1]

C-M. BraunerD. JollyL. Lorenzi and P. Thiebaut, Heterogeneous viral environment in an HIV spatial model, Disc. Cont. Dyn. Syst. Ser. B., 15 (2011), 545-572.  doi: 10.3934/dcdsb.2011.15.545.

[2]

K. Deimling, Nonlinear Functional Analysis, Berlin: Springer-Verlag, 1988. doi: 10.1007/978-3-662-00547-7.

[3]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, New York, 1993.

[4]

M. LewisJ. Renclawowicz and P. van den Driessche, Traveling waves and spread rates for a West Nile virus model, Bull. Math. Biol., 68 (2006), 3-23.  doi: 10.1007/s11538-005-9018-z.

[5]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.  doi: 10.1007/s11538-010-9503-x.

[6]

Y. Lou and X.-Q. 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.

[7]

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.

[8]

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.

[9]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal., 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.03.002.

[10]

A. MuraseT. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.  doi: 10.1007/s00285-005-0321-y.

[11]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. 

[12]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96.

[13]

H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Providence (RI): American Mathematical Society Providence, 41 (1995), 174.

[14]

H. L. Smith and X-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[15]

H. ShuL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.  doi: 10.1137/120896463.

[16]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[17]

F- B WangY. 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.

[18]

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.

[19]

W. Wang and X-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[20]

W. Wang and X-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.

[21]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[22]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theoretical Biology, 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001.

[23]

H. Yang and J. Wei, Global behaviours of an in-host viral model with general incidence terms, Appl. Anal., 97 (2018), 2431-2449.  doi: 10.1080/00036811.2017.1376246.

[24]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Disc. Cont. Dyn. Syst. Ser. B., 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.

[25]

H. Yang and J. Wei, Analyzing global stability of a viral model with general incidence rate and cytotoxic T lymphocytes immune response, Nonlinear Dynam., 82 (2015), 713-722.  doi: 10.1007/s11071-015-2189-8.

[26]

X. Yu and X-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Diff. Equ., 261 (2016), 340-372.  doi: 10.1016/j.jde.2016.03.014.

[27]

Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Anal., 15 (2014), 118-139.  doi: 10.1016/j.nonrwa.2013.06.005.

[28]

X-Q. Zhao, Dynamical Systems in Population Biology, New York: Springer, 2003. doi: 10.1007/978-0-387-21761-1.

[29]

X-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012), 787-808.  doi: 10.1007/s00285-011-0482-9.

show all references

References:
[1]

C-M. BraunerD. JollyL. Lorenzi and P. Thiebaut, Heterogeneous viral environment in an HIV spatial model, Disc. Cont. Dyn. Syst. Ser. B., 15 (2011), 545-572.  doi: 10.3934/dcdsb.2011.15.545.

[2]

K. Deimling, Nonlinear Functional Analysis, Berlin: Springer-Verlag, 1988. doi: 10.1007/978-3-662-00547-7.

[3]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, New York, 1993.

[4]

M. LewisJ. Renclawowicz and P. van den Driessche, Traveling waves and spread rates for a West Nile virus model, Bull. Math. Biol., 68 (2006), 3-23.  doi: 10.1007/s11538-005-9018-z.

[5]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.  doi: 10.1007/s11538-010-9503-x.

[6]

Y. Lou and X.-Q. 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.

[7]

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.

[8]

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.

[9]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal., 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.03.002.

[10]

A. MuraseT. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.  doi: 10.1007/s00285-005-0321-y.

[11]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. 

[12]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96.

[13]

H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Providence (RI): American Mathematical Society Providence, 41 (1995), 174.

[14]

H. L. Smith and X-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[15]

H. ShuL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.  doi: 10.1137/120896463.

[16]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[17]

F- B WangY. 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.

[18]

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.

[19]

W. Wang and X-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[20]

W. Wang and X-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.

[21]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[22]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theoretical Biology, 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001.

[23]

H. Yang and J. Wei, Global behaviours of an in-host viral model with general incidence terms, Appl. Anal., 97 (2018), 2431-2449.  doi: 10.1080/00036811.2017.1376246.

[24]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Disc. Cont. Dyn. Syst. Ser. B., 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.

[25]

H. Yang and J. Wei, Analyzing global stability of a viral model with general incidence rate and cytotoxic T lymphocytes immune response, Nonlinear Dynam., 82 (2015), 713-722.  doi: 10.1007/s11071-015-2189-8.

[26]

X. Yu and X-Q. Zhao, A nonlocal spatial model for Lyme disease, J. Diff. Equ., 261 (2016), 340-372.  doi: 10.1016/j.jde.2016.03.014.

[27]

Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Anal., 15 (2014), 118-139.  doi: 10.1016/j.nonrwa.2013.06.005.

[28]

X-Q. Zhao, Dynamical Systems in Population Biology, New York: Springer, 2003. doi: 10.1007/978-0-387-21761-1.

[29]

X-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012), 787-808.  doi: 10.1007/s00285-011-0482-9.

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