American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global phase portraits and bifurcation diagrams for reversible equivariant hamiltonian systems of linear plus quartic homogeneous polynomials
  • DCDS-B Home
  • This Issue
  • Next Article
    A nonisothermal thermodynamical model of liquid-vapor interaction with metastability
doi: 10.3934/dcdsb.2020271

Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion

Wei Wang 1,, , Wanbiao Ma 2, and  Xiulan Lai 3,

1. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2. 

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
3. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author: Wei Wang

Received  January 2020 Revised  July 2020 Published  September 2020

Fund Project: This work is supported by the NNSF of China (11901360) to W. Wang and supported by the National Key R-D Program of China (No. 2017YFF0207401) and the NNSF of China (No. 11971055) W. Ma

In this paper, we study the global dynamics of a viral infection model with spatial heterogeneity and nonlinear diffusion. For the spatially heterogeneous case, we first derive some properties of the basic reproduction number $ R_0 $. Then for the auxiliary system with quasilinear diffusion, we establish the comparison principle under some appropriate conditions. Some sufficient conditions are derived to ensure the global stability of the virus-free steady state. We also show the existence of the positive non-constant steady state and the persistence of virus. For the spatially homogeneous case, we show that $ R_0 $ is the only determinant of the global dynamics when the derivative of the function $ g $ with respect to $ V $ (the rate of change of infected cells for the repulsion effect) is small enough. Our simulation results reveal that pyroptosis and Beddington-DeAngelis functional response function play a crucial role in the controlling of the spreading speed of virus, which are some new phenomena not presented in the existing literature.

Keywords: Nonlinear diffusion, spatial heterogeneity, quasilinearly parabolic system, basic reproduction number, global dynamics.
Mathematics Subject Classification: Primary: 34D20, 35C07; Secondary: 35Q92, 92D3.
Citation: Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020271
References:
[1]

H. Amann, Dynamical 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: Function spaces, differential operators and nonlinear analysis, (Friedrichroda, 1992), vol 133. Teubner-Texte zur Mathematik. Teubner, Stuttgart, 1993, pp. 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[4]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models of infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[6]

V. Doceul, M. 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

[7]

G. Huang, W. Ma and T. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[8]

X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

[9]

H. Li and M. Ma, Global dynamics of a virus infection model with repulsive effect, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4783-4797.  doi: 10.3934/dcdsb.2019030.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Anal., 24 (2004), 817-831.  doi: 10.1111/j.0272-4332.2004.00481.x.  Google Scholar

[13]

S. Pankavich and C. Parkinson, Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1237-1257.  doi: 10.3934/dcdsb.2016.21.1237.  Google Scholar

[14]

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

[15]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

S. Tang, Z. Teng and H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Comput. Math. Appl., 78 (2019), 786-806.  doi: 10.1016/j.camwa.2019.03.004.  Google Scholar

[19]

F.-B. Wang, Y. 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.  Google Scholar

[20]

W. Wang, W. Ma and Z. Feng, Complex dynamics of a time periodic nonlocal and time-delayed model of reaction-diffusion equations for modelling CD4+ T cells decline, J. Comput. Appl. Math., 367 (2020), 112430, 29 pp. doi: 10.1016/j.cam.2019.112430.  Google Scholar

[21]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[22]

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

[23]

W. 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

[24]

W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: A nonlocal spatial mathematical model, Bull. Math. Biol., 80 (2018), 540-582.  doi: 10.1007/s11538-017-0389-8.  Google Scholar

[25]

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

[26]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edn. CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[27]

G. Zhao and S. Ruan, Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954-1980.  doi: 10.1137/17M1144106.  Google Scholar

show all references

References:
[1]

H. Amann, Dynamical 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: Function spaces, differential operators and nonlinear analysis, (Friedrichroda, 1992), vol 133. Teubner-Texte zur Mathematik. Teubner, Stuttgart, 1993, pp. 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[3]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[4]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models of infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[6]

V. Doceul, M. 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

[7]

G. Huang, W. Ma and T. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.  doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[8]

X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, Bull. Math. Biol., 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

[9]

H. Li and M. Ma, Global dynamics of a virus infection model with repulsive effect, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4783-4797.  doi: 10.3934/dcdsb.2019030.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Anal., 24 (2004), 817-831.  doi: 10.1111/j.0272-4332.2004.00481.x.  Google Scholar

[13]

S. Pankavich and C. Parkinson, Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1237-1257.  doi: 10.3934/dcdsb.2016.21.1237.  Google Scholar

[14]

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

[15]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

S. Tang, Z. Teng and H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Comput. Math. Appl., 78 (2019), 786-806.  doi: 10.1016/j.camwa.2019.03.004.  Google Scholar

[19]

F.-B. Wang, Y. 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.  Google Scholar

[20]

W. Wang, W. Ma and Z. Feng, Complex dynamics of a time periodic nonlocal and time-delayed model of reaction-diffusion equations for modelling CD4+ T cells decline, J. Comput. Appl. Math., 367 (2020), 112430, 29 pp. doi: 10.1016/j.cam.2019.112430.  Google Scholar

[21]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[22]

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

[23]

W. 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

[24]

W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: A nonlocal spatial mathematical model, Bull. Math. Biol., 80 (2018), 540-582.  doi: 10.1007/s11538-017-0389-8.  Google Scholar

[25]

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

[26]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edn. CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[27]

G. Zhao and S. Ruan, Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954-1980.  doi: 10.1137/17M1144106.  Google Scholar

Figure 1.  a. Initial distribution $ \omega_0(x) $. b. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. c. The contour of b
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Summary of model parameters
Parameters Descriptions
$ \xi(x) $ Generation of uninfected cells
$ \beta(x) $ Infection rate
$ q(x) $ Pyroptosis effect of inflammatory cytokines on uninfected cells
$ \alpha_1(x) $ Death rate due to pyroptosis
$ \alpha_2(x) $ Production rate of inflammatory cytokines
$ k(x) $ Production rate of virus
$ d_U(x) $ Death rate of uninfected cells
$ d_V(x) $ Death rate of infected cells
$ d_M(x) $ Death rate of inflammatory cytokines
$ d_{\omega}(x) $ Death rate of virus
$ D_0 $ Diffusion rate of cells (uninfected cells and infected cells)
$ D_1 $ Diffusion rate of inflammatory cytokines
$ D_2 $ Diffusion rate of virus
$ a $ Rate of the inhibitory effect on virus
$ b $ Rate of the inhibitory effect on inflammatory cytokines
Table Options

Download as excel

Parameters Descriptions
$ \xi(x) $ Generation of uninfected cells
$ \beta(x) $ Infection rate
$ q(x) $ Pyroptosis effect of inflammatory cytokines on uninfected cells
$ \alpha_1(x) $ Death rate due to pyroptosis
$ \alpha_2(x) $ Production rate of inflammatory cytokines
$ k(x) $ Production rate of virus
$ d_U(x) $ Death rate of uninfected cells
$ d_V(x) $ Death rate of infected cells
$ d_M(x) $ Death rate of inflammatory cytokines
$ d_{\omega}(x) $ Death rate of virus
$ D_0 $ Diffusion rate of cells (uninfected cells and infected cells)
$ D_1 $ Diffusion rate of inflammatory cytokines
$ D_2 $ Diffusion rate of virus
$ a $ Rate of the inhibitory effect on virus
$ b $ Rate of the inhibitory effect on inflammatory cytokines
[1]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[3]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[4]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[5]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[6]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[7]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
[8]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082
[9]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[10]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[11]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[12]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[13]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[14]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426
[15]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[16]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[17]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267
[18]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268
[19]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[20]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences