• 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
    On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions
doi: 10.3934/dcdsb.2020271

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

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.

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

[5]

O. DiekmannJ. 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. DoceulM. HollinsheadL. 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. HuangW. 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. RenY. TianL. 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. TangZ. 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. 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.  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. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[5]

O. DiekmannJ. 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. DoceulM. HollinsheadL. 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. HuangW. 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. RenY. TianL. 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. TangZ. 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. 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.  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 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 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 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
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
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]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[2]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[3]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[4]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

[5]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[6]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[7]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198

[8]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067

[9]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194

[10]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[11]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[12]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042

[13]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[14]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[15]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[16]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1

[17]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

[18]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[19]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[20]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]