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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

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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

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

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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

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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

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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

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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

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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

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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
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