# American Institute of Mathematical Sciences

July  2021, 26(7): 3989-4011. 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  July 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. 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. [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. [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. [4] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] M. H. Protter and H. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] H. Amann, Dynamical 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: 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. [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. [4] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] M. H. Protter and H. Weinberger, Maximum Principles in Differential Equations, Spring-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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
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
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
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
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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