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On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies
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 |
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.
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. |
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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.
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[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. |
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. |
[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. |




Parameters | Descriptions |
Generation of uninfected cells | |
Infection rate | |
Pyroptosis effect of inflammatory cytokines on uninfected cells | |
Death rate due to pyroptosis | |
Production rate of inflammatory cytokines | |
Production rate of virus | |
Death rate of uninfected cells | |
Death rate of infected cells | |
Death rate of inflammatory cytokines | |
Death rate of virus | |
Diffusion rate of cells (uninfected cells and infected cells) | |
Diffusion rate of inflammatory cytokines | |
Diffusion rate of virus | |
Rate of the inhibitory effect on virus | |
Rate of the inhibitory effect on inflammatory cytokines |
Parameters | Descriptions |
Generation of uninfected cells | |
Infection rate | |
Pyroptosis effect of inflammatory cytokines on uninfected cells | |
Death rate due to pyroptosis | |
Production rate of inflammatory cytokines | |
Production rate of virus | |
Death rate of uninfected cells | |
Death rate of infected cells | |
Death rate of inflammatory cytokines | |
Death rate of virus | |
Diffusion rate of cells (uninfected cells and infected cells) | |
Diffusion rate of inflammatory cytokines | |
Diffusion rate of virus | |
Rate of the inhibitory effect on virus | |
Rate of the inhibitory effect on inflammatory cytokines |
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