# American Institute of Mathematical Sciences

December  2018, 23(10): 4223-4242. doi: 10.3934/dcdsb.2018134

## Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate

 1 Department of Mathematics, Yunnan Normal University, Kunming 650092, China 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Haihong Liu

Received  September 2017 Revised  December 2017 Published  December 2018 Early access  April 2018

The aim of this paper is to study the dynamics of a new chronic HBV infection model that includes spatial diffusion, three time delays and a general incidence function. First, we analyze the well-posedness of the initial value problem of the model in the bounded domain. Then, we define a threshold parameter $R_{0}$ called the basic reproduction number and show that our model admits two possible equilibria, namely the infection-free equilibrium $E_{1}$ as well as the chronic infection equilibrium $E_{2}$. Further, by constructing two appropriate Lyapunov functionals, we prove that $E_{1}$ is globally asymptotically stable when $R_{0}<1$, corresponding to the viruses are cleared and the disease dies out; if $R_{0}>1$, then $E_{1}$ becomes unstable and the equilibrium point $E_{2}$ appears and is globally asymptotically stable, which means that the viruses persist in the host and the infection becomes chronic. An application is provided to confirm the main theoretical results. Additionally, it is worth saying that, our results suggest theoretically useful method to control HBV infection and these results can be applied to a variety of possible incidence functions presented in a series of other papers.

Citation: Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134
##### References:
 [1] WHO, Hepatitis B: Fact sheet: No. 204. 2015. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. [2] X. Chen, L. Min, Y. Zheng, Y. Kuang and Y. Ye, Dynamics of acute hepatitis B virus infection in chimpanzees, Mathematics and Computers in Simulation, 96 (2014), 157-170.  doi: 10.1016/j.matcom.2013.05.003. [3] S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol., 247 (2007), 23-35.  doi: 10.1016/j.jtbi.2007.02.017. [4] A. Elaiw and N. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. RWA, 26 (2015), 161-190.  doi: 10.1016/j.nonrwa.2015.05.007. [5] A. Elaiw and S. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci, 36 (2013), 383-394.  doi: 10.1002/mma.2596. [6] W. Fitzgibbon, Semilinear functional differential equations in Banach spaces, J. Diff. Eqns., 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2. [7] I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, CA, 2000. doi: 10.1016/B978-0-12-294760-5.50029-5. [8] F. L. Guerhier, A. Thermet, S. Guerret, M. Chevallier, C. Jamard, C. S. Gibbs, C. Trépo, L. Cova and F. Zoulim, Antiviral effect of adefovir in combination with a DNA vaccine in the duck hepatitis B virus infection model, J. Hepatol., 38 (2003), 328-334.  doi: 10.1016/S0168-8278(02)00425-7. [9] H. Guo, D. Jiang, T. Zhou, A. Cuconati, T. M. Block and J. T. Guo, Characterization of the intracellular deproteinized relaxed circular DNA of hepatitis B virus: An intermediate of covalently closed circular DNA formation, Journal of Virology, 81 (2007), 12472-12484.  doi: 10.1128/JVI.01123-07. [10] J. Hale and S. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 69 (2015), 31-40.  doi: 10.1016/j.camwa.2014.11.010. [12] K. Hattaf and N. Yousfi, Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response, Comp. Appl. Math., 34 (2015), 807-818.  doi: 10.1007/s40314-014-0143-x. [13] K. Hattaf, N. Yousfi and A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190.  doi: 10.1007/s12591-013-0167-5. [14] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. RWA, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/BFb0089647. [16] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. Math. Biol., 60 (2010), 573-590.  doi: 10.1007/s00285-009-0278-3. [17] R. Hirsch, D. Loeb, J. Pollack and D. Ganem, Cis-acting sequences required for encapsidation of duck hepatitis B virus pregenomic RNA, J. Virol., 65 (1991), 3309-3316. [18] T. B. Lentz and D. D. Loeb, Development of cell cultures that express hepatitis B virus to high levels and accumulate cccDNA, J. Virol. Methods, 169 (2010), 52-60.  doi: 10.1016/j.jviromet.2010.06.015. [19] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: rational combination chemotherapy based on viral kinetic and animal model studies, Antiviral Research, 55 (2002), 381-396.  doi: 10.1016/S0166-3542(02)00071-2. [20] D. Li and W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691.  doi: 10.1016/j.jmaa.2007.02.006. [21] K. Manna and S. Chakrabarty, Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids, Comp. Appl. Math., (2015), 1-12.  doi: 10.1007/s40314-015-0242-3. [22] K. Manna and S. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: Modeling and analysis, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 383-395.  doi: 10.1016/j.cnsns.2014.08.036. [23] C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. RWA, 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003. [24] L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic model of virus infection with application to HBV infection, Rocky Mountain J. Math., 38 (2008), 1573-1585.  doi: 10.1216/RMJ-2008-38-5-1573. [25] J. Murray, R. Purcell and S. Wieland, The half-life of hepatitis B virions, Hepatology, 44 (2006), 1117-1121.  doi: 10.1002/hep.21364. [26] J. Murray, S. Wieland, R. Purcell and F. Chisari, Dynamics of hepatitis B virus clearance in chimpanzees, Proc. Natl. Acad. Sci., 102 (2005), 17780-17785.  doi: 10.1073/pnas.0508913102. [27] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci., 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [28] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967. doi: 10.1007/978-1-4612-5282-5. [29] R. M. Ribeirom, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microb. Infect., 4 (2002), 829-835.  doi: 10.1016/S1286-4579(02)01603-9. [30] C. Seeger and W. Mason, Hepatitis B virus biology, Microbiol. Mol. Biol. Rev., 64 (2000), 51-68.  doi: 10.1128/MMBR.64.1.51-68.2000. [31] K. Simon, V. Lingappa and D. Ganem, Secreted hepatitis B surface antigen polypeptides are derived from a transmembrane precursor, J. Cell Biol., 107 (1988), 2163-2168.  doi: 10.1083/jcb.107.6.2163. [32] J. Summers, A. Connell and I. Millman, Genome of hepatitis B virus: Restriction enzyme cleavage and structure of DNA extracted from Dane particles, Proc. Natl. Acad. Sci., 72 (1975), 4597-4601.  doi: 10.1073/pnas.72.11.4597. [33] C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3. [34] K. Wang, A. Fan and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal. RWA, 11 (2010), 3131-3138.  doi: 10.1016/j.nonrwa.2009.11.008. [35] F. 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. [36] Y. Wang and X. Liu, Dynamical behaviors of a delayed HBV infection model with logistic hepatocyte growth, cure rate and CTL immune response, JPN J. Ind. Appl. Math., 32 (2015), 575-593.  doi: 10.1007/s13160-015-0184-6. [37] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [38] S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, Electron. J. Qual. Theory Differ. Equ., 9 (2012), 1-9. [39] Y. Yang and Y. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930.  doi: 10.1016/j.camwa.2016.01.009. [40] 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. [41] L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China, J. Theor. Biol., 262 (2010), 330-338.  doi: 10.1016/j.jtbi.2009.09.035.

show all references

##### References:
 [1] WHO, Hepatitis B: Fact sheet: No. 204. 2015. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/. [2] X. Chen, L. Min, Y. Zheng, Y. Kuang and Y. Ye, Dynamics of acute hepatitis B virus infection in chimpanzees, Mathematics and Computers in Simulation, 96 (2014), 157-170.  doi: 10.1016/j.matcom.2013.05.003. [3] S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol., 247 (2007), 23-35.  doi: 10.1016/j.jtbi.2007.02.017. [4] A. Elaiw and N. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. RWA, 26 (2015), 161-190.  doi: 10.1016/j.nonrwa.2015.05.007. [5] A. Elaiw and S. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci, 36 (2013), 383-394.  doi: 10.1002/mma.2596. [6] W. Fitzgibbon, Semilinear functional differential equations in Banach spaces, J. Diff. Eqns., 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2. [7] I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, CA, 2000. doi: 10.1016/B978-0-12-294760-5.50029-5. [8] F. L. Guerhier, A. Thermet, S. Guerret, M. Chevallier, C. Jamard, C. S. Gibbs, C. Trépo, L. Cova and F. Zoulim, Antiviral effect of adefovir in combination with a DNA vaccine in the duck hepatitis B virus infection model, J. Hepatol., 38 (2003), 328-334.  doi: 10.1016/S0168-8278(02)00425-7. [9] H. Guo, D. Jiang, T. Zhou, A. Cuconati, T. M. Block and J. T. Guo, Characterization of the intracellular deproteinized relaxed circular DNA of hepatitis B virus: An intermediate of covalently closed circular DNA formation, Journal of Virology, 81 (2007), 12472-12484.  doi: 10.1128/JVI.01123-07. [10] J. Hale and S. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 69 (2015), 31-40.  doi: 10.1016/j.camwa.2014.11.010. [12] K. Hattaf and N. Yousfi, Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response, Comp. Appl. Math., 34 (2015), 807-818.  doi: 10.1007/s40314-014-0143-x. [13] K. Hattaf, N. Yousfi and A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190.  doi: 10.1007/s12591-013-0167-5. [14] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. RWA, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/BFb0089647. [16] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. Math. Biol., 60 (2010), 573-590.  doi: 10.1007/s00285-009-0278-3. [17] R. Hirsch, D. Loeb, J. Pollack and D. Ganem, Cis-acting sequences required for encapsidation of duck hepatitis B virus pregenomic RNA, J. Virol., 65 (1991), 3309-3316. [18] T. B. Lentz and D. D. Loeb, Development of cell cultures that express hepatitis B virus to high levels and accumulate cccDNA, J. Virol. Methods, 169 (2010), 52-60.  doi: 10.1016/j.jviromet.2010.06.015. [19] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: rational combination chemotherapy based on viral kinetic and animal model studies, Antiviral Research, 55 (2002), 381-396.  doi: 10.1016/S0166-3542(02)00071-2. [20] D. Li and W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691.  doi: 10.1016/j.jmaa.2007.02.006. [21] K. Manna and S. Chakrabarty, Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids, Comp. Appl. Math., (2015), 1-12.  doi: 10.1007/s40314-015-0242-3. [22] K. Manna and S. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: Modeling and analysis, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 383-395.  doi: 10.1016/j.cnsns.2014.08.036. [23] C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. RWA, 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003. [24] L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic model of virus infection with application to HBV infection, Rocky Mountain J. Math., 38 (2008), 1573-1585.  doi: 10.1216/RMJ-2008-38-5-1573. [25] J. Murray, R. Purcell and S. Wieland, The half-life of hepatitis B virions, Hepatology, 44 (2006), 1117-1121.  doi: 10.1002/hep.21364. [26] J. Murray, S. Wieland, R. Purcell and F. Chisari, Dynamics of hepatitis B virus clearance in chimpanzees, Proc. Natl. Acad. Sci., 102 (2005), 17780-17785.  doi: 10.1073/pnas.0508913102. [27] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci., 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [28] M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967. doi: 10.1007/978-1-4612-5282-5. [29] R. M. Ribeirom, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microb. Infect., 4 (2002), 829-835.  doi: 10.1016/S1286-4579(02)01603-9. [30] C. Seeger and W. Mason, Hepatitis B virus biology, Microbiol. Mol. Biol. Rev., 64 (2000), 51-68.  doi: 10.1128/MMBR.64.1.51-68.2000. [31] K. Simon, V. Lingappa and D. Ganem, Secreted hepatitis B surface antigen polypeptides are derived from a transmembrane precursor, J. Cell Biol., 107 (1988), 2163-2168.  doi: 10.1083/jcb.107.6.2163. [32] J. Summers, A. Connell and I. Millman, Genome of hepatitis B virus: Restriction enzyme cleavage and structure of DNA extracted from Dane particles, Proc. Natl. Acad. Sci., 72 (1975), 4597-4601.  doi: 10.1073/pnas.72.11.4597. [33] C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3. [34] K. Wang, A. Fan and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal. RWA, 11 (2010), 3131-3138.  doi: 10.1016/j.nonrwa.2009.11.008. [35] F. 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. [36] Y. Wang and X. Liu, Dynamical behaviors of a delayed HBV infection model with logistic hepatocyte growth, cure rate and CTL immune response, JPN J. Ind. Appl. Math., 32 (2015), 575-593.  doi: 10.1007/s13160-015-0184-6. [37] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [38] S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, Electron. J. Qual. Theory Differ. Equ., 9 (2012), 1-9. [39] Y. Yang and Y. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930.  doi: 10.1016/j.camwa.2016.01.009. [40] 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. [41] L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China, J. Theor. Biol., 262 (2010), 330-338.  doi: 10.1016/j.jtbi.2009.09.035.
Schematic view of the replication process of HBV
Diagrammatic representation of the mathematical model for HBV infection
The numerical approximations of system (21)-(23) with parameters $s = 2.6\times10^{7}$, $\mu = 0.01$, $\delta = 0.053$, $a = 150$, $\beta = 0.87$, $c = 3.8$, $d_{v} = 0.01$, $b_{1} = b_{2} = 0.01$, $\alpha_{1} = 0.2$, $\alpha_{2} = 0.28$, $\alpha_{3} = 0.1$, $\tau_{1} = 10$, $\tau_{2} = 0$, $\tau_{3} = 0$ and $k = 3\times10^{-5}$, showing that solution trajectories converge to the infection-free equilibrium $E_{1}: (H_{1}, I_{1}, D_{1}, V_{1}) = (2.6\times10^{9}, 0, 0, 0)$
The numerical approximations of system (21)-(23) with parameters $s = 2.6\times10^{7}$, $\mu = 0.01$, $\delta = 0.053$, $a = 150$, $\beta = 0.87$, $c = 3.8$, $d_{v} = 0.01$, $b_{1} = b_{2} = 0.01$, $\alpha_{1} = 0.2$, $\alpha_{2} = 0.28$, $\alpha_{3} = 0.1$, $\tau_{1} = 10$, $\tau_{2} = 0$, $\tau_{3} = 0$ and $k = 1.67\times10^{-4}$, showing that solution trajectories converge to the chronic infection equilibrium $E_{2}: (H_{2}, I_{2}, D_{2}, V_{2}) = (1.61\times10^{9}, 2.53\times10^{7}, 4.12\times10^{9}, 9.43\times10^{8})$
The graphs of the basic reproduction number $R_{0}$ in terms of some parameters: (a) $R_{0}$ in terms of $\alpha_{1}$ and $\alpha_{2}$, (b) $R_{0}$ as a function of $\alpha_{1}$ and $\alpha_{3}$, and (c) $R_{0}$ in terms of $\alpha_{2}$ and $\alpha_{3}$. Here, $s = 2.6\times10^{7}$, $\mu = 0.01$, $\delta = 0.053$, $a = 150$, $\beta = 0.87$, $c = 3.8$, $b_{1} = 0.01$, $\tau_{1} = 5.8$, $\tau_{2} = 6$, $\tau_{3} = 4$ and $k = 2.4\times10^{-3}$
The plots of the basic reproduction number $R_{0}$ as a function of three delays $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$. Here, $s = 2.6\times10^{7}$, $\mu = 0.01$, $\delta = 0.053$, $a = 150$, $\beta = 0.87$, $c = 3.8$, $b_{1} = 0.01$, $\alpha_{1} = 0.2$, $\alpha_{2} = 0.28$, $\alpha_{3} = 0.1$ and $k = 2.4\times10^{-3}$. (a) $\tau_{3} = 4$, (b) $\tau_{2} = 6$, and (c) $\tau_{1} = 5.8$
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