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March  2018, 23(2): 861-885. doi: 10.3934/dcdsb.2018046

## Hopf bifurcation of an age-structured virus infection model

 1 Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box:16315-1618, Tehran, Iran 2 Department of Mathematics, University of Louisiana, Lafayette, LA, USA 3 Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran

* Corresponding author

The first author is supported by The Department of Iranian Student Affairs

Received  December 2016 Revised  September 2017 Published  December 2017

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number $R_{0}$. In particular, when $R_{0}≤1$, the infection free equilibrium is globally asymptotically stable, and conversely if $R_{0}> 1$, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.

Citation: Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
##### References:
 [1] R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.Google Scholar [2] I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214. Google Scholar [3] B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183.Google Scholar [4] C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372. doi: 10.1016/j.nonrwa.2014.10.004. Google Scholar [5] C. J. Browne, Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909. doi: 10.3934/mbe.2016022. Google Scholar [6] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999. Google Scholar [7] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar [8] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. Google Scholar [9] M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. Google Scholar [10] P. Dustin and D. Wodarz, Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28. doi: 10.1016/j.mbs.2015.03.001. Google Scholar [11] J. K. Hale, J. P. Lasalle and M. Slemrod, Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191. doi: 10.1016/0022-247X(72)90233-8. Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. Google Scholar [13] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. Google Scholar [14] S. Hongying, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302. doi: 10.1137/120896463. Google Scholar [15] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58. doi: 10.1137/110826588. Google Scholar [16] L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204.Google Scholar [17] D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390. doi: 10.1007/BF02458312. Google Scholar [18] 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 [19] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng, 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. Google Scholar [20] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267. Google Scholar [21] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. Google Scholar [22] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582. Google Scholar [23] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. Google Scholar [24] H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar [25] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM. J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. Google Scholar [26] Libin Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. Google Scholar [27] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. Google Scholar [28] R. V. Ursache, Y. E. Thomassen, G. Van Eikenhorst, P. J. T. Verheijen and W. A. M. Bakker, Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555. doi: 10.1007/s00449-014-1294-2. Google Scholar [29] Y. Wang, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3. Google Scholar [30] G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection, A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30. doi: 10.1080/17513758.2015.1090632. Google Scholar [31] Y. Yang, S. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877. doi: 10.3934/mbe.2015.12.859. Google Scholar [32] J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222. doi: 10.1016/0092-8674(90)90802-L. Google Scholar

show all references

##### References:
 [1] R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.Google Scholar [2] I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214. Google Scholar [3] B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183.Google Scholar [4] C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372. doi: 10.1016/j.nonrwa.2014.10.004. Google Scholar [5] C. J. Browne, Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909. doi: 10.3934/mbe.2016022. Google Scholar [6] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999. Google Scholar [7] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar [8] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. Google Scholar [9] M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. Google Scholar [10] P. Dustin and D. Wodarz, Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28. doi: 10.1016/j.mbs.2015.03.001. Google Scholar [11] J. K. Hale, J. P. Lasalle and M. Slemrod, Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191. doi: 10.1016/0022-247X(72)90233-8. Google Scholar [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. Google Scholar [13] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. Google Scholar [14] S. Hongying, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302. doi: 10.1137/120896463. Google Scholar [15] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58. doi: 10.1137/110826588. Google Scholar [16] L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204.Google Scholar [17] D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390. doi: 10.1007/BF02458312. Google Scholar [18] 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 [19] C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng, 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. Google Scholar [20] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267. Google Scholar [21] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. Google Scholar [22] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582. Google Scholar [23] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. Google Scholar [24] H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar [25] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM. J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. Google Scholar [26] Libin Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. Google Scholar [27] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. Google Scholar [28] R. V. Ursache, Y. E. Thomassen, G. Van Eikenhorst, P. J. T. Verheijen and W. A. M. Bakker, Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555. doi: 10.1007/s00449-014-1294-2. Google Scholar [29] Y. Wang, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3. Google Scholar [30] G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection, A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30. doi: 10.1080/17513758.2015.1090632. Google Scholar [31] Y. Yang, S. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877. doi: 10.3934/mbe.2015.12.859. Google Scholar [32] J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222. doi: 10.1016/0092-8674(90)90802-L. Google Scholar
A numerical solution of system (40) tends to the infection-free equilibrium $E_0$, as time tends to infinity, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = [0.038, 0.0045, 4.6137, 1, 0.093, 0.4, 0.028]$. In this case $R_0 = 0.6518$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$
A numerical solution of system (40) approaches to $\overline{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] =$ $[0.03285, 0.01, 4.6137, 1.3, 0.045, 0.1, 0.0351]$. In this case $R_0 = 16.0999$, $\bar{E} = (\bar{T},\bar{V},\bar{I}) =[0.2212, 3.1275, 0.1971]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space.
A numerical solution of system (40) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] =$ $[0.03285, 0.01, 4.6137, 1.3, 0.03, 0.1, 0.0351]$. In this case $R_0 = 22.4437$, $(\bar{T},\bar{V},\bar{I}) =[ 0.1115, 3.2056, 0.1018]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$
A numerical solution of system (41)-(44) tends to the DFE, as time tends to infinity, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] =$ $[.1,100000,0.0000005,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =0.9905$ and $(\bar{T},\bar{V},\bar{I}) =[ 10^5, 0 , 0]$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
A numerical solution of system (41)-(44) tends to the $\bar{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] =$ $[.1,100000,0.0000008,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =1.5812$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 6.3209\times 10^4, 4.5989\times 10^4, 7.4321\times 10^3]$. The probability of re-infection of infected cells during eclipse phase (during age $0\leq a \leq \tau$) calculated at $\bar{E}$ is $\pi(\tau) = 0.23$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
A numerical solution of system (41)-(44) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] =$ $[1,100000, 0.000005,200, 13, 0.000001, 2, 0.05, 0.7]$. In this case $R_0 =9.5750$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 1.0119\times 10^4, 1.7976\times 10^5, 2.9066\times 10^4]$. The probability of re-infection of infected cells during eclipse phase calculated at $\bar{E}$ is $\pi(\tau) = 0.2882$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
Parameter definition and values from literatures.
 Parameter Value Description Reference $e$ day$^{-1}$ Maximum proliferation rate See text $g$ 0.008 day$^{-1}$ Death rate of uninfected cells [27] $T_{\text{max}}$ mm$^{-3}$ Density of $T$ cell at which proliferation shouts off See text $k$ $5 \times 10^{-7}$ ml virion day$^{-1}$ Infection rate of target cells by virus [27] $\delta$ 0.8 day$^{-1}$ Death rate of infected cells [32] p Varied Virion production rate of an infected cell See text $d$ 3 day$^{-1}$ clearance rate of free virus [22] $\gamma$ day$^{-1}$ Reinfection rate of infected cells by virus See text
 Parameter Value Description Reference $e$ day$^{-1}$ Maximum proliferation rate See text $g$ 0.008 day$^{-1}$ Death rate of uninfected cells [27] $T_{\text{max}}$ mm$^{-3}$ Density of $T$ cell at which proliferation shouts off See text $k$ $5 \times 10^{-7}$ ml virion day$^{-1}$ Infection rate of target cells by virus [27] $\delta$ 0.8 day$^{-1}$ Death rate of infected cells [32] p Varied Virion production rate of an infected cell See text $d$ 3 day$^{-1}$ clearance rate of free virus [22] $\gamma$ day$^{-1}$ Reinfection rate of infected cells by virus See text
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