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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. [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. [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. [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. [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. [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. [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. [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. [9] M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. [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. [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. [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. [13] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [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. [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. [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. [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. [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. [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. [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. [21] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

show all references

##### References:
 [1] R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003. [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. [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. [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. [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. [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. [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. [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. [9] M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. [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. [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. [12] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. [13] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [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. [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. [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. [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. [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. [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. [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. [21] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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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