Hopf bifurcation of an age-structured virus infection model

Previous Article
Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method
DCDS-B Home
This Issue
Next Article
Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis

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

Hopf bifurcation of an age-structured virus infection model

Hossein Mohebbi ^1,, , Azim Aminataei ^1, , Cameron J. Browne ^2, and Mohammad Reza Razvan ^3,

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.

Keywords: Mathematical modeling, global analysis, Lyaponov function, poliovirus, Hopf bifurcation.

Mathematics Subject Classification: Primary: 37N25, 37B25; Secondary: 92B05.

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.

Figure 1. 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 $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. 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.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. 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 $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. 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.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. 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.

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. 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.

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. 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

Table Options

Download as excel

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

[1]	Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[2]	Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347
[3]	Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[4]	Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[5]	Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[6]	Bashar Ibrahim. Mathematical analysis and modeling of DNA segregation mechanisms. Mathematical Biosciences & Engineering, 2018, 15 (2) : 429-440. doi: 10.3934/mbe.2018019
[7]	Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[8]	Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[9]	John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[10]	Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[11]	Loïc Louison, Abdennebi Omrane, Harry Ozier-Lafontaine, Delphine Picart. Modeling plant nutrient uptake: Mathematical analysis and optimal control. Evolution Equations & Control Theory, 2015, 4 (2) : 193-203. doi: 10.3934/eect.2015.4.193
[12]	Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[13]	Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[14]	Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[15]	Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[16]	R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[17]	Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[18]	Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[19]	Yuting Ding, Jinli Xu, Jun Cao, Dongyan Zhang. Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 943-958. doi: 10.3934/dcdss.2017049
[20]	Arni S. R. Srinivasa Rao, Kurien Thomas, Kurapati Sudhakar, Philip K. Maini. HIV/AIDS epidemic in India and predicting the impact of the national response: Mathematical modeling and analysis. Mathematical Biosciences & Engineering, 2009, 6 (4) : 779-813. doi: 10.3934/mbe.2009.6.779

2017 Impact Factor: 0.972

Tools

Metrics

Tools

Article outline

Figures and Tables

2017 Impact Factor: 0.972

Tools

Figures and Tables

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences