A DELAY-DIFFERENTIAL EQUATION MODEL OF HIV INFECTION OF CD4+ T-CELLS

Title & Authors
A DELAY-DIFFERENTIAL EQUATION MODEL OF HIV INFECTION OF CD4+ T-CELLS
SONG, XINYU; CHENG, SHUHAN;

Abstract
In this paper, we introduce a discrete time to the model to describe the time between infection of a CD4$\small{^{+}}$ T-cells, and the emission of viral particles on a cellular level. We study the effect of the time delay on the stability of the endemically infected equilibrium, criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. We also obtain the condition for existence of an orbitally asymptotically stable periodic solution.
Keywords
Hopf bifurcation;Periodic solution;Stability;HIV;HCV;
Language
English
Cited by
1.
STABILITY PROPERTIES OF A DELAYED VIRAL INFECTION MODEL WITH LYTIC IMMUNE RESPONSE,;;;

Journal of applied mathematics & informatics, 2011. vol.29. 5_6, pp.1117-1127
2.
A DELAY DYNAMIC MODEL FOR HIV INFECTED IMMUNE RESPONSE,;;;

Journal of applied mathematics & informatics, 2015. vol.33. 5_6, pp.559-578
1.
Stability and Hopf bifurcation in a delayed model for HIV infection of CD4+T cells, Chaos, Solitons & Fractals, 2009, 42, 1, 1
2.
Stability and Hopf bifurcation for a viral infection model with delayed non-lytic immune response, Journal of Applied Mathematics and Computing, 2010, 33, 1-2, 251
3.
Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 2014, 38, 21-22, 5067
4.
Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Analysis: Real World Applications, 2011, 12, 4, 2058
5.
Qualitative Analysis of Delayed SIR Epidemic Model with a Saturated Incidence Rate, International Journal of Differential Equations, 2012, 2012, 1
6.
Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, Journal of Mathematical Analysis and Applications, 2011, 373, 2, 345
7.
A delay differential equation model of HIV infection of CD4+ T-cells with cure rate, Journal of Applied Mathematics and Computing, 2009, 31, 1-2, 51
8.
Effect of Time Delay on Spatial Patterns in a Airal Infection Model with Diffusion, Mathematical Modelling and Analysis, 2016, 21, 2, 143
9.
A differential equation model of HIV infection of CD4+ T-cells with cure rate, Journal of Mathematical Analysis and Applications, 2008, 342, 2, 1342
10.
A Delayed Model for HIV Infection Incorporating Intracellular Delay, International Journal of Applied and Computational Mathematics, 2016
11.
Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay, Applied Mathematical Modelling, 2014, 38, 21-22, 5047
12.
Dynamic behaviors of a delayed HIV model with stage-structure, Communications in Nonlinear Science and Numerical Simulation, 2012, 17, 12, 4753
13.
Stability analysis of delay seirepidemic model, International Journal of ADVANCED AND APPLIED SCIENCES, 2016, 3, 7, 46
14.
Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Applied Mathematical Modelling, 2010, 34, 6, 1511
15.
Stability and Hopf bifurcation on a model for HIV infection of CD4+ T cells with delay, Chaos, Solitons & Fractals, 2009, 42, 3, 1838
16.
Analysis of stability and Hopf bifurcation for a delay-differential equation model of HIV infection of CD4+ T-cells, Chaos, Solitons & Fractals, 2008, 38, 2, 447
17.
Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Applied Mathematics and Computation, 2008, 199, 1, 23
18.
Analysis of the dynamics of a delayed HIV pathogenesis model, Journal of Computational and Applied Mathematics, 2010, 234, 2, 461
19.
DYNAMICS OF A NON-AUTONOMOUS HIV-1 INFECTION MODEL WITH DELAYS, International Journal of Biomathematics, 2013, 06, 05, 1350030
20.
Global stability analysis of HIV-1 infection model with three time delays, Journal of Applied Mathematics and Computing, 2015, 48, 1-2, 293
References
1.
D. Ho, A. Neumann, A. Perelson, W. Chen, J. Leonard, and M. Markowitz, Rapid turnover of plasma virions and \$CD4^+\$ lymphocytes in HIV-1 infection,Nature 373 (1995), 123-126

2.
X. Wei, S. Ghosh, M. Taylor, V. Johnson, E. Emini, P. Deutsch, J. Lifson, S. Bonhoeffer, M. Nowak, B. Hahn, S. Saag, and G. Shaw, Viral dynamics in human immunodeficiency virus type 1 infection, Nature 373 (1995), 117

3.
A. Perelson, A. Neumann, M. Markowitz, J. Leonard, and D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582

4.
A. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz, and D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387 (1997), 188

5.
A. Neumann, N. Lam, H. Dahari, D. Gretch, T. Wiley, T. Layden, and A. Perel- son, Hepatitis C viral dynamics in vivo and antiviral efficacy of the interferon-ff therapy, Science 282 (1998), 103-107

6.
A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), no. 1, 3-44

7.
V. Herz, S. Bonhoeffer, R. Anderson, R. May, and M. Nowak, Viral dynamics in vivo: limitations on estimations on intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996), 7247-7251

8.
Z. Grossman, M. Feinberg, V. Kuznetsov, D. Dimitrov, and W. Paul, HIV infection: how effective is drug combination treatment, Immunol. Today 19 (1998), 528

9.
Z. Grossman, M. Polis, M. Feinberg, I. Levi, S. Jankelevich, R. Yarchoan, J. Boon, F. de Wolf, J. Lange, J. Goudsmit, D. Dimitrov, and W. Paul, Ongoing HIV dissemination during HAART, Nat. Med. 5 (1999), 1099

10.
J. Mittler, B. Sulzer, A. Neumann, and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998), 143

11.
J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol. 16 (1999), 29

12.
P. Nelson, J. Murray, and A. Perelson,A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), 201

13.
J. Mittler, M. Markowitz, D. Ho, and A. Perelson,Refined estimates for HIV-1 clearance rate and intracellular delay, AIDS, 13 (1999), 1415

14.
P. Nelson and A. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci. 179 (2002), 73-94

15.
H. I. Freedman and V. Sree Hari Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45 (1983), 991-1003

16.
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977

17.
J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), 388-396

18.
H. Smith, Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (1987), 420-442

19.
K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dordrecht/Norwell, MA

20.
Hsiu-Rong Zhu and H. Smith, Stable periodic orbits for a class three dimentional competitive systems, J. Differential Equations 110 (1994), 143-156

21.
E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002), 1144-1165