GLOBAL STABILITY OF THE VIRAL DYNAMICS WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE

Zhou, Xueyong; Cui, Jingan

doi:10.4134/BKMS.2011.48.3.555

Title & Authors

GLOBAL STABILITY OF THE VIRAL DYNAMICS WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE
Zhou, Xueyong; Cui, Jingan;

Abstract

It is well known that the mathematical models provide very important information for the research of human immunodeciency virus type. However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T-cells and the viral particles. In this paper, a differential equation model of HIV infection of ${$CD4^+$}$ T-cells with Crowley-Martin function response is studied. We prove that if the basic reproduction number ${$R_0$}$ < 1, the HIV infection is cleared from the T-cell population and the disease dies out; if ${$R_0$}$ > 1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if ${$R_0$}$ > 1. Numerical simulations are presented to illustrate the results.

Keywords

HIV infection;permanence;globally asymptotical stability;

Language

English

Cited by

36time(s) in CrossRef

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Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 2012, 13, 4, 1866

30.

Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Analysis: Real World Applications, 2016, 27, 55

31.

Global dynamics of a virus dynamical model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 2014, 16, 17

32.

An SEIR Epidemic Model with Relapse and General Nonlinear Incidence Rate with Application to Media Impact, Qualitative Theory of Dynamical Systems, 2017

33.

Qualitative Analysis of a Predator–Prey Model with Crowley–Martin Functional Response, International Journal of Bifurcation and Chaos, 2015, 25, 09, 1550110

34.

A generalized virus dynamics model with cell-to-cell transmission and cure rate, Advances in Difference Equations, 2016, 2016, 1

35.

Global properties of a discrete viral infection model with general incidence rate, Mathematical Methods in the Applied Sciences, 2016, 39, 5, 998

36.

Global properties for an age-structured within-host model with Crowley–Martin functional response, International Journal of Biomathematics, 2017, 10, 02, 1750030

References

1.

S. Bonhoeffer, R. M. May, G. M. Shaw, and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA 94 (1997), 6971-6976.

2.

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), no. 1, 23-35.

3.

P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragon y population, J. North. Am. Benth. Soc. 8 (1989), 211-221.

4.

R. V. Culshaw and S. G. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci. 165 (2000), 27-39.

5.

R. V. Culshaw, S. G. Ruan, and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol. 48 (2004), no. 5, 545-562.

6.

F. R. Gantmacher, The Theory of Matrices, Chelsea Publ. Co., New York, 1959.

7.

M. W. Hirsch, Systems of differential equations which are competitive or cooperative IV, SIAM J. Math. Anal. 21 (1990), 1225-1234.

8.

P. De Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math. 63 (2003), no. 4, 1313-1327.

9.

D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007), no. 1, 683-691.

10.

A. L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data, Proc. R. Soc. Lond. B 268 (2001), 847-854.

11.

A. R. McLean and T. B. L. Kirkwood, A model of human immunodeciency virus infection in T helper cell clones, J. Theor. Biol. 147 (1990), 177-203.

12.

A. R. McLean, M. M. Rosado, F. Agenes, R. Vasconcellos, and A. A. Freitas, Resource competition as a mechanism for B cell homeostasis, Proc. Natl Acad. Sci. USA 94 (1997), 5792-5797.

13.

L. Q. Min, Y. M. Su, and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mountain J. Math. 38 (2008), no. 5, 1573-1585.

14.

J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, In uence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998), 143-163.

15.

J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990), no. 4, 857-872.

16.

N. Nagumo, Uber die lage der integralkurven gewohnlicher differential gleichungen, Proc. Phys-Math. Soc. Japan 24 (1942), 551-559.

17.

P. W. Nelson, J. D. Murray, and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), no. 2, 201-215.

18.

A. S. Perelson, D. E. Kirschner, and R. de Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci. 114 (1993), 81-125.

19.

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Rev. 41 (1999), 3-44.

20.

A. S. Perelson, A. U. Neumann, M. Markowitz, et al., HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582-1586.

21.

H. L. Smith, Monotone dynamical systems: An Introduction to the theory of competitive and cooperative systems, Trans. Amer. Math. Soc., vol. 41, 1995.

22.

H. L. Smith , Systems of ordinary differential equations which generate an order preserving flow, SIAM Rev. 30 (1998), 87-98.

23.

X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl. 329 (2007), no. 1, 281-297.

24.

H. R. Thieme, Persistence under relaxed point-dissipativity (with applications to an endemic model), SIAMJ. Math. Anal. 24 (1993), 407-435.

25.

X. Y. Zhou, X. Y. Song, and X. Y. Shi, A differential equation model of HIV infection of CD4+CD4+ T-cells with cure rate, J. Math. Anal. Appl. 342 (2008), no. 2, 1342-1355.

26.

X. Y. Zhou, X. Y. Song, and X. Y. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput. 199 (2008), no. 1, 23-38.

27.

H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations 110 (1994), no. 1, 143-156.