Geodesic
Stability of discrete-time HIV dynamics models with long-lived chronically infected cells
Elaiw, A. M. 1 ; Alshaikh, M. A. 2
1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia;Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia
Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 7, p. 420-439.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This paper studies the global dynamics for discrete-time HIV infection models. The models integrate both long-lived chronically infected and short-lived infected cells. The HIV-susceptible incidence rate is taken as bilinear, saturation and general function. We discretize the continuous-time models by using nonstandard finite difference scheme. The positivity and boundedness of solutions are established. The basic reproduction number is derived. By using Lyapunov method, we prove the global stability of the models. Numerical simulations are presented to illustrate our theoretical results.
DOI : 10.22436/jnsa.012.07.02
Classification : 34D20, 34D23, 37N25, 92B05
Keywords: HIV infection, short-lived infected cells, long-lived infected cells, global stability, Lyapunov function

Elaiw, A. M.  1 ; Alshaikh, M. A.  2

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia;Department of Mathematics, Faculty of Science, Taif University, Saudi Arabia 
@article{JNSA_2019_12_7_a1,
     author = {Elaiw, A. M.  and Alshaikh, M. A. },
     title = {Stability of discrete-time {HIV} dynamics models with long-lived chronically infected cells},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {420-439},
     publisher = {mathdoc},
     volume = {12},
     number = {7},
     year = {2019},
     doi = {10.22436/jnsa.012.07.02},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.07.02/}
}
TY  - JOUR
AU  - Elaiw, A. M. 
AU  - Alshaikh, M. A. 
TI  - Stability of discrete-time HIV dynamics models with long-lived chronically infected cells
JO  - Journal of nonlinear sciences and its applications
PY  - 2019
SP  - 420
EP  - 439
VL  - 12
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.07.02/
DO  - 10.22436/jnsa.012.07.02
LA  - en
ID  - JNSA_2019_12_7_a1
ER  - 
%0 Journal Article
%A Elaiw, A. M. 
%A Alshaikh, M. A. 
%T Stability of discrete-time HIV dynamics models with long-lived chronically infected cells
%J Journal of nonlinear sciences and its applications
%D 2019
%P 420-439
%V 12
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.07.02/
%R 10.22436/jnsa.012.07.02
%G en
%F JNSA_2019_12_7_a1
Elaiw, A. M. ; Alshaikh, M. A. . Stability of discrete-time HIV dynamics models with long-lived chronically infected cells. Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 7, p. 420-439. doi : 10.22436/jnsa.012.07.02. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.07.02/

[1] Callaway, D. S.; Perelson, A. S. HIV-1 infection and low steady state viral loads, Bull. Math. Biol., Volume 64 (2002), pp. 29-64 | Zbl | DOI

[2] Culshaw, R. V.; Ruan, S. G. A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., Volume 165 (2000), pp. 27-39 | DOI

[3] Culshaw, R. V.; Ruan, S. G.; Webb, G. A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., Volume 46 (2003), pp. 425-444 | DOI | Zbl

[4] Ding, D. Q.; Ding, X. H. Dynamic consistent non-standard numerical scheme for a dengue disease transmission model , J. Difference Equ. Appl., Volume 20 (2014), pp. 492-505 | Zbl | DOI

[5] Ding, D.; Qin, W.; Ding, X. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model, Discrete Contin. Dyn. Syst. Ser. B, Volume 20 (2015), pp. 1971-1981 | DOI | Zbl

[6] A. Dutta; Gupta, P. K. A mathematical model for transmission dynamics of HIV/AIDS with effect of weak CD4+ T cells, Chinese J. Phys., Volume 56 (2018), pp. 1045-1056 | DOI

[7] Elaiw, A. M. Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., Volume 11 (2010), pp. 2253-2263 | DOI

[8] A. M. Elaiw Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., Volume 69 (2012), pp. 423-435 | Zbl

[9] Elaiw, A. M.; Almuallem, N. A. Global properties of delayed-HIV dynamics models with differential drug efficacy in cocirculating target cells, Appl. Math. Comput., Volume 265 (2015), pp. 1067-1089 | DOI | Zbl

[10] Elaiw, A. M.; Almuallem, N. A. Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Math. Methods Appl. Sci., Volume 39 (2016), pp. 4-31 | Zbl | DOI

[11] Elaiw, A. M.; AlShamrani, N. H. Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Math. Methods Appl. Sci., Volume 40 (2017), pp. 699-719 | DOI | Zbl

[12] Elaiw, A. M.; Hassanien, I. A.; Azoz, S. A. Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., Volume 49 (2012), pp. 779-794 | DOI

[13] Elaiw, A. M.; Raezah, A. A. Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Methods Appl. Sci., Volume 40 (2017), pp. 5863-5880 | Zbl | DOI

[14] Elaiw, A. M.; Azoz, S. A. Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., Volume 36 (2013), pp. 383-394 | DOI | Zbl

[15] Elaydi, S. An introduction to Difference Equations, Third ed., Springer, New York, 2005

[16] Enatsu, Y.; Nakata, Y.; Muroya, Y.; Izzo, G.; A. Vecchio Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, J. Difference Equ. Appl., Volume 18 (2012), pp. 1163-1181 | DOI | Zbl

[17] Geng, Y.; Xu, J. H.; Hou, J. Y. Discretization and dynamic consistency of a delayed and diffusive viral infection model, Appl. Math. Comput., Volume 316 (2018), pp. 282-295 | DOI

[18] Hattaf, K.; Lashari, A. A.; Boukari, B. El; Yousfi, N. Effect of discretization on dynamical behavior in an epidemiological model, Differ. Equ. Dyn. Syst., Volume 23 (2015), pp. 403-413 | DOI | Zbl

[19] Hattaf, K.; Yousfi, N. A generalized virus dynamics model with cell-to-cell transmission and cure rate, Adv. Difference Equ., Volume 2016 (2016), pp. 1-11 | DOI | Zbl

[20] Hattaf, K.; Yousfi, N. A numerical method for delayed partial differential equations describing infectious diseases, Comput. Math. Appl., Volume 72 (2016), pp. 2741-2750 | Zbl | DOI

[21] Hattaf, K.; Yousfi, N. A numerical method for a delayed viral infection model with general incidence rate, J. King Saud Uni.-Sci., Volume 28 (2016), pp. 368-374 | DOI

[22] Hattaf, K.; N. Yousfi Global properties of a discrete viral infection model with general incidence rate, Math. Methods Appl. Sci., Volume 39 (2016), pp. 998-1004 | DOI | Zbl

[23] Huang, G.; Takeuchi, Y.; Ma, W. Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., Volume 70 (2010), pp. 2693-2708 | DOI

[24] Korpusik, A. A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection, Commun. Nonlinear Sci. Numer. Simul., Volume 43 (2017), pp. 369-384 | DOI

[25] Lasalle, J. P. The stability and control of discrete processes, Springer-Verlag, New York, 1986 | DOI

[26] Li, B.; Chen, Y. M.; Lu, X. J.; Liu, S. Q. A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., Volume 13 (2016), pp. 135-157 | DOI | Zbl

[27] Li, M. Y.; C.Wang, L. Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., Volume 17 (2014), pp. 147-160 | DOI | Zbl

[28] Liu, J. L.; Peng, B. Y.; Zhang, T. L. Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Appl. Math. Lett., Volume 39 (2015), pp. 60-66 | Zbl | DOI

[29] Manna, K.; Chakrabarty, S. P. Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids, J. Difference Equ. Appl., Volume 21 (2015), pp. 918-933 | DOI | Zbl

[30] Mickens, R. E. Nonstandard Finite Difference Models of Differential equations, World Scientific Publishing Co., River Edge, 1994

[31] Mickens, R. E. Dynamics consistency: a fundamental principle for constructing nonstandard finite difference scheme for differential equation, J. Difference Equ. Appl., Volume 11 (2005), pp. 645-653 | DOI

[32] Nelson, P. W.; Murray, J. D.; Perelson, A. S. A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., Volume 163 (2000), pp. 201-215 | Zbl | DOI

[33] Nowak, M. A.; C. R. M. Bangham Population dynamics of immune responses to persistent viruses, Science, Volume 272 (1996), pp. 74-79 | DOI

[34] Pinto, C. M. A.; Carvalho, A. R. M. A latency fractional order model for HIV dynamics, J. Comput. Appl. Math., Volume 312 (2017), pp. 240-256 | Zbl | DOI

[35] Qin, W. D.; Wang, L. S.; Ding, X. H. A non-standard finite difference method for a hepatitis b virus infection model with spatial diffusion , J. Difference Equ. Appl., Volume 20 (2014), pp. 1641-1651 | DOI | Zbl

[36] Shi, P. L.; Dong, L. Z. Dynamical behaviors of a discrete HIV-1 virus model with bilinear infective rate, Math. Methods Appl. Sci., Volume 37 (2014), pp. 2271-2280 | DOI | Zbl

[37] Shi, X. Y.; Zhou, X. Y.; X. Y. Song Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Anal. Real World Appl., Volume 11 (2010), pp. 1795-1809 | DOI

[38] Song, X. Y.; Neumann, A. U. Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., Volume 329 (2007), pp. 281-297 | DOI

[39] Teng, Z. D.; Wang, L.; Nie, L. F. Global attractivity for a class of delayed discrete SIRS epidemic models with general nonlinear incidence, Math. Methods Appl. Sci., Volume 38 (2015), pp. 4741-4759 | Zbl | DOI

[40] Wang, L. C.; M. Y. Li Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., Volume 200 (2006), pp. 44-57 | DOI

[41] Wang, J. P.; Teng, Z. D.; Miao, H. Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response, Adv. Difference Equ., Volume 2016 (2016), pp. 1-19 | DOI | Zbl

[42] Xu, J.; Geng, Y.; Hou, J. A non-standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate, Comput. Math. Appl., Volume 74 (2017), pp. 1782-1798 | DOI | Zbl

[43] Xu, J. H.; Hou, J. Y.; Geng, Y.; Zhang, S. X. Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence, Adv. Difference Equ., Volume 2018 (2018), pp. 1-17 | Zbl | DOI

[44] Yang, Y.; Ma, X. S.; Y. H. Li Global stability of a discrete virus dynamics model with Holling type-II infection function, Math. Methods Appl. Sci., Volume 39 (2016), pp. 2078-2082 | Zbl | DOI

[45] Yang, Y.; Zhou, J. L. Global stability of a discrete virus dynamics model with diffusion and general infection function, Int. J. Comput. Math., Volume 2018 (2018), pp. 1-11 | DOI

[46] Yang, Y.; Zhou, J. L.; Ma, X. S.; Zhang, T. H. Nonstandard finite difference scheme for a diffusive within-host virus dynamics model both virus-to-cell and cell-to-cell transmissions, Comput. Math. Appl., Volume 72 (2016), pp. 1013-1020 | Zbl | DOI

[47] Zhao, Y.; Dimitrov, D. T.; Liu, H.; Kuang, Y. Mathematical insights in evaluating state dependent effectiveness of HIV prevention interventions, Bull. Math. Biol., Volume 75 (2013), pp. 649-675 | DOI | Zbl

[48] Zhou, J. L.; Yang, Y. Global dynamics of a discrete viral infection model with time delay, virus-to-cell and cell-to-cell transmissions, J. Difference Equ. Appl., Volume 23 (2017), pp. 1853-1868 | DOI | Zbl

Cité par Sources :