# Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses

Volume 10, Issue 10, pp 5201--5218
Publication Date: October 14, 2017 Submission Date: April 25, 2017
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### Authors

Yoichi Enatsu - Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan. Jinliang Wang - School of Mathematical Science, Heilongjiang University, Harbin 150080, China. Toshikazu Kuniya - Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan.

### Abstract

In this paper, we study the dynamical behavior of a virus model into which cell-mediated and humoral immune responses are incorporated. The global stability of an infection-free equilibrium and four infected equilibria is established via a Lyapunov functional approach. The present construction methods are applicable to a wide range of incidence rates that are monotone increasing with respect to concentration of uninfected cells and concave with respect to the concentration of free virus particles. In addition, when the incidence rate is monotone increasing with respect to concentration of free virus particles, the functional approach plays an important role in determining the global stability of each of the four infected equilibria. This implies that the dynamical behavior of virus prevalence would be determined by basic reproduction numbers when the saturation effect" for free virus particles appears. We point out that the incidence rate includes not only separable incidence rate but also non-separable incidence rate such as standard incidence and Beddington-DeAngelis functional response.

### Share and Cite

##### ISRP Style

Yoichi Enatsu, Jinliang Wang, Toshikazu Kuniya, Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5201--5218

##### AMA Style

Enatsu Yoichi, Wang Jinliang, Kuniya Toshikazu, Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses. J. Nonlinear Sci. Appl. (2017); 10(10):5201--5218

##### Chicago/Turabian Style

Enatsu, Yoichi, Wang, Jinliang, Kuniya, Toshikazu. "Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5201--5218

### Keywords

• Virus infection model
• delay
• global stability
• incidence rate

•  34K20
•  34K25
•  92D30

### References

• [1] R. Arnaout, M. Nowak, D. Wodarz , HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B., 267 (2000), 1347–1354.

• [2] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency , J. Animal Ecol., 44 (1975), 331–340.

• [3] S. Bonhoeffer, J. M. Coffin, M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275–3278.

• [4] M. S. Ciupe, B. L. Bivort, D. M. Bortz, P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1–27.

• [5] D. L. DeAngelis, R. A. Goldstein, R. V. O’Neill, A model for trophic interaction , Ecology, 56 (1975), 881–892.

• [6] R. J. De Boer, A. S. Perelson , Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567–576.

• [7] R. J. De Boer, A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison , J. Theoret. Biol., 190 (1998), 201–214.

• [8] K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. Real World Appl., 13 (2012), 1866–1872.

• [9] G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections , SIAM Journal on Appl. Math., 70 (2010), 2693–2708.

• [10] Y. Ji, M. Zheng , Dynamics analysis of a viral infection model with a general standard incidence rate, Abst. Appl. Anal., 2014 (2014), 6 pages.

• [11] T. Kajiwara, T. Sasaki, Y. Takeuchi , Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Anal. Real World Appl., 13 (2012), 1802–1826.

• [12] A. Korobeinikov , Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879–883.

• [13] Y. Kuang , Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston (1993)

• [14] C. C. McCluskey, Y. Yang , Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64–78.

• [15] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14–27.

• [16] M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79.

• [17] R. Ouifki, G. Witten, Stability analysis of a model for HIV infection with RTI and three intracellular delays, BioSystems, 95 (2009), 1–6.

• [18] H. Peng, Z. Guo, Global stability for a viral infection model with saturated incidence rate, Abst. Appl. Anal., 2014 (2014), 9 pages.

• [19] A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44.

• [20] J. Prüss, R. Zacher, R. Schnaubelt , Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126–142.

• [21] J.-L. Wang, M. Guo, X.-N. Liu, Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cellmediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149–161.

• [22] T. Wang, Z.-X. Hu., F. Liao, W. Ma , Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comp. Simulation, 89 (2013), 13–22.

• [23] J.-L. Wang, S.-Q. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 263–272.

• [24] J.-L. Wang, J.-M. Pang, T. Kuniya, Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cellmediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298–316.

• [25] K. Wang, W. Wang, H. Pang, X.-N. Liu, Complex dynamic behavior in a viral model with delayed immune response, Phys. D, 226 (2007), 197–208.

• [26] Z.-P. Wang, R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 964–978.

• [27] S. Wang, D. Zou , Global stability of in-host viral models with humoral immunity and intracellular delays , Appl. Math. Model., 36 (2012), 1313–1322.

• [28] Y.-C. Yan, W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 401–416.

• [29] Z.-H. Yuan, X.-F. Zou , Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosci. Eng., 10 (2013), 483–498.

• [30] H. Zhu, Y. Luo, M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091–3102.