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Discontinuity, Nonlinearity, and Complexity 8(3) (2019) 313--323 | DOI:10.5890/DNC.2019.09.007

Marouane Mahrouf$^{1}$, Khalid Hattaf$^{1}$,$^{2}$, Noura Yousfi$^{1}$

$^{1}$ Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955 Sidi Othman, Casablanca, Morocco

$^{2}$ Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

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Abstract

In this paper, we propose a stochastic viral infection model with general incidence rate. In the proposed model, the stochastic perturbations are modeled by general functions. Further, the global existence and positivity of solutions are investigated. In addition, the stochastic stability of the model is established by using the direct Lyapunov method. Finally, an application of the hepatitis B virus (HBV) is given to validate our theoretical results.

Acknowledgments

We would like to express our gratitude to the editor and the two anonymous reviewers for their constructive comments and suggestions, which helped to enrich this paper.

References

1. [1]	Huang, Z., Yang, Q., and Cao, J. (2011), Complex dynamics in a stochastic internal HIV model, Chaos, Solitons and Fractals, 44(11), 954-963.

2. [2]	Zhao, Y. and Jiang, D. (2014), The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett., 34, 90-93.

3. [3]	Wang, Y. and Jiang, D. (2017), Stationary distribution and extinction of a stochastic viral infection model, Discrete Dynamics in Nature and Society, 2017.

4. [4]	Jiang, D., Liu, Q., Shi, N., Hayat, T., Alsaedi, A., and Xia, P. (2017), Dynamics of a stochastic HIV-1 infection model with logistic growth, Physica A: Statistical Mechanics and its Applications, 469, 706-717.

5. [5]	Mahrouf, M., Adnani, J., and Yousfı, N. (2017), Stability analysis of a stochastic delayed SIR epidemic model with general incidence rate, Applicable Analysis, 1-9.

6. [6]	Djordjevic, J., Cristiana, J.S., and Delfim FM Torres, D.F. (2018), A stochastic SICA epidemic model for HIV transmission, Applied Mathematics Letters, 84, 168-175.

7. [7]	Zhang, X.B., Shi, Q., Ma, S.H., Huo, H.F., and Li, D. (2018), Dynamic behavior of a stochastic SIQS epidemic model with Levy jumps, Nonlinear Dynamics, 1-13.

8. [8]	Hattaf, K., Mahrouf, M., Adnani, J., and Yousfı, N. (2018), Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Physica A: Statistical Mechanics and its Applications, 490, 591-600.

9. [9]	DDalal, N., Greenhalgh, D., and Mao, X. (2008), A stochastic model for internal HIV dynamics, Journal of Mathematical Analysis and Applications, 341(2), 1084-1101.

10. [10]	Beretta, E. , Kolmanovskii, V., and Shaikhet, L. (1998), Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation, 45, 269-277.

11. [11]	Carletti, M. (2002), On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175, 117-131.

12. [12]	Mahrouf,M., Lotfi, E.M., Hattaf, K., Maziane, M., and Yousfı, N. (2017), Stability Analysis for Stochastic Differential Equations in Virology, British Journal of Mathematics and Computer Science, 20(1), 1-12.

13. [13]	Adnani, J., Hattaf, K., and Yousfı, N. (2013), Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate, International Journal of Stochastic Analysis, 2013.

14. [14]	Lahrouz, A., Omari, L., Kiouach, D., and Belmaâi, A. (2011), Deterministic and stochastic stability of a mathematical model of smoking, Statistics and Probability Letters, 81(8), 1276-1284.

15. [15]	Beddington, J.R. and May, R.M. (1977), Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465.

16. [16]	Carletti, M., Burrage, K., and Burrage, P.M. (2004), Numerical simulation of stochastic ordinary differential equations in biomathematical modeling, Math. Comput. Simulation, 64, 271-277.

17. [17]	Hattaf, K., Tridane, A., and Yousfı, N. (2012),Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal. RWA, 13, 1866-1872.

18. [18]	Hattaf, K., Yousfı, N., and Tridane, A. (2013), Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 221, 514-521.

19. [19]	Beddington, J.R. (1975), Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44, 331-340.

20. [20]	DeAngelis, D.L., Goldstein, R.A., and O’Neill, R.V. (1975), A model for trophic interaction, Ecology, 881-892.

21. [21]	Crowley, P.H. and Martin, E.K. (1989), Functional responses and interference within and between year classes of a dragonfly population, J. North. Am. Benth. Soc., 8, 211-221.

22. [22]	Lotfi, E.M., Maziane, M., Hattaf, K., and Yousfı, N. (2014), Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations, 2014.

23. [23]	Mahrouf, M., Hattaf, K., and Yousfı, N. (2017), Dynamics of a Stochastic Viral Infection Model with Immune Response, Mathematical Modelling of Natural Phenomena, 12(5), 15-32.

24. [24]	Mahrouf, M., Lotfi, E.M., Maziane, M., Hattaf, K., and Yousfı, N. (2016), A stochastic viral infection model with general functional response, Nonlinear Analysis and Differential Equations, 4(9), 435-445.

25. [25]	Mao, X. (1997), Stochastic Differential Equations and Applications, Horwood, Chichester.

26. [26]	Afanasiev, V.N., Kolmanowskii, V.B., and Nosov, V.R. (1996), Mathematical Theory of Control Systems Design, Kluwer Academic, Dordrecht, Netherlands.

27. [27]	Higham, D.J. (2001), An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM review, 43(3), 525-546.

28. [28]	Hattaf, K. and Yousfı, N. (2015),A generalizedHBVmodelwith diffusion and two delays, Computers andMathematics with Applications, 69(1), 31-40.