An integer-order SIS epidemic model having variable population and fear effect: comparing the stability with fractional order
Tóm tắt
This paper investigates the dynamics of an integer-order and fractional-order SIS epidemic model with birth in both susceptible and infected populations, constant recruitment, and the effect of fear levels due to infectious diseases. The existence, uniqueness, non-negativity, and boundedness of the solutions for both proposed models have been discussed. We have established the existence of various equilibrium points and derived sufficient conditions that ensure the local stability under two cases in both integer- and fractional-order models. Global stability has been vindicated using Dulac–Bendixson criterion in the integer-order model. The forward transcritical bifurcation near the disease-free equilibrium has been investigated. The effect of fear level on infected density has also been observed. We have done numerical simulation by MATLAB to verify the theoretical results, found the impact of fear level on the dynamic behaviour of the infected population, and obtained a bifurcation diagram concerning the constant recruitment and fear level. Finally, we have compared the stability of the population in integer and fractional-order systems.
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Tài liệu tham khảo
Hamer, W.: Epidemic diseases in England- the evidence of variability and of persistency of type. Lancet 1, 733–739 (1906)
Ross, R.: The Prevention of Malaria. John Murray, London (1911)
Kermack, W., McKendric, A.: A contribution to the mathematical theory of epidemics. P. Roy. Soc. Long A Mat. 115, 700–721 (1927)
Hethcote, H.W., Yorke, J.A.: Gonorrhea-1 transmission dynamics and control. Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 56 (1984)
Busenberg, S., Cooke, K.L.: Vertically Transmitted Diseases, Biomathematics, vol. 23. Springer-Verlag, Berlin (1993)
Zhang, J., Sun, J.: Stability analysis of an SIS epidemic model with feedback mechanism on networks. Phys. A 39, 24–32 (2014)
El-Saka, H. A. A.: The fractional-order SIS epidemic model with variable population. J. Egypt. Math. Soc. 22(1), 50–54 (2014)
Zhang, X., Jiang, D., Hayat, T., Ahmad, B.: Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps. Phys. A Stat. Mech. Appl. 471, 767–777 (2017)
Liu, N., Fang, J., Deng, W., Sun, J.W.: Stability analysis of a fractional-order SIS model on complex networks with linear treatment function. Adv. Differ. Equ. 1-10 (2019)
Wang, Y., Cao, J., Alofi, A., Al-Mazrooei, A., Elaiw, A.: Revisiting node-based SIR models in complex networks with degree correlations. Phys. A 437, 75–88 (2015)
Huo, J.-J., Zhao, H.-Y.: Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks. Phys. A 448, 41–45 (2016)
Zhou, J. S.: An SIS disease transmission model with recruitment-birth-death emographics. Math. Compzlt. Model. 21(11), 1–11 (1995)
Li, J., Ma, Z.: Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Math. Comput. Model. 20, 1235–43 (2002)
Funk, S., Marcel, S., Vincent, A.A.J.: Modelling the influence of human behaviour on the spread of infectious diseases a review. J. Royal Soc. Interface 7, 1247–56 (2010)
Polgar, S.: Health and human behavior : areas of interest common to the social and medical sciences. Curr. Anthropol. 3(2), 159–205 (1962)
Morse, S.S.: Factors in the emergence of infectious diseases. In: Plagues and pol- itics, pp. 8-26. Palgrave Macmillan (2001)
World Heath Organization Severe acute respiratory syndrome. Accessed 27 Jan 2010
Johnston, A.C., Warkentin, M.: Fear appeals and information security behaviour: an empirical study. MIS Q. 34, 549–66 (2010)
Wang, X., Zanette, L., Zou, X.: Modelling the fear effect in predator-prey interactions. J. Math. Biol. 73(5), 1–26 (2016)
Wang, X., Zou, X.: Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators. Bull. Math. Biol. 79(6), 1–35 (2017)
Zhang, J., Song, X., Saka, El.: Analysis of an SEIR epidemic model with saturated incidence and saturated treatment function. Sci. World J. 1-11(2014)
Capasso, V., Serio, G.: A generalization of the Kermack-Mckendrick deterministic epidemic model. Math. Biosci. 42(1–2), 43–61 (1978)
El.Saka, H. A. A.: Backward bifurcations in fractional order vaccination models. J. Egypt. Math. Soc. 23(1), 49–55 (2015)
Ameen, P.: Novati, The solution of fractional order epidemic model by implicit Adams methods. J. Appl. Math. Model. 43, 78–84 (2017)
Banerjee, S.K.: Analysis of fractional order SIS epidemic with constant recruitment rate and variable population size. ASIO-JCPMAS 1(2), 1–4 (2016)
Chen, C., Kang, Y.: Dynamics of a Stochastic SIS epidemic model with saturated incidence. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/723825
Diethelm, K., Ford, N.J.: Analysis of fractional differential equation. J. Math. Anal. Appl. 265, 229–248 (2002)
Jing, H.Z., Yan, C.H., Zhidong, T.: Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. J. Appl. Math. comput. 54, 435–449 (2015)
Odibat, Z., Shawagfeh, N.: Generalised Taylors formula. Appl. Math. comput. 186, 286–293 (2007)
Matington, D.: Stability result on fractional differential equations with application to control processing, In: IMACS-SMC proceeding, Lille, France, 963–968 (1996)
Petras, I.: Fractional-Order Nonlinear Systems: Modelling, Analysis and Simulation. Springer, Beijing (2011)
Birkhoff, G., Rota, G.C.: Ordinary Differential Equation. Ginn, Boston (1982)