Analyses of the SIR Epidemic Model Including Treatment and Immigration

Year 2024, Volume: 7 Issue: 1, 1 - 13, 08.05.2024

Özlem Ak Gümüş , George Maria Selvam , Narin Kılınç , Janagaraj Rajendran

https://doi.org/10.33187/jmsm.1341741

Abstract

This paper aims to examine the dynamics of a variation of a nonlinear SIR epidemic model. We analyze the complex dynamic nature of the discrete-time SIR epidemic model by discretizing a continuous SIR epidemic model subject to treatment and immigration effects with the Euler method. First of all, we show the existence of equilibrium points in the model by reducing the three-dimensional system to the two-dimensional system. Next, we show the stability conditions of the obtained positive equilibrium point and the visibility of flip bifurcation. A feedback control strategy is applied to control the chaos occurring in the system after a certain period of time. We also perform numerical simulations to support analytical results. We do all these analyses for models with and without immigration and show the effect of immigration on dynamics.

Keywords

Flip bifurcation, Immigration, SIR model, Stability

References

• [1] F. Brauer, C. Castillo-Cavez, Mathematical Models in Population Biology and Epidemology, Texts in Applied Mathematics, 2001.
• [2] R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1992.
• [3] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015.
• [4] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
• [5] A. G. Perez, E. Avila-Vales, G. E. Garcia-Almeida, Bifurcation analysis of an SIR model with logistic growth, nonlinear incidence, and saturated treatment, Complexity, (2019), 1–21.
• [6] G. Li, W. Wang, Z. Jin, Global stability of an SEIR epidemic model with constant immigration, Chaos Solitons Fractals, 30 (4) (2006), 1012-1019.
• [7] L. Jian-quan, Z. Juan, M. Zhi-en, Global analysis of some epidemic models with general contact rate and constant immigration, Appl. Math. Mech., 25 (4) (2004), 396-404.
• [8] Z. A. Khan, A. L. Alaoui, A. Zeb, M. Tilioua, S. Djilali, Global dynamics of a SEI epidemic model with immigration and generalized nonlinear incidence functional, Results Phys., 27 (2021), 104477.
• [9] A. Zeb, S. Djilali, T. Saeed, M. S. Alhodaly, N. Gul, Global proprieties of an SIR epidemic model with nonlocal diffusion and immigration, Results Phys., 39 (2022), 105758.
• [10] A. G. M. Selvam, R. Janagaraj, S. Britto Jacob, D. Vignesh, Stability and bifurcations of a discrete-time Prey–predator system with constant prey refuge, J. Phys. Conf. Ser., 2070 012068 (2021), 1-13.
• [11] A. G. M. Selvam, R. Janagaraj, A. Hlafta, Bifurcation behaviour of a discrete differential algebraic Prey-predator system with Holling type II functional response and prey refuge, AIP Conf. Proc., 2282, 020011 (2020), 1-13.
• [12] A. G. M. Selvam, R. Janagaraj, M. Jacintha, Stability, bifurcation, chaos: discrete prey predator model with step size, Int. J. Eng. Innov. Technol., 9 (1) (2019), 3382-3387.
• [13] O. A. Gumus¸ A. G. M. Selvam, R. Janagaraj, Stability of modified Host-Parasitoid model with Allee effect, Appl. Appl. Math., 15 (2) (2020), 1032-1045.
• [14] O. A. Gumus, A. G. M. Selvam, D. A. Vianny, Bifurcation and stability analysis of a discrete time SIR epidemic model with vaccination, Int. J. Anal. Appl., 17 (5) (2019), 809-820.
• [15] O. A. Gumus, S. Acer, Period-doubling bifurcation analysis and stability of epidemic model, J. Sci. Arts, 49 (4) (2019), 905-914.
• [16] O. A. Gumus, M. Feckan, Stability, Neimark-Sacker bifurcation and chaos control for a prey-predator system with harvesting effect on predator, Miskolc Math. Notes, 22 (2) (2021), 663-679.
• [17] O. A. Gumus, Neimark-Sacker bifurcation and stability of a prey-predator model, Miskolc Math. Notes, 21 (2) (2020), 873-885.
• [18] Q. Din, O. A. Gumus, H. Khalil, Neimark-sacker bifurcation and chaotic behaviour of a modified host parasitoid model, Z. Naturforsch. A, 72 (1) (2017), 25-37.
• [19] Q. Din, Stability, Bifurcation analysis and chaos control for a predator-prey system, J. Vib. Control, 25 (3) (2019), 612-626.
• [20] O. A. Gumus, A.G.M. Selvam, R. Dhineshbabu, Bifurcation analysis and Chaos control of the population model with harvest, Int. J. Nonlinear Anal. Appl., 13 (1) (2021), 115-125.
• [21] O. A. Gumus, Q. Cui, A.G.M.Selvam, D.A. Vianny, Global stability and bifurcation analysis of a discrete-time sir epidemic model, Miskolc Math. Notes, 22 (2023), 193-210.
• [22] O. A. Gumus, A.G.M. Selvam, R. Janagaraj, Dynamics of the mathematical model related to COVID-19 pandemic with treatment, Thai J. Math, 20 (2) (2022), 957-970.
• [23] O. A. Gumus, H. Baran, Dynamics of SIR Epidemic model with treatment function, Int. Battalgazi Sci. Stud. Cong., (2021), 140-153.
• [24] Y. Enatsu, Y. Nakata, Y. Muroya, Global stability for a discrete SIS epidemic model with immigration of infectives, J. Difference Equ. Appl., 18 (2012), 1913-1924.
• [25] S. Yildiz, S. Bilazeroglu, H. Merdan, Stability and bifurcation analyses of a discrete Lotka–Volterra type predator–prey system with refuge effect, J. Comput. Appl. Math., 422 (2023) 114910.
• [26] O. A. Gumus, A.G.M. Selvam, D. Vignesh, The effect of allee factor on a nonlinear delayed population model with harvesting, J. Sci. Arts, 22 (1) (2022), 159-176.
• [27] Z. Hu, Z. Teng, L. Zhang, Stability and flip bifurcation of a discrete SIS epidemic model, J. Xinjiang Univ. (Natural Sci. Edit.), 28 (2011), 446-453.
• [28] Z. Teng, H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal. RWA, 13 (2012), 2017-2033.
• [29] Q. Chen, Z. Teng, L. Wang, H. Jiang, The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence, Nonlinear Dynam., 71 (2013), 55-73.
• [30] A.Q. Khan, M. Tasneem, B. Younis, T.F. Ibrahim, Dynamical analysis of a discrete-time COVID-19 epidemic model, Math. Meth. Appl. Sci., 46 (2022), 4789–4814.
• [31] M.H. DarAssi, S. Damrah, Y. AbuHour, A mathematical study of the omicron variant in a discrete-time Covid-19 model, Eur. Phys. J. Plus, 138 (2023), 601.
• [32] R. George, N. Gul, A. Zeb, Z. Avazzadeh, S. Djilali, S. Rezapour, Bifurcations analysis of a discrete time SIR epidemic model with nonlinear incidence function, Results Phys., 38 (2022), 105580.
• [33] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, 1998.
• [34] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 2003.
• [35] N. Kilinc, O.A. Gumus, Analysis of the epidemic model depending on saturated and mass action incidence rates with treatment, 7th Int. Erciyes Sci. Res. Cong., (2022), 229-316.
• [36] S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996.
• [37] X. Liu, D. Xiao, Complex dynamic behaviors of a discrete time predator–prey system, Chaos Solitons Fractals, 32 (2007), 80-94.
• [38] Q. Din, Dynamics of a discrete lotka-volterra model, Adv. Difference Equ., 2013 (2013), 1-13.
• [39] S. Kapcak, Discrete dynamical systems with sage math, The Electron. J. Math. & Tech., 12(2) (2018), 292-308.
• [40] U. Ufuktepe, S. Kapcak, Applications of discrete dynamical systems with mathematica, Kurenai, 1909 (2014), 207-216.