Qualitative Analysis of Both Hyperbolic and Non-hyperbolic Equilibria of a SIRS Model with Logistic Growth Rate of Susceptibles and Inhibitory Effect in the Infection
Ghosh Jayanta Kumar 1, Ghosh Uttam 2, Sarkar Susmita 2
1 Boalia Junior High School, Nadia, West Bengal, India
E-mail: jayantaghosh.326@rediffmail.com2 Department of Applied Mathematics, University of Calcutta, Kolkata, India
E-mail: uttam_math@yahoo.co.in, susmita62@yahoo.co.in
Received:
Received: 20 May 2018; revised: 19 December 2018; accepted: 20 December 2018; published online: 31 December 2018
DOI: 10.12921/cmst.2018.0000029
Abstract:
This paper describes a SIRS model with the logistic growth rate of susceptible class. The effect of an inhibitory factor in the infection is also taken into consideration. We have analysed local as well as global stabilities of the equilibrium points (both hyperbolic and non-hyperbolic) of the system and investigated the Transcritical bifurcation at the disease free equilibrium point with respect to the inhibitory factor. The occurrence of Hopf bifurcation of the system is examined and it was observed that this Hopf bifurcation is either supercritical or subcritical depending on parameters. Some numerical simulations are carried out for the validity of theoretical results.
Key words:
centre manifold theory, global stability, Hopf bifurcation, inhibitory factors, logistic growth, losses immunity, transcritical bifurcation
References:
[1] W.Kermack, A. Mckendric, A contribution to mathematical theory of epidemics, Proc. Roy. Soc. Lond. A Mat. 115, 700-721 (1927).
[2] W.Kermack, A. Mckendric, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology 53, 33-55 (1991).
[3] O. Diekman, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Disease, Wiley, New York, 2000.
[4] N.T.J. Bailey,The Mathematical Theory of Infectiou sDiseases, Griffin, London, 1975.
[5] J.D. Murray, Mathematical Biology, Springer, New York, 1993.
[6] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1998.
[7] Z. Ma, J. Li (eds.), Dynamical Modelling and Analysis of Epidemics, World Scientific, 2009.
[8] F. Brauer,C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2011.
[9] Y.Enatsu,E.Messina,Y.Muroya,Y.Nakata,E.RussoandA. Vecchio, Stability analysis of delayed SIR epidemic models with
a class of nonlinear incidence rates, Applied Mathematics and Computation 218, 5327-5336 (2012).
[10] J.J. Wang, J.Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal. RWA 11, 2390-2402 (2009).
[11] J. Wang, S. Liu, B. Zhen and Y. Takeuchi, Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Mathematical and Computer Modelling 55, 710-722 (2012).
[12] T.K. Kar, P. Mandal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear analysis: Real World Applications 12, 2058-2068 (2011).
[13] W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol 23, 187-204 (1986).
[14] J.Z. Zhang, Z. Jin, Q.X. Liu, Z.Y. Zhang, Analysis of a delayed SIR model with non-linear incidence rate, Discrete Dynamics in Nature and Society, (2008).
[15] L.Cai,S.Guo,X.Li,M.Ghosh,Globaldynamicsofadengue epidemic mathematical model, Chaos Solitons Fractals 42, 22972304 (2009).
[16] Z.X. Liu, S. Liu, H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Anal. RWA 348, 433–443 (2008).
[17] L. Zhou, M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. RWA 13, 312-324 (2012).
[18] D. Xiao, S. Ruan, Global analysis of an epidemic model with non-monotone incidence rate, Math. Biosci. 208, 419-429 (2007).
[19] X. Zhang, X.N. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal. RWA 10, 565–575 (2009).
[20] V. Capasso, G. Serio, A generalization of the Kermack– Mckendrick deterministic epidemic model, Math. Biosci. 42, 43–61 (1978).
[21] P. Van den Driessche and J. Watmough, Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Trans-mission, Mathematical Biosciences 180, 29-48 (2002).
[22] J.K. Hale, Ordinary Differential Equations 2nd ed., Krieger, Basel, 1980.
[23] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2000.
[24] T.C. Gard, Persistence in Food Webs: Holling-Type Food Chains, Math Biosci 49, 61-67 (1980).
[25] M.B. Trawicki, Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity, Mathematics 5(1), 7 (2017).
[26] L. Wang, D. Zhou, Z. Liu, D. Xu, X. Zhang, Media alert in an SIS epidemic model with logistic growth, Journal of Biological Dynamics 11, 120–137 (2017).
This paper describes a SIRS model with the logistic growth rate of susceptible class. The effect of an inhibitory factor in the infection is also taken into consideration. We have analysed local as well as global stabilities of the equilibrium points (both hyperbolic and non-hyperbolic) of the system and investigated the Transcritical bifurcation at the disease free equilibrium point with respect to the inhibitory factor. The occurrence of Hopf bifurcation of the system is examined and it was observed that this Hopf bifurcation is either supercritical or subcritical depending on parameters. Some numerical simulations are carried out for the validity of theoretical results.
Key words:
centre manifold theory, global stability, Hopf bifurcation, inhibitory factors, logistic growth, losses immunity, transcritical bifurcation
References:
[1] W.Kermack, A. Mckendric, A contribution to mathematical theory of epidemics, Proc. Roy. Soc. Lond. A Mat. 115, 700-721 (1927).
[2] W.Kermack, A. Mckendric, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology 53, 33-55 (1991).
[3] O. Diekman, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Disease, Wiley, New York, 2000.
[4] N.T.J. Bailey,The Mathematical Theory of Infectiou sDiseases, Griffin, London, 1975.
[5] J.D. Murray, Mathematical Biology, Springer, New York, 1993.
[6] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1998.
[7] Z. Ma, J. Li (eds.), Dynamical Modelling and Analysis of Epidemics, World Scientific, 2009.
[8] F. Brauer,C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2011.
[9] Y.Enatsu,E.Messina,Y.Muroya,Y.Nakata,E.RussoandA. Vecchio, Stability analysis of delayed SIR epidemic models with
a class of nonlinear incidence rates, Applied Mathematics and Computation 218, 5327-5336 (2012).
[10] J.J. Wang, J.Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal. RWA 11, 2390-2402 (2009).
[11] J. Wang, S. Liu, B. Zhen and Y. Takeuchi, Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Mathematical and Computer Modelling 55, 710-722 (2012).
[12] T.K. Kar, P. Mandal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear analysis: Real World Applications 12, 2058-2068 (2011).
[13] W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol 23, 187-204 (1986).
[14] J.Z. Zhang, Z. Jin, Q.X. Liu, Z.Y. Zhang, Analysis of a delayed SIR model with non-linear incidence rate, Discrete Dynamics in Nature and Society, (2008).
[15] L.Cai,S.Guo,X.Li,M.Ghosh,Globaldynamicsofadengue epidemic mathematical model, Chaos Solitons Fractals 42, 22972304 (2009).
[16] Z.X. Liu, S. Liu, H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Anal. RWA 348, 433–443 (2008).
[17] L. Zhou, M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. RWA 13, 312-324 (2012).
[18] D. Xiao, S. Ruan, Global analysis of an epidemic model with non-monotone incidence rate, Math. Biosci. 208, 419-429 (2007).
[19] X. Zhang, X.N. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal. RWA 10, 565–575 (2009).
[20] V. Capasso, G. Serio, A generalization of the Kermack– Mckendrick deterministic epidemic model, Math. Biosci. 42, 43–61 (1978).
[21] P. Van den Driessche and J. Watmough, Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Trans-mission, Mathematical Biosciences 180, 29-48 (2002).
[22] J.K. Hale, Ordinary Differential Equations 2nd ed., Krieger, Basel, 1980.
[23] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2000.
[24] T.C. Gard, Persistence in Food Webs: Holling-Type Food Chains, Math Biosci 49, 61-67 (1980).
[25] M.B. Trawicki, Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity, Mathematics 5(1), 7 (2017).
[26] L. Wang, D. Zhou, Z. Liu, D. Xu, X. Zhang, Media alert in an SIS epidemic model with logistic growth, Journal of Biological Dynamics 11, 120–137 (2017).
