Global Stability Analysis of Logistically Grown SIR Model with Loss of Immunity, Inhibitory Effect, Crowding Effect and its Protection Measure
Department of Applied Mathematics, University of Calcutta, Kolkata, India
∗E-mail: uttam_math@yahoo.co.in
Received:
Received: 12 December 2016; revised: 30 March 2018; accepted: 04 April 2018; published online: 30 May 2018
DOI: 10.12921/cmst.2016.0000071
Abstract:
In this paper we have considered an SIR model with logistically grown susceptible in which the rate of incidence is directly affected by the inhibitory factors of both susceptible and infected populations and the protection measure for the infected class. Permanence of the solutions, global stability and bifurcation analysis in the neighborhood of equilibrium points has been investigated here. The Center manifold theory is used to find the direction of bifurcations. Finally numerical simulation is carried out to justify the theoretical findings.
Key words:
center manifold theory, Hopf bifurcation, inhibition effect, logistic growth rate, loss immunity, normal forms
References:
[1] Z. Hu, Z. Teng, L. Zhang, Stability and bifurcation analysis in
a discrete SIR epidemic model, Mathematics and computers
in Simulation 97, 80–93 (2014).
[2] W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear inci-
dence rates upon the behavior of SIRS epidemiological models,
J. Math. Biol 23, 187–204 (1986).
[3] H.W. Hethcote, M.A. Lewis, P. van den Driessche, An epi-
demiological model with delay and a nonlinear incidence rate,
J. Math. Biol. 27, 49–64, (1989).
[4] D. Xiao, S. Ruan, Global analysis of an epidemic model with
nonmonotone incidence rate, Math. Biosci. 208, 419–429
(2007).
[5] H.N. Agiza, E.M. Elabbasy, EL-Metwally, A.A. Elsadany,
Chaotic dynamics of a discrete prey-predator model with
Holling type-II, Non-linear Annal. Real World Appl. 10, 116–
129 (2009).
[6] W.O. Kermack, A.G. McKendrick, Contribution to mathe-
matical theory of epidemics, P. Roy. Soc. Lond. A Mat. 115,
700–721 (1927).
[7] N.T.J. Bailey, The Mathematical Theory of Infectious Dis-
eases, Griffin, London, 1975.
[8] J.D. Murray, Mathematical Biology, Springer, New York,
1993.
[9] Z. Ma, J. Li (eds.), Dynamical Modeling and Analysis of
Epidemics, World Scientific, 2009.
[10] R.M. Anderson, R.M. May, Infectious Diseases of Humans:
Dynamics and Control. Oxford University Press, 1998.
[11] F. Brauer, C. Castillo-Chavez, Mathematical Models in Popu-
lation Biology and Epidemiology, Springer, 2001.
[12] M.Y. Li, L. Wang, Global stability in some SEIR epidemic
models, IMA Volumes in Mathematics and its Application,
Spring-Verlag 126, 295–311 (2002).
[13] Q. Zou, S. Gao, Q. Zhong, pulse vaccination strategy in an
epidemic model with time delay and nonlinear, Advanced
Studies in Biology 7(1), 307–321 (2009).
[14] T. Zhang, Z. Teng, Global asymptotic stability of a delayed
SEIRS epidemic model with saturation incidence, Chaos Soli-
ton and Fractals 37, 1450–1468 (2008).
[15] L. Zhou, M. Fan, Dynamics of an SIR epidemic model with
limited medical resources revisited, Nonlinear Anal RWA 13,
312–324 (2012).
[16] F. Brauer, P. van den Driessche, Models for transmission of
disease with immigration of infectives, Math Biosci 171(2),
143–154 (2001).
[17] U. Ghosh, S. Sarkar, D.K. Khan, Modelling of infectious
disease in Presence of Vaccination and Delay, International
Journal of Epidemiology and Infection 2(3), 50–57 (2014).
[18] Y. Xue, T. Li, Stability and Hopf Bifurcation for a Delayed SIR
Epidemic Model with Logistic Growth, Hindawi Publishing
Corporation 1–11 (2013).
[19] Y. Enatsu, E. Messina, Y. Muroya, Y. Nakata, E. Russo and A
Vecchio, Stability analysis of delayed SIR epidemic model with
a class of nonlinear incidence rates, Applied Mathematics
and Computation 218, 5327–5336 (2012).
[20] R. Ghosh, U. Ghosh, Bifurcation analysis of SIR Model with
Logistically Grown Susceptibles and Effect of Loss of Immu-
nity of the Recovered Class, Proceeding: International seminar:
Recent trends in Mathematics, Calcutta University, Applied
Mathematics, Springer Proceedings 227–235 (2015).
[21] A. Kaddar, Stability analysis of delayed SIR epidemic model
with saturated incidence rate, Nonlinear Analysis 3, 299–306
(2010).
[22] A. Karobeinikov, P.K. Maini, Nonlinear incidence and stabil-
ity of infectious disease models, Math. Med. Bil. 22, 113–128
(2005).
[23] B. Dubey, A. Parta, P.K. Srivastava, U.S. Dubey, Modeling
and analysis of an SEIR model with different type of non-
linear treatment rates, Journal of Biological System 2(3),
1–25 (2013).
[24] B. Dubey, P. Dubey, U.S. Dubey, Dynamics of an SIR epi-
demic model with non-linear incidence rate and treatment
rate, Applications and Applied Mathematics 10(2), 718–737
(2015).
[25] T.K. Kar, P.K. Mondal, Global dynamics and bifurcation in
delayed SIR epidemic model, Nonlinear Analysis: Real World
Applications 55, 710–722 (2011).
[26] M. Agarwal, V. Verma, Modeling and Analysis of the Spread
of an Infectious Disease Cholera with Environmental Fluctua-
tions, Applications and Applied Mathematics 7(1), 406–425
(2012)
[27] P.K. Mondal, T. K. Kar, Global Dynamics of a Water-Borne
Disease Model with Multiple Transmission Pathways, Appli-
cations and Applied Mathematics 8(1), 99–115 (2013).
[28] C. Wei, L. Chen, A Delayed Epidemic Model with Pulse Vac-
cination, Discrete Dynamics in Nature and Society (2008).
[29] J.Z. Zhang, Z. Jin, Q.X. Liu, Z.Y. Zhang, Analysis of a De-
layed SIR Model with Nonlinear Incidence Rate, Discrete
Dynamics in Nature and Society, (2008).
[30] M. N’zi, G. Kanga, Global analysis of a deterministic and
stochastic nonlinear SIRS epidemic model with saturated inci-
dence rate, Random Oper. Stoch. Equ 24, 1–21. (2016).
[31] J. R.Beddington, Mutual Interference Between Parasites or
Predators and its Effect on Searching Efficiency, Journal of
Animal Ecology 44, 331–340 (1975).
[32] D.L.DeAngelis, R.A Goldstein, R.V. O’Neill, A Model for
Tropic Interaction, Ecology 56, 881–892 (1975).
[33] A.M. Elaiw, S.A. Azoz, Global Properties of a Class of HIV
Infection Models with Beddington–DeAngelis Functional Re-
sponse, Mathematical Methods in the Applied Science 36,
383–394 (2013).
[34] G.Huang, W. Ma, Y. Takeuchi, Global Analysis for Delay
Virus Dynamics Model with Beddington–DeAngelis Func-
tional response, Applied Mathematics Letters 24, 1199–1203
(2011).
[35] P. Feng, Analysis of delayed predator-prey model with ratio
dependent functional responce and quadratic harvesting, Jour-
nal of Applied Mathematics and Computing 44(1), 251–262
(2014).
[36] J. Hale, Theory of Functional Differential Equations, Springer-
Verlag, Heidelberg, 1977.
[37] S. Guo, J. Wu, Bifurcation theory of functional differential
equations, Springer, 2013.
[38] P. Van den Driessche, J. Watmough, Reproduction Numbers
and Sub-Threshold Endemic Equilibria for Compartmental
Models of Disease Transmission, Mathematical Biosciences
180, 29–48 (2002).
[39] L. Perko, Differential Equations and Dynamical Systems,
Springer, New York, 2000.
[40] S. Wiggins, Introduction to applied nonlinear dynamical sys-
tems and chaos, Springer, New York, 2003.
In this paper we have considered an SIR model with logistically grown susceptible in which the rate of incidence is directly affected by the inhibitory factors of both susceptible and infected populations and the protection measure for the infected class. Permanence of the solutions, global stability and bifurcation analysis in the neighborhood of equilibrium points has been investigated here. The Center manifold theory is used to find the direction of bifurcations. Finally numerical simulation is carried out to justify the theoretical findings.
Key words:
center manifold theory, Hopf bifurcation, inhibition effect, logistic growth rate, loss immunity, normal forms
References:
[1] Z. Hu, Z. Teng, L. Zhang, Stability and bifurcation analysis in
a discrete SIR epidemic model, Mathematics and computers
in Simulation 97, 80–93 (2014).
[2] W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear inci-
dence rates upon the behavior of SIRS epidemiological models,
J. Math. Biol 23, 187–204 (1986).
[3] H.W. Hethcote, M.A. Lewis, P. van den Driessche, An epi-
demiological model with delay and a nonlinear incidence rate,
J. Math. Biol. 27, 49–64, (1989).
[4] D. Xiao, S. Ruan, Global analysis of an epidemic model with
nonmonotone incidence rate, Math. Biosci. 208, 419–429
(2007).
[5] H.N. Agiza, E.M. Elabbasy, EL-Metwally, A.A. Elsadany,
Chaotic dynamics of a discrete prey-predator model with
Holling type-II, Non-linear Annal. Real World Appl. 10, 116–
129 (2009).
[6] W.O. Kermack, A.G. McKendrick, Contribution to mathe-
matical theory of epidemics, P. Roy. Soc. Lond. A Mat. 115,
700–721 (1927).
[7] N.T.J. Bailey, The Mathematical Theory of Infectious Dis-
eases, Griffin, London, 1975.
[8] J.D. Murray, Mathematical Biology, Springer, New York,
1993.
[9] Z. Ma, J. Li (eds.), Dynamical Modeling and Analysis of
Epidemics, World Scientific, 2009.
[10] R.M. Anderson, R.M. May, Infectious Diseases of Humans:
Dynamics and Control. Oxford University Press, 1998.
[11] F. Brauer, C. Castillo-Chavez, Mathematical Models in Popu-
lation Biology and Epidemiology, Springer, 2001.
[12] M.Y. Li, L. Wang, Global stability in some SEIR epidemic
models, IMA Volumes in Mathematics and its Application,
Spring-Verlag 126, 295–311 (2002).
[13] Q. Zou, S. Gao, Q. Zhong, pulse vaccination strategy in an
epidemic model with time delay and nonlinear, Advanced
Studies in Biology 7(1), 307–321 (2009).
[14] T. Zhang, Z. Teng, Global asymptotic stability of a delayed
SEIRS epidemic model with saturation incidence, Chaos Soli-
ton and Fractals 37, 1450–1468 (2008).
[15] L. Zhou, M. Fan, Dynamics of an SIR epidemic model with
limited medical resources revisited, Nonlinear Anal RWA 13,
312–324 (2012).
[16] F. Brauer, P. van den Driessche, Models for transmission of
disease with immigration of infectives, Math Biosci 171(2),
143–154 (2001).
[17] U. Ghosh, S. Sarkar, D.K. Khan, Modelling of infectious
disease in Presence of Vaccination and Delay, International
Journal of Epidemiology and Infection 2(3), 50–57 (2014).
[18] Y. Xue, T. Li, Stability and Hopf Bifurcation for a Delayed SIR
Epidemic Model with Logistic Growth, Hindawi Publishing
Corporation 1–11 (2013).
[19] Y. Enatsu, E. Messina, Y. Muroya, Y. Nakata, E. Russo and A
Vecchio, Stability analysis of delayed SIR epidemic model with
a class of nonlinear incidence rates, Applied Mathematics
and Computation 218, 5327–5336 (2012).
[20] R. Ghosh, U. Ghosh, Bifurcation analysis of SIR Model with
Logistically Grown Susceptibles and Effect of Loss of Immu-
nity of the Recovered Class, Proceeding: International seminar:
Recent trends in Mathematics, Calcutta University, Applied
Mathematics, Springer Proceedings 227–235 (2015).
[21] A. Kaddar, Stability analysis of delayed SIR epidemic model
with saturated incidence rate, Nonlinear Analysis 3, 299–306
(2010).
[22] A. Karobeinikov, P.K. Maini, Nonlinear incidence and stabil-
ity of infectious disease models, Math. Med. Bil. 22, 113–128
(2005).
[23] B. Dubey, A. Parta, P.K. Srivastava, U.S. Dubey, Modeling
and analysis of an SEIR model with different type of non-
linear treatment rates, Journal of Biological System 2(3),
1–25 (2013).
[24] B. Dubey, P. Dubey, U.S. Dubey, Dynamics of an SIR epi-
demic model with non-linear incidence rate and treatment
rate, Applications and Applied Mathematics 10(2), 718–737
(2015).
[25] T.K. Kar, P.K. Mondal, Global dynamics and bifurcation in
delayed SIR epidemic model, Nonlinear Analysis: Real World
Applications 55, 710–722 (2011).
[26] M. Agarwal, V. Verma, Modeling and Analysis of the Spread
of an Infectious Disease Cholera with Environmental Fluctua-
tions, Applications and Applied Mathematics 7(1), 406–425
(2012)
[27] P.K. Mondal, T. K. Kar, Global Dynamics of a Water-Borne
Disease Model with Multiple Transmission Pathways, Appli-
cations and Applied Mathematics 8(1), 99–115 (2013).
[28] C. Wei, L. Chen, A Delayed Epidemic Model with Pulse Vac-
cination, Discrete Dynamics in Nature and Society (2008).
[29] J.Z. Zhang, Z. Jin, Q.X. Liu, Z.Y. Zhang, Analysis of a De-
layed SIR Model with Nonlinear Incidence Rate, Discrete
Dynamics in Nature and Society, (2008).
[30] M. N’zi, G. Kanga, Global analysis of a deterministic and
stochastic nonlinear SIRS epidemic model with saturated inci-
dence rate, Random Oper. Stoch. Equ 24, 1–21. (2016).
[31] J. R.Beddington, Mutual Interference Between Parasites or
Predators and its Effect on Searching Efficiency, Journal of
Animal Ecology 44, 331–340 (1975).
[32] D.L.DeAngelis, R.A Goldstein, R.V. O’Neill, A Model for
Tropic Interaction, Ecology 56, 881–892 (1975).
[33] A.M. Elaiw, S.A. Azoz, Global Properties of a Class of HIV
Infection Models with Beddington–DeAngelis Functional Re-
sponse, Mathematical Methods in the Applied Science 36,
383–394 (2013).
[34] G.Huang, W. Ma, Y. Takeuchi, Global Analysis for Delay
Virus Dynamics Model with Beddington–DeAngelis Func-
tional response, Applied Mathematics Letters 24, 1199–1203
(2011).
[35] P. Feng, Analysis of delayed predator-prey model with ratio
dependent functional responce and quadratic harvesting, Jour-
nal of Applied Mathematics and Computing 44(1), 251–262
(2014).
[36] J. Hale, Theory of Functional Differential Equations, Springer-
Verlag, Heidelberg, 1977.
[37] S. Guo, J. Wu, Bifurcation theory of functional differential
equations, Springer, 2013.
[38] P. Van den Driessche, J. Watmough, Reproduction Numbers
and Sub-Threshold Endemic Equilibria for Compartmental
Models of Disease Transmission, Mathematical Biosciences
180, 29–48 (2002).
[39] L. Perko, Differential Equations and Dynamical Systems,
Springer, New York, 2000.
[40] S. Wiggins, Introduction to applied nonlinear dynamical sys-
tems and chaos, Springer, New York, 2003.
