Local and global bifurcations in an SIRS epidemic model

This paper studies the existence and stability of the disease-free equilibrium and endemic equilibria for the SIRS epidemic model with the saturated incidence rate, considering the factor of population dynamics such as the disease-related, the natural mortality and the constant recruitment of population. Analytical techniques are used to show, for some parameter values, the periodic solutions can arise through the Hopf bifurcation, which is important to carry different strategies for the controlling disease. Then the codimension-two bifurcation, i.e. BT bifurcation, is investigated by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Moreover, several numerical simulations are given to support the theoretical analysis.

[1]  Wendi Wang,et al.  Epidemic models with nonlinear infection forces. , 2005, Mathematical biosciences and engineering : MBE.

[2]  V. Capasso.  Mathematical Structures of Epidemic Systems , 1993, Lecture Notes in Biomathematics.

[3]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[4]  S. Levin,et al.  Dynamical behavior of epidemiological models with nonlinear incidence rates , 1987, Journal of mathematical biology.

[5]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[6]  Vincenzo Capasso,et al.  Analysis of a Reaction-Diffusion System Modeling Man-Environment-Man Epidemics , 1997, SIAM J. Appl. Math..

[7]  Shigui Ruan,et al.  Dynamical behavior of an epidemic model with a nonlinear incidence rate , 2003 .

[8]  H. Hethcote,et al.  Some epidemiological models with nonlinear incidence , 1991, Journal of mathematical biology.

[9]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[10]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[11]  Herbert W. Hethcote,et al.  An epidemiological model with a delay and a nonlinear incidence rate , 1989, Journal of mathematical biology.

[12]  K. Cooke,et al.  The population dynamics of two vertically transmitted infections. , 1988, Theoretical population biology.

[13]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[14]  R. I. Bogdanov.  Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues , 1975 .

[15]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[16]  H. Hethcote.  PERIODICITY AND STABILITY IN EPIDEMIC MODELS: A SURVEY , 1981 .

[17]  Y. Iwasa,et al.  Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models , 1986, Journal of mathematical biology.

[18]  Huaiping Zhu,et al.  Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response , 2003, SIAM J. Appl. Math..

[19]  Shigui Ruan,et al.  Global analysis of an epidemic model with nonmonotone incidence rate , 2006, Mathematical Biosciences.

[20]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[21]  G. Serio,et al.  A generalization of the Kermack-McKendrick deterministic epidemic model☆ , 1978 .

[22]  Dongmei Xiao,et al.  Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action , 2006, Chaos, Solitons & Fractals.

[23]  Chen Lan-sun,et al.  Nonlinear incidence rate of a pest management SI model with biological and chemical control concern , 2007 .