Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. (English) Zbl 1406.92637

Summary: We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratio \(\mathcal{R}_0 < 1\), we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. If \(\mathcal{R}_0 > 1\), we obtain sufficient conditions under which the endemic equilibrium \(E^*\) of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.


92D30 Epidemiology
92C60 Medical epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI


[1] Jia, J. W.; Li, Q. Y., Qualitative analysis of an SIR epidemic model with stage structure, Applied Mathematics and Computation, 193, 1, 106-115 (2007) · Zbl 1193.34113
[2] Li, J.-Q.; Ma, Z.-E.; Zhang, J., Global analysis of some epidemic models with general contact rate and constant immigration, Applied Mathematics and Mechanics, 25, 4, 396-404 (2004) · Zbl 1070.92042
[3] Wang, J. J.; Zhang, J. Z.; Jin, Z., Analysis of an SIR model with bilinear incidence rate, Nonlinear Analysis: Real World Applications, 11, 4, 2390-2402 (2010) · Zbl 1203.34136
[4] Huang, G.; Takeuchi, Y., Global analysis on delay epidemiological dynamic models with nonlinear incidence, Journal of Mathematical Biology, 63, 1, 125-139 (2011) · Zbl 1230.92048
[5] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bulletin of Mathematical Biology, 69, 6, 1871-1886 (2007) · Zbl 1298.92101
[6] Xu, R.; Ma, Z. E., Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Analysis: Real World Applications, 10, 5, 3175-3189 (2009) · Zbl 1183.34131
[7] Enatsu, Y.; Messina, E.; Muroya, Y.; Nakata, Y.; Russo, E.; Vecchio, A., Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates, Applied Mathematics and Computation, 218, 9, 5327-5336 (2012) · Zbl 1244.92048
[8] d’Onofrio, A.; Manfredi, P.; Manfredi, P., Bifurcation thresholds in an SIR model with information-dependent vaccination, Mathematical Modelling of Natural Phenomena, 2, 1, 26-43 (2007) · Zbl 1337.92223
[9] d’Onofrio, A.; Manfredi, P.; Salinelli, E., Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71, 3, 301-317 (2007) · Zbl 1124.92029
[10] Buonomo, B.; d’Onofrio, A.; Lacitignola, D., Global stability of an SIR epidemic model with information dependent vaccination, Mathematical Biosciences, 216, 1, 9-16 (2008) · Zbl 1152.92019
[11] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33, 5, 1144-1165 (2002) · Zbl 1013.92034
[12] Song, X. Y.; Wang, S. L.; Dong, J., Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, Journal of Mathematical Analysis and Applications, 373, 2, 345-355 (2011) · Zbl 1208.34128
[13] Hale, J.; Lunel, S. M. V., Introduction to the Theory of Functional Differential Equations Methods and Applications (1993), Spring
[14] Hassard, B.; Kazarinoff, N.; Wan, Y., Theory and Applications of Hopf Bifurcation (1981), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0474.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.