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Rihan, Fathalla A.; Anwar, M. Naim

Qualitative analysis of delayed SIR epidemic model with a saturated incidence rate. (English) Zbl 1269.34088

Int. J. Differ. Equ. 2012, Article ID 408637, 13 p. (2012).
Summary: We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag \(\tau\) in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system.

Cited in 16 Documents

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations

Keywords:

asymptotic stability; Hopf bifurcation; numerical simulations

Software:

DDE-BIFTOOL; RADAR5
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\textit{F. A. Rihan} and \textit{M. N. Anwar}, Int. J. Differ. Equ. 2012, Article ID 408637, 13 p. (2012; Zbl 1269.34088)
Full Text: DOI

References:

[1] R. M. Anderson and R. M. May, “Population biology of infectious diseases: part I,” Nature, vol. 280, no. 5721, pp. 361-367, 1979.
[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1998.
[3] V. Capasso, Mathematical Structure of Epidemic Systems, vol. 97 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1993. · Zbl 0798.92024
[4] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Disease, John Wiley & Sons, London, UK, 2000. · Zbl 0997.92505
[5] H. W. Hethcote and D. W. Tudor, “Integral equation models for endemic infectious diseases,” Journal of Mathematical Biology, vol. 9, no. 1, pp. 37-47, 1980. · Zbl 0433.92026
[6] H.-F. Huo and Z.-P. Ma, “Dynamics of a delayed epidemic model with non-monotonic incidence rate,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 459-468, 2010. · Zbl 1221.34197
[7] C. C. McCluskey, “Complete global stability for an SIR epidemic model with delay-distributed or discrete,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 55-59, 2010. · Zbl 1185.37209
[8] D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419-429, 2007. · Zbl 1119.92042
[9] R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3175-3189, 2009. · Zbl 1183.34131
[10] R. Xu and Z. Ma, “Stability of a delayed SIRS epidemic model with a nonlinear incidence rate,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2319-2325, 2009. · Zbl 1198.34098
[11] W. O. Kermack and A. McKendrick, “Contributions to the mathematical theory epidemics,” Proceedings of the Royal Society A, vol. 115, pp. 700-721, 1927. · JFM 53.0517.01
[12] F. Berezovsky, G. Karev, B. Song, and C. Castillo-Chavez, “A simple epidemic model with surprising dynamics,” Mathematical Biosciences and Engineering, vol. 2, no. 1, pp. 133-152, 2005. · Zbl 1061.92052
[13] L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2297-2304, 2009. · Zbl 1198.34075
[14] A. d’Onofrio, P. Manfredi, and E. Salinelli, “Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases,” Theoretical Population Biology, vol. 71, no. 3, pp. 301-317, 2007. · Zbl 1124.92029
[15] A. d’Onofrio, P. Manfredi, and E. Salinelli, “Bifurcation thresholds in an SIR model with information-dependent vaccination,” Mathematical Modelling of Natural Phenomena, vol. 2, no. 1, pp. 26-43, 2007. · Zbl 1337.92223
[16] L. Esteva and M. Matias, “A model for vector transmitted diseases with saturation incidence,” Journal of Biological Systems, vol. 9, no. 4, pp. 235-245, 2001.
[17] D. Greenhalgh, “Some results for an SEIR epidemic model with density dependence in the death rate,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 9, no. 2, pp. 67-106, 1992. · Zbl 0805.92025
[18] S. Hsu and A. Zee, “Global spread of infectious diseases,” Journal of Biological Systems, vol. 12, pp. 289-300, 2004. · Zbl 1073.92044
[19] F. A. Rihan, M.-N. Anwar, M. Sheek-Hussein, and S. Denic, “SIR model of swine influenza epidemic in Abu Dhabi: Estimation of vaccination requirement,” Journal of Public Health Frontier, vol. 1, no. 4, 2012.
[20] S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135-163, 2003. · Zbl 1028.34046
[21] M. Safan and F. A. Rihan, “Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation,” Mathematics and Computers in Simulation. In press.
[22] N. Yi, Z. Zhao, and Q. Zhang, “Bifurcations of an SEIQS epidemic model,” International Journal of Information & Systems Sciences, vol. 5, no. 3-4, pp. 296-310, 2009.
[23] H. W. Hethcote and P. van den Driessche, “An SIS epidemic model with variable population size and a delay,” Journal of Mathematical Biology, vol. 34, no. 2, pp. 177-194, 1995. · Zbl 0836.92022
[24] E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, “Global asymptotic stability of an SIR epidemic model with distributed time delay,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 47, pp. 4107-4115, 2001. · Zbl 1042.34585
[25] X. Song and S. Cheng, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Journal of the Korean Mathematical Society, vol. 42, no. 5, pp. 1071-1086, 2005. · Zbl 1078.92042
[26] N. Guglielmi and E. Hairer, “Implementing Radau IIA methods for stiff delay differential equations,” Journal of Computational Mathematics, vol. 67, no. 1, pp. 1-12, 2001. · Zbl 0986.65069
[27] Y. Kuang and H. I. Freedman, “Uniqueness of limit cycles in Gause-type models of predator-prey systems,” Mathematical Biosciences, vol. 88, no. 1, pp. 67-84, 1988. · Zbl 0642.92016
[28] Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 42, no. 6, pp. 931-947, 2000. · Zbl 0967.34070
[29] K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330, Department of Computer Science, K.U.Leuven, Leuven, Belgium, 2001.
[30] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[31] F. A. Rihan, E. H. Doha, M. I. Hassan, and N. M. Kamel, “Numerical treatments for Volterra delay integro-differential equations,” Computational Methods in Applied Mathematics, vol. 9, no. 3, pp. 292-308, 2009. · Zbl 1184.65122
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