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 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. 1. Introduction Epidemics have ever been a great concern of human kind, because the impact of infectious diseases on human and animal is enormous, both in terms of suffering and social and economic consequences. Mathematical modeling is an essential tool in studying a diverse range of such diseases to gain a better understanding of transmission mechanisms, and make predictions; determine and evaluate control strategies. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission (see [1–10] and references therein). The basic elements for the description of infectious diseases have been considered by three epidemiological classes: that measures the susceptible portion of population, the infected, and the removed ones. Kermack and McKendrick [11] described the simplest SIR model which computes the theoretical number of people infected with a contagious illness in a closed population over time. Transmission of a disease is a dynamical process driven by the interaction between susceptible and infective. The behaviour of the SIR models are greatly affected by the way in which transmission between infected and susceptible individuals are modelled. The simplest model in which recovery does not give immunity is the SIS model, since individuals move from the susceptible class to the infective class and then back to the susceptible class upon recovery. If individuals recover with permanent immunity, then the simplest model is an SIR model. If individuals recover with temporary immunity so that they eventually become susceptible again, then the simplest model is an SIRS model. If individuals do not recover, then the simplest model is an SI model. In general, SIR (epidemic and endemic) models are appropriate for viral agent diseases such as measles, mumps, and smallpox, while SIS models are appropriate for some bacterial agent diseases such as meningitis, plague, and sexually transmitted diseases, and for protozoan agent diseases such
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.
[4]
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Disease, John Wiley & Sons, London, UK, 2000.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]
S. Hsu and A. Zee, “Global spread of infectious diseases,” Journal of Biological Systems, vol. 12, pp. 289–300, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[28]
Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay epidemic model with finite incubation times,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 42, no. 6, pp. 931–947, 2000.
[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.
[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.
