Stability analysis of an SIR model with immunity and modified transmission function


  • Nidhi Nirwani
  • V.H. Badshah
  • R. Khandelwal





Basic Reproduction Number, Disease Frees Equilibrium, Endemic Equilibrium, Epidemiology, Non-Monotonic Incidence, SIR Model, Stability.


This paper examines an SIR epidemic model with a non-monotonic incidence rate. We analyzed the model by considering after infection, only a fraction of transmitted part is shifted to infectious and remaining part gets recovered without becoming infectious. We also analyze the dynamical behavior of the model and derive the stability conditions for the disease-free and the endemic equilibrium. We have found a threshold condition, in terms of basic reproduction number  which is, less than one, the disease free equilibrium is globally attractive and if more than one, the endemic equilibrium exists and is globally stable. We illustrate theoretical results by carrying numerical simulation.


[1] Anderson RM. Transmission dynamics and control of infectious disease agents. In: Anderson RM, May RM, eds.Population Biology of Infectious Diseases. Springer-Verlag, Berlin, 1982, pp. 149–76.

[2] Anderson RM, May RM, Infectious Diseases of Humans. Dynamics and Control. Oxford University Press, Oxford, 1991.

[3] Diekmann O, Heesterbeek JAP.Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley and Sons, Chichester, 2000.

[4] D. Xiao and S. Ruan: Math. Biosci. 208, 2007, 419-429.

[5] H.W. Hethcote: The Mathematics of Infectious Disease, SIAM Rew. 42, 2000, 599-633.

[6] H.W. Hethcote and P. Van Den Driessche: J. math. Boil. 29, 1991, 271-287.

[7] S Khekare et al., Global Dynamics of an Epidemic Model with a Non-monotonic Incidence Rate, IJOSR-JM, 2014, 71-77.

[8] L.Perko,Differential Equations and Dynamical Systems, Texts in Applied Mathematics,Vol. 7,Springer-Verlag,New work, 1991

[9] S. Ruan and W. Wang: J. Differential equations, 188, 2003, 135-163.

[10] V. Capasso and G. Serio: Math. Biosci. 42, 1978, 43-61.

[11] W. M. Liu, S. A. Levin and Y. Iwasa: J. Math. Biol., 1986, 187-204.

[12] W. M. Liu, H. W. Hethcote and S. Levin: J. Math. Biol., 25, 1987, 359-380.

[13] W.M. Liu, P. van den Driessche, Math. Biosci. 128, 1995, 57-69.

[14] W.R. DERRICK AND P.VAN DEN DRIESSCHE: J MATH. BIOL. 31, 1993, 495-512.

[15] W. Wang: Math.Biosci. 201, 2006, 58-71.

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