Analysis of an $SIR$ Model with Nonlinear Incidence Rate‏ and the ‎Full‎ Impact of‎‏‎ Vaccination

Document Type : Original Paper

Authors

Faculty of Mathematics, Department of Mathematical Sciences, University of Kashan, Kashan, Iran

Abstract

‎In this article‎, ‎we present an $SIR$ model with a general nonlinear incidence rate‎, ‎assuming a $100\%$ effective vaccine‎.
‎This system has a disease-free equilibrium point‎, ‎corresponding to which a basic reproduction number $\mathscr{R}_0$‎​ ‎is obtained‎. ‎For $\mathscr{R}_0 > 1$‎, ‎the system will also have an endemic equilibrium point‎. ‎We study the local and global stability of these equilibrium points‎. ‎Considering the change in the stability status of the equilibrium points with the change of one of the parameters‎, ‎we will examine the existence of a transcritical bifurcation‎. ‎Additionally‎, ‎we calculate the sensitivity index of $\mathscr{R}_0$​‎, ‎which essentially determines the susceptibility of the system to the existing parameters‎. ‎Finally‎, ‎we examine the obtained results with numerical examples‎.

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References

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History

  • Receive Date: 14 July 2024
  • Revise Date: 04 September 2024
  • Accept Date: 09 September 2024

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