A mathematical and numerical study of a SIR epidemic model with time delay, nonlinear incidence and treatment rates

  • Kanica Goel
  • NilamEmail author
Original Article


A novel nonlinear time-delayed susceptible–infected–recovered epidemic model with Beddington–DeAngelis-type incidence rate and saturated functional-type treatment rate is proposed and analyzed mathematically and numerically to control the spread of epidemic in the society. Analytical study of the model shows that it has two equilibrium points: disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability of the model at DFE is discussed with the help of basic reproduction number, denoted by \({R_0}\), and it is shown that if the basic reproduction number \({R_0}\) is less than one, the DFE is locally asymptotically stable and unstable if \({R_0}\) is greater than one. The stability of the model at DFE for \({R_0}=1\) is analyzed using center manifold theory and Castillo-Chavez and Song theorem which reveals a forward bifurcation. We also derived the conditions for the stability and occurrence of Hopf bifurcation of the model at endemic equilibrium. Further, to illustrate the analytical results, the model is simulated numerically.


Epidemic model Beddington–DeAngelis-type incidence rate Saturated treatment rate Stability Bifurcation Center manifold theory 

Mathematics Subject Classification

34D20 92B05 37M05 



The authors acknowledged the support of Delhi Technological University, Delhi, India, for giving monetary help to complete this research work. They are also indebted to the anonymous reviewers and the handling editor for their constructive comments and suggestions which have enhanced the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia

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