Stability behavior of a nonlinear mathematical epidemic transmission model with time delay

  • Kanica Goel
  • NilamEmail author
Original Paper


In this article, we study a time-delayed susceptible–infected–recovered mathematical model along with nonlinear incidence rate and Holling functional type II treatment rate for epidemic transmission. The mathematical study of the model demonstrates that the model exhibits two equilibria, to be specific, disease-free equilibrium (DFE) and endemic equilibrium (EE). We obtain the basic reproduction number \(R_0\) and investigate that the model is locally asymptotically stable at DFE if \(R_0<1\) and unstable if \(R_0>1\) for the time lag \(\nu >0\). The stability of DFE at \(R_0=1\) is also investigated for the time lag \(\nu \ge 0\), and we show that for \(\nu >0\), the DFE is linearly neutrally stable, whereas for \(\nu =0\), the model exhibits backward bifurcation whereby the DFE will coexists with two endemic equilibria, when \(R_0<1\). We also investigate the stability of the model at the EE and find that oscillatory solution may appear via Hopf bifurcation, taking the delay as a bifurcation parameter. Further, global stability of the model equilibria has also been analyzed. Finally, numerical simulations have been presented to illustrate the analytical studies.


Time delay Nonlinear incidence Holling functional type II Basic reproduction number (BRN) Stability Bifurcations 

Mathematics Subject Classification

34D20 92B05 37M05 



The authors are grateful to the Delhi Technological University, Delhi, India, for providing financial support to carry this research work. They are also indebted to the anonymous reviewers and the handling editor for their constructive comments and suggestions which led to improvements of our manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Ethical standard

The authors state that this research complies with ethical standards. This research does not involve either human participants or animals.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsDelhi Technological UniversityDelhiIndia

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