Nonlinear Dynamics

, Volume 96, Issue 4, pp 2351–2368 | Cite as

Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates

  • Ranjit Kumar UpadhyayEmail author
  • Ashok Kumar Pal
  • Sangeeta Kumari
  • Parimita Roy
Original Paper


The control of highly contagious diseases is very important today. In this paper, we proposed an SEIR model with Crowley–Martin-type incidence rate and Holling type II and III treatment rates. Dynamics of the spread of infection and its control are performed for both the cases of treatment functions. We have performed the stability and bifurcation analyses of the model system. The sensitivity analysis of all the parameters with respect to the basic reproduction number has been performed. Furthermore, we discussed the optimal control strategy using Pontryagin’s maximum principle and determined the effect of control parameter u on the model dynamics. Moreover, we validate the theoretical results using numerical simulations. Between both the treatment functions, we observe that the implementation of Holling type II treatment is most effective to prevent the spread of diseases. Thus, we conclude that the pervasive effect of treatment not only reduces the basic reproduction number as the control parameter u increases with nonlinear treatment, h(I) but also controls the spread of disease infection among the population.


Epidemic model Holling type II and III treatment functions Stability analysis Optimal control 



We are thankful to Council of Scientific & Industrial Research (CSIR) India for providing financial support through Project No.- CSIR-25(0277)/17/EMR-II to the first author (RKU).

Compliance with ethical standards

Conflicts 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 MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.School of MathematicsThapar Institute of Engineering and TechnologyPatialaIndia

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