Global Dynamics of an SEIR Epidemic model with saturated incidence under treatment

M. Badole¹ , S. K. Tiwari² , V. Gupta³ , A. Agrawal⁴

1. School of Studies in Mathematics, Vikram University, Ujjain, India.
2. School of Studies in Mathematics, Vikram University, Ujjain, India.
3. Department of Mathematics, Govt. Kalidas Girl’s College, Ujjain, India.
4. Department of Mathematics, Acropolis Institute of Technology and Research, Indore, India.

Section:Research Paper, Product Type: Isroset-Journal
Vol.5 , Issue.3 , pp.48-57, Jun-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.4857

Online published on Jun 30, 2018

Copyright © M. Badole, S. K. Tiwari, V. Gupta, A. Agrawal . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
An SEIR epidemic model with saturated incidence rate under a limited resource for treatment function which is proposed by W. Wang (2006) is investigated in this paper. We have assumed that treatment rate is proportional to the number of infective when it is below the capacity and is a constant when the number of infective is larger than the capacity. The existing threshold conditions of all kinds of the equilibrium points are obtained. The local and global stability of the disease free equilibrium and the endemic equilibrium of the model are discussed. The local asymptotical stability of equilibrium is verified by analyzing the eigen values and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the disease free equilibrium by using, Lyapunov function and endemic equilibrium by autonomous convergence theorem. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease. Finally, numerical simulations are given to illustrate the validity of the proposed results.

Key-Words / Index Term :
SEIR epidemic model, Treatment, Basic reproduction number, Lyapunov function, Stability analysis.

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