QUALITATIVE ANALYSIS OF AN SEIR EPIDEMIC MODEL WITH A RELAPSE RATE

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Published: 2015-09-14

Page: 190-200


AMINE BERNOUSSI

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, P.O.Box 242, Kenitra, Morocco

SAID ASSERDA

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, P.O.Box 242, Kenitra, Morocco

ABDELILAH KADDAR *

Department of Economics, Faculty of Juridical, Economic and Social Sciences of Sale, Mohammed V University in Rabat, P.O.Box 8007, Rabat, Morocco

*Author to whom correspondence should be addressed.


Abstract

In all previous works, the dynamics of the SEIR and SEIRI epidemic models demonstrate a threshold phenomenon as follows: If the basic reproduction number R0 is below unity the diseasefree equilibrium is globally asymptotically stable and the disease always dies out. If R0 > 1; a unique endemic equilibrium is globally asymptotically stable and the disease will persist at the endemic equilibrium if it is initially present. In this paper, we show that the de nition of the basic reproduction number can modify this threshold phenomenon as follows: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than a critical value (di erent to unity) and the unique endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than this critical value. The global stability results are obtained by constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle. Our results show that the latent period plays a positive role to control disease development. Also some numerical simulations are given to compare the results of the SEIR and SEIRI epidemic models.

Keywords: SEIR epidemic model, SEIRI epidemic model, latent period, relapse, Lyapunov function, LaSalle invariance principle, global stability


How to Cite

BERNOUSSI, A., ASSERDA, S., & KADDAR, A. (2015). QUALITATIVE ANALYSIS OF AN SEIR EPIDEMIC MODEL WITH A RELAPSE RATE. Asian Journal of Mathematics and Computer Research, 7(3), 190–200. Retrieved from https://ikprress.org/index.php/AJOMCOR/article/view/402

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