Research in Applied Mathematics vol. 1 (2017), Article ID 101266, 16 pages doi:10.11131/2017/101266 AgiAl Publishing House http://www.agialpress.com/ Research Article Stability Analysis of an SEIR Model with Treatment Soufiane Elkhaiar 1 and Abdelilah Kaddar 2 1 Chouaib Doukkali University, Faculty of Sciences, Department of Mathematics and Computer Science, P.B. 20, El Jadida, Morocco 2 Mohammed V University of Rabat, Faculty of Law, Economics and Social Sciences, Salé, Morocco Abstract. We study the dynamics of a SEIR epidemic model with nonlinear treatment function, that takes into account the limited availability of resources in community. Under some conditions we prove the existence of two possible equilibria: the disease-free equilibrium and the endemic equilibrium. Using Lyapunov’s method and Li’s geometrical approach, We also show that the reproduction number  0 is a threshold parameter: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity and the unique endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than this critical value. In the end, we give some concluding remarks concerning the role of treatment on the epidemic propagation. Keywords: SEIR epidemic model; generalized incidence rates; local and global asymptotic stability; treatment; Lyapunov-LaSalle’s principle; geometric approach; compound matrix. Mathematics Subject Classification: 34C23, 34D23, 92D30. 1. Introduction Infectious diseases remain to be one of the main sources of deaths for the human beings. The goal of research in epidemiology is to develop vaccines, treatments and intervention strategies for stopping the spread of infectious diseases and hence reducing the deaths. One important approach to understand transmission mechanisms of infectious diseases is mathematical modeling. In this optic, the differential equations play a crucial role because such equations describe the impact of principal parameters on the spread of diseases. For example, we cite the SIR epidemic model and the SEIR epidemic model which provide good descriptions of infectious diseases (see [5, 11, 16, 17]). Moreover, the epidemiological models describe the effect of treatment on transmission of infection. The modeling of this effect may be taken into account by introducing a treatment function in an epidemi- ological model. However, this function changes from one work to another. In this work, we propose to study the role of treatment on the epidemics transmission. For this, we propose the following SEIR epidemic model with treatment function: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩   =  − (, ) − ,   = (, ) − ( + ),   =  − ( + ) − (),   =  −  + (). (1.1) How to cite this article: Soufiane Elkhaiar and Abdelilah Kaddar, “Global Asymptotic Stability of an SEIR Model with Treatment,” Research in Applied Mathematics, vol. 1, Article ID 101266, 16 pages, 2017. doi:10.11131/2017/101266 Page 1 Corresponding Author Abdelilah Kaddar a.kaddar@yahoo.fr Editor Jianlong Qiu Dates Received 16 November 2016 Accepted 16 April 2017 Copyright © 2017 Soufiane Elkhaiar and Abdelilah Kaddar. 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.