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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Global Stability Analysis for a Generalized SEIR Epidemic Model with Vaccination and Treatment

Journal of Applied Nonlinear Dynamics 14(1) (2025) 67--85 | DOI:10.5890/JAND.2025.03.005

Amine Bernoussi$^1$, Khalid Hattaf$^{2,3}$, Brahim El Boukari$^4$

$^1$ Laboratory: '{E}quations aux d'{e}riv'{e}es partielles, Alg`{e}bre et G'{e}om'{e}trie spectrales, Faculty of Science, Ibn Tofail University, BP 133, 14000 Kenitra, Morocco

$^{2}$ Equipe de Recherche en Mod'elisation et Enseignement des Math'ematiques (ERMEM) Centre R'egional des M'etiers de l'Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

$^{3}$ Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca, P.O Box 7955 Sidi Othman, Casablanca, Morocco

$^4$ Laboratory of Applied Mathematics and Scientific Calculus (LMACS), Higher school of technology, Sultan Moulay Slimane University, 23000 B'eni Mellal, Morocco

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Abstract

The aim of this work is to propose and investigate the global stability of a delayed SEIR epidemic model with a generalized incidence function. The proposed model also includes general treatment function and vaccination term. Using the Lyapunov functions in the absence of delay, we show that the disease-free steady state is globally asymptotically stable if $R_{0} \leq 1$, and the disease-endemic steady state is globally asymptotically stable if $R_{0} > 1$, where $R_{0}$ is the basic reproduction number. For specific functions which are given to the treatment function and the incidence function, we show that the vaccination and early treatment play an important role in healing. Moreover, numerical simulations are given to illustrate and confirm our main analytical results.

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