Global stability analysis for a generalized delayed SIR model with vaccination and treatment

A Elazzouzi; A Lamrani Alaoui; M Tilioua; A Tridane

doi:10.1186/s13662-019-2447-z

Global stability analysis for a generalized delayed SIR model with vaccination and treatment

Adv Differ Equ. 2019;2019(1):532. doi: 10.1186/s13662-019-2447-z. Epub 2019 Dec 21.

Authors

A Elazzouzi¹, A Lamrani Alaoui², M Tilioua², A Tridane³

Affiliations

¹ 1LSI Laboratory, FP Taza, Department of MPI, Sidi Mohamed Ben Abdellah University, Taza, Morocco.
² 2MAMCS Group, M2I Laboratory, FST Errachidia, Moulay Ismaïl University of Meknès, Errachidia, Morocco.
³ 3Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates.

Abstract

In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if $R_{0} \leq 1$ , where $R_{0}$ is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when $R_{0} > 1$ . For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters.

Keywords: Distributed delay; Generalized nonlinear incidence; Lyapunov function; SIR epidemic model; Treatment; Vaccination.