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

• Published in: Advances in difference equations 2019-12, Vol.2019 (1), p.532-19, Article 532
• Main Authors: Elazzouzi, A., Lamrani Alaoui, A., Tilioua, M., Tridane, A.
• Format: Article
• Language: English
• Subjects: Analysis; Asymptotic properties; Difference and Functional Equations; Disease control; Distributed delay; Epidemics; Functional Analysis; Generalized nonlinear incidence; Immunization; Incidence; Liapunov functions; Lyapunov function; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; SIR epidemic model; Stability analysis; Treatment; Vaccination
• ISSN: 1687-1839; 1687-1847; 1687-1847
• DOI: 10.1186/s13662-019-2447-z

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 ≤ 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.