Global asymptotic stability for a distributed delay differential–difference system of a Kermack–McKendrick SIR model

ABSTRACT

We investigate a system of distributed delay differential–difference equations describing an epidemic model of susceptible, infected, recovered and temporary protected population dynamics. A nonlocal term (distributed delay) appears in this model to describe the temporary protection period of the susceptible individuals. We investigate the mathematical properties of the model. We obtain the global asymptotic stability of the two steady states: disease-free and endemic. We construct appropriate Lyapunov functionals where the basic reproduction number appears as a threshold for the global asymptotic behavior of the solution between disease extinction and persistence.

KEYWORDS:

• SIR epidemic model
• delay differential–difference system
• Lyapunov–Krasovskii functional
• local and global asymptotic stability

AMS CLASSIFICATIONS:

• 34K60
• 37N25
• 92D30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

A. Chekroun thanks grant, PRFU: C00L03UN220120180004, from DGRSDT of Algeria. T. Kuniya is supported by JSPS Grant-in-Aid for Early-Career Scientists [grant number 19K14594] and the Japan Agency for Medical Research and Development [grant number JP20fk0108535].