Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection

Mostafa Adimy (1, 2, 3) , Abdennasser Chekroun (4) , Charlotte Dugourd-Camus (1, 2, 3) , Hanene Meghelli (4)

1 ICJ - Institut Camille Jordan
2 DRACULA - Modélisation multi-échelle des dynamiques cellulaires : application à l'hématopoïese
3 MMCS - Modélisation mathématique, calcul scientifique
4 Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen13000, Algeria

We study the local and global asymptotic stability of the two steady-states, disease-free and endemic, of hybrid differential-difference SIR and SIS epidemic models with a nonlinear force of infection and a temporary phase of protection against the disease, e.g. by vaccination or medication. The initial model is an age-structured system that is reduced using the method of characteristic lines to a hybrid system, coupled between differential equations and a time continuous difference equation. We first prove that the solutions of the original system can be obtained from the reduced one. We then focus on the reduced system to obtain new results on the asymptotic stability of the two steady-states. We determine the local asymptotic stability of the two steady-states by studying the associated characteristic equation. We then discuss their global asymptotic stability in various situations (SIR, SIS, mass action, nonlinear force of infection), by constructing appropriate Lyapunov functions.

Mots clés: Local and global asymptotic stability; Lyapunov function; age-structured model; SIR and SIS epidemic models; nonlinear force of infection; hybrid differential-difference system; 37N25; 92D30; 34K06

Domaines: Mathématiques [math]; Sciences du Vivant [q-bio]

Mostafa Adimy, Abdennasser Chekroun, Charlotte Dugourd-Camus, Hanene Meghelli. Global Asymptotic Stability of a Hybrid Differential–Difference System Describing SIR and SIS Epidemic Models with a Protection Phase and a Nonlinear Force of Infection. Qualitative Theory of Dynamical Systems, 2023, 23 (34), pp.32. ⟨10.1007/s12346-023-00891-z⟩. ⟨hal-04271594⟩