# American Institute of Mathematical Sciences

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## Global dynamics of an age-structured model with relapse

 1 Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Département de Mathématiques, Université Aboubekr Belkaïd Tlemcen, 13000 Tlemcen, Algeria 2 Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33000, Bordeaux, France

* Corresponding author

Revised  February 2019 Published  September 2019

The aim of this paper is to study a general class of $SIRI$ age infection structured model where infectivity depends on the age since infection and where some individuals from the $R$ class, also called quarantaine class in this work, can return to the infectiousness class after a while. Using classical technics we compute a basic reproductive number $R_0$ and show that the disease dies out when $R_0 < 1$ and persists if $R_0 > 1$. Some Lyapunov suitable functions are derived to prove global stability for the disease free equilibrium (DFE) when $R_0 < 1$ and for the endemic equilibrium (EE) when $R_0 > 1$. Using numerical results we show that the non homogeneous infectivity combined with the feedback to the infectiousness class of a part of the quarantaine population modifies drastically the behavior of the epidemic.

Citation: Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019226
##### References:

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##### References:
A schematic diagram of the epidemic model with quarantine
The functions $\beta,\theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
The functions $\beta, \theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
The functions $\beta,\theta$ and $\delta$ with respect to age $a$
The evolution of solution $S$ with respect to time $t$
The evolution of solutions $i$ and $q$ with respect to time $t$ and age $a$
The functions $\beta,\theta$ and $\delta$ with respect to age $a$ : $\delta \equiv 0$ such that $R_0 < 1$, $\delta \not\equiv 0$ such that $R_0 < 1$ and $\delta \not\equiv 0$ such that $R_0 > 1$
The evolution of solution S with respect to time t : δ ≡ 0 such that R0 < 1, $\delta \not \equiv 0$ such that $\delta \not \equiv 0$ and $\delta \not \equiv 0$ such that R0 > 1
The evolution of solution i with respect to time t and age a : δ ≡ 0 such that R0 < 1, $\delta \not \equiv 0$ such that R0 < 1 and $\delta \not \equiv 0$ such that R0 > 1
The evolution of solution q with respect to time t and age a : δ ≡ 0 such that R0 < 1
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