Bifurcation analysis for a double age dependence epidemic model with two delays

In recent research, we are interested in analyzing a double age dependence SIRS model. Our main interest is to seek the influence of the incubation period and the immunity period on the outbreak of contagious disease, these two periods can be considered as delays. These ones can generate different behaviors for the dynamical system, we focus in this paper on the Hopf bifurcation, which is discussed in detail. Further, the global behavior of solutions is studied using a suitable Lyapunov functional. We also expect that the incubation period duration may have an important impact on the value of the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}, hence it can be considered as a control of the outbreak of the studied infectious disease. Numerically the obtained results are checked utilizing graphical representations.