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The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

Many problems of growing interest in science, engineering, biology, and medicine are modeled with systems of differential equations involving delay terms. In general, the presence of the delay in a model increases its reliability in describing the relevant real phenomena and predicting its behavior. Besides, the introduction of history in the evolution law of a system also augments its complexity since, opposite to Ordinary Differential Equations (ODEs), Delay Differential Equations (DDEs) represent infinite dimensional dynamical systems. Thus their time integration and the study of their stability properties require much more effort, together with efficient numerical methods. Since the introduction of the delay terms in the differential equations may drastically change the system dynamics, inducing dangerous instability and loss of performance as well as improving stability, analyzing the asymptotic stability of either an equilibrium or a periodic solution of nonlinear DDEs is a crucial requirement. Several monographs have been written on this subject and the theory is well developed. By the Principle of Linearized Stability, the stability questions can be reduced to the analysis of linear(ized) DDEs. In the literature, a great number of analytical, geometrical, and numerical techniques have been proposed to answer such questions. Part of these techniques aim at analyzing the distribution in the complex plane of the eigenvalues of certain infinite dimensional linear operators, in particular the solution operators associated to the linear(ized) problem and their infinitesimal generator. This monograph does not aim to be a survey, but presents the authors' recent work on the numerical methods for the stability analysis of the zero solution of linear DDEs, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator. The eigenvalues of the resulting matrices are then used to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from applications and, above all, MATLAB programs implementing the proposed algorithms. MATLAB is a high-level language and interactive environment, which is nowadays well developed and widely used for a variety of mathematical problems arising from vii

Abstract. We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations.

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