Approximating reproduction numbers: a general numerical method for age-structured models.

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

Efficient numerical computation of the basic reproduction number for structured populations.

As widely known, the basic reproduction number plays a key role in weighing birth/infection and death/recovery processes in several models of population dynamics. In this general setting, its characterization as the spectral radius of next generation operators is rather elegant, but simultaneously poses serious obstacles to its practical determination. In this work we address the problem numerically by reducing the relevant operators to matrices through a pseudospectral collocation, eventually computing the sought quantity by solving finite-dimensional eigenvalue problems. The approach is illustrated for two classes of models, respectively from ecology and epidemiology. Several numerical tests demonstrate experimentally important features of the method, like fast convergence and influence of the smoothness of the models' coefficients. Examples of robust analysis of instances of specific models are also presented to show potentialities and ease of application.

Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations.

We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.

A numerical approach for investigating the stability of equilibria for structured population models.

We are interested in the asymptotic stability of equilibria of structured populations modelled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on studying the characteristic equation of the linearized system is often involved or even unattainable. Therefore, we propose and investigate a numerical method to compute the eigenvalues of the associated infinitesimal generator. The latter is discretized by using a pseudospectral approach, and the eigenvalues of the resulting matrix are the sought approximations. An algorithm is presented to explicitly construct the matrix from the model coefficients and parameters. The method is tested first on academic examples, showing its suitability also for a class of mathematical models much larger than that mentioned above, including neutral- and mixed-type equations. Applications to cannibalism and consumer-resource models are then provided in order to illustrate the efficacy of the proposed technique, especially for studying bifurcations.

Stability analysis of age-structured population equations by pseudospectral differencing methods.

In this paper a numerical scheme to investigate the stability of linear models of age-structured population dynamics is studied. The method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques, hence following the approach recently proposed in Breda et al. [SIAM J Sci Comput 27(2): 482-495, 2005] for delay differential equations. The method computes the rightmost characteristic roots and it is shown to converge with spectral accuracy behavior.