Abstract
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.
Highlights
In the wide field of population dynamics, including both ecological and epidemic models, the basic reproduction number R0 is a key quantity in tackling important evolutionary aspects, see, e.g., [3,18] as starting references
As already remarked after (5), the numerical approach we study in Sect. 4 is based on the assumption that the basic reproduction number R0 is a generalized eigenvalue, i.e., a solution λ of (5) for some eigenfunction φ
While we omit to plot the convergence to the eigenvalue since unaffected, it can be seen that the convergence to the eigenfunction is slowed down as f increases, still being the error spectrally accurate. This is in perfect agreement with the convergence analysis: the convergence is spectral since D is smooth, yet the error constant is proportional () to the growth of the derivatives of D, and to f. In this respect see the proof of Proposition 4 and the dependence on the derivatives of A(λ)η for A(λ) in (30). (T2A) The results reported in Fig. 2 about the error with respect to the reference value R0,Nshow spectral accuracy for (T2.1A), where compactness of the Next-Generation Operator (NGO) is ensured according to Proposition 1
Summary
In the wide field of population dynamics, including both ecological and epidemic models, the basic reproduction number R0 is a key quantity in tackling important evolutionary aspects, see, e.g., [3,18] as starting references. Together with [5], the current research represents a framework of reference for the numerical computation of R0 in both ecology and epidemiology In this respect, the general outcome is a quite reliable tool, with faster convergence ideally of infinite order, a feature known as spectral accuracy, see, e.g., [37]. This advantage translates into much more accurate approximations obtained with much smaller matrices, leading to a reduced computational load, in terms of both time and memory This is a favorable feature when stability and bifurcation analyses are the final target in presence of varying or uncertain model parameters, as is frequently the case in realistic contexts.
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