Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations
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Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations Dimitri Breda1
· Toshikazu Kuniya2 · Jordi Ripoll3 · Rossana Vermiglio1
Received: 5 June 2020 / Accepted: 8 October 2020 / Published online: 31 October 2020 © The Author(s) 2020
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. Keywords Pseudospectral collocation · Spectral approximation · Spectral radius · Next-generation operator · Basic reproduction number · Stability analysis of equilibria · Structured population dynamics Mathematics Subject Classification 65J10 · 65L03 · 65L15 · 65M70 · 37N25 · 47D06 · 47A75 · 92D25 · 92D30 · 92D40
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Dimitri Breda [email protected] Toshikazu Kuniya [email protected] Jordi Ripoll [email protected] Rossana Vermiglio [email protected]
1
CDLab – Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle scienze 206, 33100 Udine, Italy
2
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
3
Department of Computer Science, Applied Mathematics and Statistics, University of Girona, Campus Montilivi, 17003 Girona, Spain
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Journal of Scientific Computing (2020) 85:40
1 Introduction 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. In mathematical epidemiology, for instance, R0 measures the average number of secondary cases produced by a typical infected individual in a fully susceptible population, thus indicating the intensity of an epidemic in its initial stage. Estimating R0 is thus a primary target during serious emerging infectious diseases. In this respect, coronavirus disease 2019 (COVID-19) is a last testament of such a critical scenario [26,27,32]. As a second instance, within-host dynamics gives rise to rich ecological models which are underneath the infection processes. The computation of R0 is also important for these type of models as long as the evolutionary standpoint is concerned [9]. A novel and efficient numerical method for computing R0 for structured population dynamics is introduced i
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