Pseudospectral discretization of delay differential equations in &lt;i&gt;sun-star&lt;/i&gt; formulation: Results and conjectures

Diekmann, Odo; Scarabel, Francesca; Vermiglio, Rossana

doi:10.3934/dcdss.2020196

Cited by 16 publications

(11 citation statements)

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“…In the sequel we assume the following. The collocation nodes are the Chebyshev extrema in [0, l ], i.e., Under (A3) both and can be obtained rather easily 8 (see, e.g., [ 13 , 37 , 39 ]).…”

Section: Discretization and Convergence Analysismentioning

confidence: 99%

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

et al. 2020

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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.

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“…In the sequel we assume the following. The collocation nodes are the Chebyshev extrema in [0, l ], i.e., Under (A3) both and can be obtained rather easily 8 (see, e.g., [ 13 , 37 , 39 ]).…”

Section: Discretization and Convergence Analysismentioning

confidence: 99%

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

et al. 2020

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“…However, the distributed DDE formulation of cyclic models has some limitations. The most striking of these is the lack of established numerical techniques for the simulation and bifurcation analysis of infinite delay models, although recent work has alleviated this limitation somewhat [17,25] Nevertheless, the equivalence established in this work allows researchers to study the the mathematical model in whichever form is most convenient, and may elucidate otherwise hidden delayed processes.…”

Section: Discussionmentioning

confidence: 99%

Distributed Delay Differential Equation Representations of Cyclic Differential Equations

Cassidy

2020

Preprint

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Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with non-linear transit rates and possibly delayed arguments are also equivalent to a scalar distributed delay differential equation. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function-without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model.

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“…5 For these and other related aspects see, e.g., [9] . See also [27] for a recent review of results on

.…”

Section: Discretizationmentioning

confidence: 99%

Efficient numerical computation of the basic reproduction number for structured populations

Breda

Florian

Ripoll

et al. 2021

Journal of Computational and Applied Mathematics

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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.

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Pseudospectral discretization of delay differential equations in <i>sun-star</i> formulation: Results and conjectures

Cited by 16 publications

References 50 publications

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

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

Distributed Delay Differential Equation Representations of Cyclic Differential Equations

Efficient numerical computation of the basic reproduction number for structured populations

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