Numerical bifurcation analysis of renewal equations via pseudospectral approximation

Francesca Scarabel; Odo Diekmann; Rossana Vermiglio

doi:10.1016/j.cam.2021.113611

Numerical bifurcation analysis of renewal equations via pseudospectral approximation

Francesca Scarabel^*, Odo Diekmann, Rossana Vermiglio

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.

Original language	English
Article number	113611
Pages (from-to)	1-21
Number of pages	21
Journal	Journal of Computational and Applied Mathematics
Volume	397
DOIs	https://doi.org/10.1016/j.cam.2021.113611
Publication status	Published - 1 Dec 2021

Bibliographical note

Funding Information:
The authors are grateful to two anonymous reviewers for their constructive comments that improved the manuscript. The research of FS was supported by the NSERC-Sanofi Industrial Research Chair in Vaccine Mathematics, Modelling and Manufacturing, Canada. FS and RV are members of the INdAM Research group GNCS and of the UMI Research group ?Modellistica Socio-Epidemiologica?.

Funding Information:
The authors are grateful to two anonymous reviewers for their constructive comments that improved the manuscript. The research of FS was supported by the NSERC-Sanofi Industrial Research Chair in Vaccine Mathematics, Modelling and Manufacturing, Canada . FS and RV are members of the INdAM Research group GNCS and of the UMI Research group “Modellistica Socio-Epidemiologica”.

Publisher Copyright:
© 2021 Elsevier B.V.

Funding

The authors are grateful to two anonymous reviewers for their constructive comments that improved the manuscript. The research of FS was supported by the NSERC-Sanofi Industrial Research Chair in Vaccine Mathematics, Modelling and Manufacturing, Canada. FS and RV are members of the INdAM Research group GNCS and of the UMI Research group ?Modellistica Socio-Epidemiologica?. The authors are grateful to two anonymous reviewers for their constructive comments that improved the manuscript. The research of FS was supported by the NSERC-Sanofi Industrial Research Chair in Vaccine Mathematics, Modelling and Manufacturing, Canada . FS and RV are members of the INdAM Research group GNCS and of the UMI Research group “Modellistica Socio-Epidemiologica”.

Keywords

Equilibria
Hopf bifurcation
Nonlinear renewal equation
Periodic solutions
Pseudospectral method
Stability analysis

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10.1016/j.cam.2021.113611

1-s2.0-S0377042721002338-mainFinal published version, 775 KBLicence: Taverne

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title = "Numerical bifurcation analysis of renewal equations via pseudospectral approximation",

abstract = "We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.",

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author = "Francesca Scarabel and Odo Diekmann and Rossana Vermiglio",

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AU - Diekmann, Odo

AU - Vermiglio, Rossana

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PY - 2021/12/1

Y1 - 2021/12/1

N2 - We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.

AB - We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.

KW - Equilibria

KW - Hopf bifurcation

KW - Nonlinear renewal equation

KW - Periodic solutions

KW - Pseudospectral method

KW - Stability analysis

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JF - Journal of Computational and Applied Mathematics

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ER -

Numerical bifurcation analysis of renewal equations via pseudospectral approximation

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Keywords

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