Software

GUI for basic reproduction number

Author: Alessandra Gambellini

A GUI written in Python for the approximation of the reproduction numbers of a class of linear age-structured population models with finite age span in the case of scalar equations. The code was developed as part of Alessandra’s MSc thesis.

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Johnson–Lindenstrauss random projection in \(L^2\)

Author: Michela Palermo

MATLAB codes developed as part of Michela’s MSc thesis.

  • JL tests the Johnson–Lindenstrauss random projection in finite dimension.
  • JL_inf_Leg randomly maps the first n normalised Legendre polynomials from \(L^2([0,1],\mathbb{R})\) to finite dimension, preserving their distances but for a certain relative error, with a certain probability.
  • Comp_JL_F_Leg compares the Johnson–Lindenstrauss random projection and the Fourier–Legendre projection from \(L^2([0,1],\mathbb{R})\) to finite dimension applied to the set of the first n normalised Legendre polynomials.

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eigTMNsd

Author: Cristian Tanase

A MATLAB code for computing the eigenvalues of the monodromy operator of the linearization around a periodic solution of a specific state-dependent DDE. The code was developed as part of Cristian’s MSc thesis.

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Exponential integrators for delay equations

Implementation of exponential Runge–Kutta methods for delay equations as described in [1] and demo examples of MATLAB scripts.

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  • [1] A. Andò and R. Vermiglio, Exponential Runge–Kutta methods for delay equations in the sun–star abstract framework, Discrete Contin. Dyn. Syst. Ser. B (2024), DOI: 10.3934/dcdsb.2024139.

RepNumASM

Author: Simone De Reggi

Package for the numerical approximation of the reproduction numbers of linear age-structured population models formulated as integro-partial differential equations with non-local boundary conditions according to the method of [2].

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  • [2] S. De Reggi, F. Scarabel and R. Vermiglio, Approximating reproduction numbers: a general numerical approach for age-structured models, Math. Biosci. Eng., 21 (2024), pp. 5360–5393, DOI: 10.3934/mbe.2024236.

LE-RE: Lyapunov exponents for renewal and coupled equations

Author: Davide Liessi

Approximation via discrete QR method of the Lyapunov exponents of renewal equations and systems of coulped renewal and delay differential equations via pseudospectral discretization as ODEs (see [3]).

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  • [3] D. Breda and D. Liessi, A practical approach to computing Lyapunov exponents of renewal and delay equations, submitted, arXiv: 2310.15400 [math.NA, math.DS].

Frattali

Author: Dimitri Breda

Codici MATLAB/Octave per il laboratorio “Dalla bisezione ai frattali di Newton” del Seminario Nazionale dei Licei Matematici in collaborazione con Paolo Giangrandi, Antonella Mereu, Chiara Milan e Marzia Toso dell’ISIS “A. Malignani” di Udine.

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AgeNonlocIG

Author: Simone De Reggi

Package for the numerical approximation of the spectrum of the reformulation in terms of the integrated age-state of the infinitesimal generator associated to linear age-structured population models with nonlocal diffusion formulated as integro-partial differential equations (see [4]).

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  • [4] D. Breda, S. De Reggi and R. Vermiglio, A numerical method for the stability analysis of linear age-structured models with nonlocal diffusion, SIAM J. Sci. Comp., to appear, arXiv: 2304.10835 [math.NA, math.DS].

Extension of MatCont for delay equations

Author: Enrico Santi

Extension of MatCont implementing the methods [5] for delay differential equations and [6] for renewal equations and enabling their use via the GUI (see below for using the methods via the command line interface: DDEs, REs). The extension, based on version 7p4 of MatCont, was developed as part of Enrico’s BSc internship at CDLab. His BSc thesis contains the description of the changes and usage details.

Download (version of 29/01/2024; obsolete, see below)

Starting from version 7p5, the pseudospectral collocation method for delay equations is included in MatCont as the Delay Equation Importer. You can download MatCont from its SourceForge page. This version is kept here for documentation purposes only and no support will be given on it.

Slides for the talks presented at IFAC TDS 2022 in Montréal:

  • [5] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1–23, DOI: 10.1137/15M1040931.
  • [6] F. Scarabel, O. Diekmann and R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, J. Comput. Appl. Math., 397 (2021), 113611, DOI: 10.1016/j.cam.2021.113611.

Software for the minicourse “Dynamical systems with delay” at SDS2022

See the dedicated page.

Information on the course.

IG2Str

Author: Simone De Reggi

Pseudospectral discretization of the reformulation in terms of the integrated states of the infinitesimal generator for population models with one and two physiological structures formulated as first-order hyperbolic PDEs (see [7]).

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  • [7] A. Andò, S. De Reggi, D. Liessi and F. Scarabel, A pseudospectral method for investigating the stability of linear population models with two physiological structures, Math. Biosci. Eng., 20 (2023), pp. 4493–4515, DOI: 10.3934/mbe.2023208.

BeeStability

Author: Dimitri Breda

Software for simulating a model of honey bee health (see [8]).

Download from Zenodo, DOI: 10.5281/zenodo.7050516

  • [8] D. Breda, D. Frizzera, G. Giordano, E. Seffin, V. Zanni, D. Annoscia, C. J. Topping, F. Blanchini and F. Nazzi, A deeper understanding of system interactions can explain contradictory field results on pesticide impact on honey bees, Nature Commun., 13, 5720 (2022), DOI: 10.1038/s41467-022-33405-7.

DoubleStructureR0

Author: Simone De Reggi

Approximation of the basic reproduction number \(R_0\) in population models with two internal structures formulated as PDEs (see [9]).

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  • [9] D. Breda, S. De Reggi, F. Scarabel, R. Vermiglio and J. Wu, Bivariate collocation for computing \(R_0\) in epidemic models with two structures, Comput. Math. Appl., 116 (2022), DOI: 10.1016/j.camwa.2021.10.026.

Efficient numerical computation of \(R_0\) for single structure population models

Author: Dimitri Breda

Approximation of the basic reproduction number \(R_0\) for some population models with one internal structure formulated as PDEs (see [10] and [11]).

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  • [10] D. Breda, F. Florian, J. Ripoll and R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021), 113165, DOI: 10.1016/j.cam.2020.113165.
  • [11] D. Breda, T. Kuniya, J. Ripoll and R. Vermiglio, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput., 85 (2020), 40, DOI: 10.1007/s10915-020-01339-1.

eigTMNpw

Author: Davide Liessi

MATLAB code extending the methods of eigTMN and eigTMNc to approximate the multipliers of linear coupled renewal equations and retarded functional differential equations with a piecewise approach (see [12] and [13]).

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  • [12] D. Breda, D. Liessi and R. Vermiglio, Piecewise discretization of monodromy operators of delay equations on adapted meshes, J. Comput. Dyn., 9 (2022), pp. 103–121, DOI: 10.3934/jcd.2022004.
  • [13] D. Breda, D. Liessi and R. Vermiglio, A practical guide to piecewise pseudospectral collocation for Floquet multipliers of delay equations in MATLAB, submitted, arXiv: 2203.12734 [math.NA].

Numerical bifurcation analysis of renewal equations (integrated approach)

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of renewal equations with finite delay using the formulation for the integrated state (main reference [14]). The bifurcation analysis of the ODE system is performed using MatCont.

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  • [14] F. Scarabel, O. Diekmann and R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, J. Comput. Appl. Math., 397 (2021), 113611, DOI: 10.1016/j.cam.2021.113611.

Numerical bifurcation analysis of nonlinear delay equations

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of delay differential equations and renewal equations with finite delay (main reference [15]), using MatCont. The codes are organized in several folders corresponding to each publication.

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  • [15] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1–23, DOI: 10.1137/15M1040931.

Numerical bifurcation analysis of structured population models formulated as PDE

Author: Francesca Scarabel

MATLAB codes for the numerical bifurcation analysis of the ODE approximating system obtained by pseudospectral discretization of structured population models formulated as PDE [16], using MatCont.

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  • [16] F. Scarabel, D. Breda, O. Diekmann, M. Gyllenberg and R. Vermiglio, Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation, Vietnam J. Math., 49 (2021), pp. 37–67, DOI: 10.1007/s10013-020-00421-3.

Computation of periodic solutions

Author: Alessia Andò

Demo examples of Python scripts implementing the continuation of periodic solutions for models defined by renewal equations or systems of coupled renewal and delay differential equations, as described in [17].

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  • [17] A. Andò, Collocation methods for complex delay problems of structured populations, PhD thesis, University of Udine, Italy, 2020.

Efficient numerical continuation of equilibria

Author: Alessia Andò

Demo examples of Python scripts implementing the internal continuation method described in [18].

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  • [18] A. Andò, D. Breda and F. Scarabel, Numerical continuation and delay equations: A novel approach for complex models of structured populations, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), pp. 2619–2640, DOI: 10.3934/dcdss.2020165.

Software for the CISM advanced school on delay

See the dedicated page.

Information on the school.

Numerical bifurcation of equations with infinite delay via pseudospectral collocation

Author: Ilaria Fontana

MATLAB codes for discretizing equations with infinite delay via pseudospectral collocation and studying their bifurcations via continuation with MatCont. The codes were developed as part of Ilaria’s MSc thesis.

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Automatic differentiation for equilibria of dynamical systems

Author: Marco Gambone

MATLAB codes based on ADiMat [19] computing equilibria of epidemiological models and studying their stability. The codes were developed as part of Marco’s BSc thesis.

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  • [19] C. H. Bischof, H. M. Bücker, B. Lang, A. Rasch and A. Vehreschild, Combining source transformation and operator overloading techniques to compute derivatives for MATLAB programs, in Proceedings of the Second IEEE International Workshop on Source Code Analysis and Manipulation (SCAM 2002), IEEE Computer Society, Los Alamitos, CA, 2002, pp. 65–72, DOI: 10.1109/SCAM.2002.1134106.

differential

Author: Francesco Florian

C++ library providing interpolatory quadrature and differentiation weights. The library was developed as part of Francesco’s MSc thesis.

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eigTMNc

Authors: Davide Liessi and Dimitri Breda

MATLAB code extending the method of eigTMN to approximate the multipliers of linear coupled renewal equations and retarded functional differential equations. See [20] for renewal equations and [21] for coupled equations.

  • [20] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), pp. 1456–1481, DOI: 10.1137/17M1140534.
  • [21] D. Liessi, Pseudospectral methods for the stability of periodic solutions of delay models, PhD thesis, University of Udine, Italy, 2018.

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LEs for DDEs

Authors: Dimitri Breda and Sara Della Schiava

MATLAB codes from a recent work inspired by Sara’s MSc thesis, see [22]. Based on the pseudospectral reduction to ODEs, with these codes one can approximate a number of Lyapunov exponents of a DDE. This version is tuned for the Mackey–Glass equation, but modification should be straightforward.

  • [22] D. Breda and S. Della Schiava, Pseudospectral reduction to compute Lyapunov exponents of delay differential equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), pp. 2727–2741, DOI: 10.3934/dcdsb.2018092.

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eigAM/eigTMN

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

MATLAB codes accompanying the recent book [23]. They are devoted to the computation of either the characteristic roots or multipliers of linear, respectively autonomous and periodic, delay differential equations. Chapter 8 of the book should be enough to learn and experimenting.

  • [23] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer, New York, 2015, DOI: 10.1007/978-1-4939-2107-2.

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LEVEL – level curves of surfaces

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

A MATLAB package for the computation of level curves of surfaces. It is a grid-like approach based on adaptive triangulation. The method is described in [24]. The package contains a brief manual and test cases.

  • [24] D. Breda, S. Maset and R. Vermiglio, An adaptive algorithm for efficient computation of level curves of surfaces, Numer. Algorithms, 52 (2009), pp. 605–628, DOI: 10.1007/s11075-009-9303-2.

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TRACE-DDE – Tool for Robust Analysis and Characteristic Equations of Delay Differential Equations

Authors: Dimitri Breda, Stefano Maset, and Rossana Vermiglio

A MATLAB GUI for computing characteristic roots and stability charts of linear autonomous systems of DDEs with several discrete and/or distributed delays, see the book chapter [25]. The GUI structure, developed at UniUD by Daniele Sechi in his BSc thesis, properly works on Mac OS X 10.3 (or superior) with MATLAB 7x, but is no longer maintained.

  • [25] D. Breda, S. Maset and R. Vermiglio, TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, in J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, eds., Topics in Time Delay Systems: Analysis, Algorithms, and Control, Lect. Notes Control Inf. Sci. 388, Springer, New York, 2009, pp. 145–155, DOI: 10.1007/978-3-642-02897-7.

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