Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology

Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics University of Udine, via delle scienze 206, 33100 Udine, Italy

Special Issues: Applications of delay differential equations in biology

A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents’ decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.
  Article Metrics

Keywords linear chain trick; stability analysis; distributed delay; SIR models with vaccination; behavioral epidemiology

Citation: Alessia Andò, Dimitri Breda, Giulia Gava. How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology. Mathematical Biosciences and Engineering, 2020, 17(5): 5059-5084. doi: 10.3934/mbe.2020273


  • 1. F. Curtain, H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995.
  • 2. O. Diekmann, M. Gyllenberg, J. A. J. Metz, Finite dimensional state representation of linear and nonlinear delay systems, J. Dynam. Differ. Equations, 30 (2018), 1439-1467.
  • 3. A. K. Erlang, Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Post Office Elec. Eng., (1917), 189-197.
  • 4. N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics 27, Springer Verlag, Berlin, 1978.
  • 5. N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge Studies in Mathematical Biology 8, Cambridge Univeristy Press, Cambridge, 1989.
  • 6. D. Fargue, Reductibilitè des systèmes héréditaires à des systèmes dynamiques, C.R. Acad. Sci. Paris Sér. A-B, 277 (1973), 471-473.
  • 7. D. Fargue, Reductibilitè des systèmes héréditaires, Int. J. Nonlin. Mech., 9 (1974), 331-338.
  • 8. T. Vogel, Théorie Des Systèmes Evolutifs, Gautier Villars, Paris, 1965.
  • 9. D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Sys., 15 (2016), 1-23.
  • 10. S. Busenberg, C. Travis, On the use of reducible-functional differential equations in biological models, J. Math. Anal. Appl., 89 (1982), 46-66.
  • 11. K. L. Cooke, Z. Grossman, Discrete delay, distributed delays and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.
  • 12. E. Beretta, D. Breda, Discrete or distributed delay? Effects on stability of population growth, Math. Biosci. Eng., 13 (2016), 19-41.
  • 13. C. Barril, A. Calsina, J. Ripoll, A practical approach to R0 in continuous-time ecological models, Math. Meth. Appl. Sci., 41 (2018), 8432-8445.
  • 14. A. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Roy. Soc. Lond. B, 268 (2001), 985-993.
  • 15. A. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theor. Popul. Biol., 60 (2001), 59-71.
  • 16. C. Bauch, A. d'Onofrio, P. Manfredi, Behavioral epidemiology of infectious diseases: An overview, in Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases (eds. P. Manfredi and A. d'Onofrio), Springer-Verlag, New York, (2013), 1-19.
  • 17. P. Manfredi, A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 2013.
  • 18. Z. Wang, T. C. Bauch, S. Bhattacharyya, A. d'Onofrio, P. Manfredi, M. Percg, et al., Statistical physics of vaccination, Phys. Rep., 664 (2016), 1-113.
  • 19. B. Buonomo, G. Carbone, A. d'Onofrio, Effect of seasonality on the dynamics of an imitationbased vaccination model with public health intervention, Math. Biosci., 15 (2018), 299-321.
  • 20. B. Buonomo, A. d'Onofrio, D. Lacitignola, Global stability of an sir epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16.
  • 21. B. Buonomo, A. d'Onofrio, D. Lacitignola, The geometric approach to global stability in behavioral epidemiology, in Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases (eds. P. Manfredi and A. d'Onofrio), Springer-Verlag, New York, (2013), 289-308.
  • 22. A. d'Onofrio, P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theoret. Biol., 256 (2009), 473-478.
  • 23. A. d'Onofrio, P. Manfredi, Vaccine demand driven by vaccine side effects: Dynamic implications for sir diseases, J. Theoret. Biol., 264 (2010), 237-252.
  • 24. A. d'Onofrio, P. Manfredi, P. Poletti, The impact of vaccine side effects on the natural history of immunization programmes: an imitation-game approach, J. Theoret. Biol., 273 (2011), 63-71.
  • 25. A. d'Onofrio, P. Manfredi, E. Salinelli, Vaccinating behaviour and the dynamics of vaccine preventabe infections, in Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases (eds. P. Manfredi and A. d'Onofrio), Springer-Verlag, New York, (2013), 267-287.
  • 26. S. Funk, M. Salathé, V. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. R. Soc. Interface, 7 (2010), 1247-1256.
  • 27. A. d'Onofrio, P. Manfredi, E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301-317.
  • 28. A. d'Onofrio, Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times, Appl. Math. Comput, 151 (2004), 181-187.
  • 29. A. Calsina, J. Ripoll, Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers, J. Math. Biol., 45 (2002), 22-36.
  • 30. D. Breda, O. Diekmann, S. Maset, R. Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7 (2013), 4-20.
  • 31. D. Breda, S. Maset, R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495.
  • 32. D. Breda, S. Maset, R. Vermiglio, Stability of Linear Delay Differential Equations-A Numerical Approach with MATLAB, Springer, New York, 2015.
  • 33. N. Olgac, R. Sipahi, Kernel and offspring concepts for the stability robustness of multiple time delayed systems (MTDS), J. Dyn. Syst. T. ASME, 129 (2006), 245-251.
  • 34. O. Diekmann, P. Getto, M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2008), 1023-1069.
  • 35. D. Breda, P. Getto, J. Sánchez Sanz, R. Vermiglio, Computing the eigenvalues of realistic Daphnia models by pseudospectral methods, SIAM J. Sci. Comput., 37 (2015), 2607-2629.
  • 36. H. A. Priestley, Introduction to Complex Analysis, Oxford University Press, New York, 1990.
  • 37. L. Fanti, P. Manfredi, The Solow's model with endogenous population: A neoclassical growth cycle model, J. Econ. Dev., 28 (2003), 103-115.
  • 38. P. Manfredi, L. Fanti, Cycles in dynamic economic modelling, Econ. Model., 21 (2004), 573-594.
  • 39. D. Breda, D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), 1456-1481.
  • 40. D. Breda, S. Maset, R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456-1483.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved