On Renewal Equations in Population Biology

Abstract

On Renewal Equations in Population Biology

Odo DiekmannВ  (Mathematical Institute, Utrecht University)

The first aim of this talk is to show, by way of examples, how ubiquitous Renewal Equations are in the mathematical formulation of models in Population Dynamics and Infectious Disease Epidemiology.

The second aim is to advocate viewing Renewal Equations as Delay Equations, on par with Delay Differential Equations.

A third aim is to draw attention to a class of numerical methods that allow to use standard ODE tools for a numerical bifurcation analysis.

[1] A. Lotka
On an integral equation in population analysis
Annals of Math. Stat. 10 (1939) 144-161

[2] W.O. Kermack, A.G. McKendrick
A contribution to the mathematical theory of epidemics
Proc. Roy. Soc. A 115 (1927) 700-721 doi : 10.1016/S0092-8240(05)80040-0

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[3] W. Feller
On the Integral Equation of Renewal Theory
Annals of Math. Stat. 12 (1941) 243-267

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[4] W. Feller An introduction to probability theory and its applications, Vol. II (Wiley, second edition, 1971)

[5] J.A.J. Metz, O. Diekmann (eds.)
The Dynamics of Physiologically Structured Populations
Springer Lecture Notes in Biomathematics 68, 1986. В  downloadable from : http://webarchive.iiasa.ac.at/Research/EEP/Metz2Book.html

[6] O.Diekmann, M.Gyllenberg, J.A.J.Metz, H.R.Thieme
On the formulation and analysis of general deterministic structured population models.
I. Linear theory J.Math.Biol. 36 (1998) 349 - 388
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[7] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz, H.R. Thieme
On the formulation and analysis of general deterministic structured population models.
II. Nonlinear theory J. Math. Biol. 43 (2001) 157 -189

[8]В  O. Diekmann, Ph. Getto, M. Gyllenberg
Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars
SIAM J. Math. Anal. 39 (2007) 1023-1069

[9] O. Diekmann, M. Gyllenberg, J.A.J. Metz, S. Nakaoka, A.M. de Roos
Daphnia revisited : local stability and bifurcation theory for physiologically structured population models explained by way of an example
J. Math. Biol. 61 (2010) 277-318

[10] O. Diekmann, M. Gyllenberg
Equations with infinite delay : Blending the abstract and the concrete

J. Diff. Equa. 252 (2012) 819-851

[11] D. Breda, O. Diekmann, W.F. de Graaf, A. Pugliese, R. Vermiglio
On the formulation of epidemic models (an appraisal of Kermack and McKendrick)
Journal of Biological Dynamics 6:sup2 (2012) 103-117В  DOI:10.1080/17513758.2012.716454

[12] O. Diekmann & K. KorvasovГЎ
Linearization of solution operators for state-dependent delay equations : a simple example
Discrete and Continuous Dynamical Systems A 36 (1) 2016 137-149В В В В В В  doi:10.3934/dcds.2016.36.137

[13] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, R. Vermiglio
Pseudospectral discretization of nonlinear delay equations : new prospects for numerical bifurcation analysis
SIAM J. Applied Dynamical Systems (2016)15(1): 1-23В В  DOI. 10.1137/15M1040931

[14]В  O. Diekmann, Ph. Getto, Y. Nakata
On the characteristic equation О» = О±1 + (О±2 + О±3О») eв€’О» and its use in the context of a cell population model
 J. Math. Biol. (2016) 72:877–908  DOI 10.1007/s00285-015-0918-8

[15] A. Calsina, O. Diekmann, J.Z. Farkas

Structured populations with distributed recruitment: from PDE to delay formulationВ 
Mathematical Methods in the Applied SciencesВ В  DOI:В 10.1002/mma.3898

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