On Renewal Equations in Population Biology

Abstract

On Renewal Equations in Population Biology Odo Diekmann (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 [3] W. Feller On the Integral Equation of Renewal Theory Annals of Math. Stat. 12 (1941) 243-267 [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. 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 [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] 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) 103-117 DOI:10.1080/17513758.2012.716454 [12] O. Diekmann & K. Korvasova Linearization of solution operators for state-dependent delay equations : a simple example Discrete and Continuous Dynamical Systems A 36 (1) 2016 137-149 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): 1-23 DOI. 10.1137/15M1040931 [14] O. Diekmann, Ph. Getto, Y. Nakata On the characteristic equation λ = α1 + (α2 + α3λ) e−λ 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 formulation Mathematical Methods in the Applied Sciences DOI: 10.1002/mma.3898