Odo Diekmann
Utrecht University
Network
Latest external collaboration on country level. Dive into details by clicking on the dots.
Publication
Featured researches published by Odo Diekmann.
Journal of Mathematical Biology | 1990
Odo Diekmann; J.A.P. Heesterbeek; J.A.J. Metz
The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.
Journal of the Royal Society Interface | 2010
Odo Diekmann; J.A.P. Heesterbeek; M. G. Roberts
The basic reproduction number ℛ0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of ℛ0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterization of ℛ0. We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number ℛ0. Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that ℛ0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that ℛ0 > 1 if and only if r > 0, and ℛ0 = 1 if and only if r = 0.
Oikos | 1995
Sido D. Mylius; Odo Diekmann
Concentrating on monomorphic populations in demographic steady state, we give three different conditions under which the evolutionarily stable life-history strategy can be characterized as the life-history strategy at which a relatively simple function is maximal. Depending on the way density dependence acts, this function, or fitness measure, can be the life-time production of offspring, the population growth rate, or another quantity from a large range of possible optimization criteria. We illustrate this by examining the optimal age at maturity for a hypothetical organism. All of this demonstrates that, when studying the evolutionary aspects of life-history characteristics, one cannot escape the task of specifying how density dependence limits population growth.
Journal of Mathematical Biology | 1978
Odo Diekmann
SummaryA nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic is derived and analysed. Particular attention is paid to the hair-trigger effect and to the travelling wave problem.
Journal of Differential Equations | 1979
Odo Diekmann
We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c0 (i.e., travelling wave solutions with speed c exist if ¦c¦ > c0 and do not exist if ¦c¦ < c0). In this paper we show that c0 is the asymptotic speed of propagation (i.e., for any c1, c2with 0 < c1 < c0 < c2 the solution tends to zero uniformly in the region ¦x¦ ⩾ c2t, whereas it is bounded away from zero uniformly in the region ¦x¦ ⩽ c1t for t sufficiently large).
Theoretical Population Biology | 2003
Odo Diekmann; Mats Gyllenberg; J.A.J. Metz
Our systematic formulation of nonlinear population models is based on the notion of the environmental condition. The defining property of the environmental condition is that individuals are independent of one another (and hence equations are linear) when this condition is prescribed (in principle as an arbitrary function of time, but when focussing on steady states we shall restrict to constant functions). The steady-state problem has two components: (i). the environmental condition should be such that the existing populations do neither grow nor decline; (ii). a feedback consistency condition relating the environmental condition to the community/population size and composition should hold. In this paper we develop, justify and analyse basic formalism under the assumption that individuals can be born in only finitely many possible states and that the environmental condition is fully characterized by finitely many numbers. The theory is illustrated by many examples. In addition to various simple toy models introduced for explanation purposes, these include a detailed elaboration of a cannibalism model and a general treatment of how genetic and physiological structure should be combined in a single model.
Nonlinear Analysis-theory Methods & Applications | 1978
Odo Diekmann; Hans G. Kaper
This investigation is concerned with the nonlinear convolution equation u(x) − (gou) * k(x) = 0 on the real line IR. The kernel k is nonnegative and integrable on IR , with ∫ IR k(x)dx = 1; the function g is real-valued and continuous on IR, g(0) = 0 , and there exists a p > 0 such that g(x) > x for x ∈ (0,p) and g(p) = p. Sufficient conditions are given for the non-existence of bounded nontrivial solutions. Implications for the solution of the inhomogeneous equation u(x) − (g o u) * k(x) = f(x), x ∈ IR , are discussed. Finally, uniqueness (modulo translation) is shown to hold. The results are applied to a problem of mathematical epidemiology.
Journal of Mathematical Biology | 1991
Odo Diekmann; M. Kretzschmar
An infectious disease may reduce or even stop the exponential growth of a population. We consider two very simple models for microparasitic and macroparasitic diseases, respectively, and study how the effect depends on a contact parameter K. The results are presented as bifurcation diagrams involving several threshold values of к. The precise form of the bifurcation diagram depends critically on a second parameter ζ, measuring the influence of the disease on the fertility of the hosts. A striking outcome of the analysis is that for certain ranges of parameter values bistable behaviour occurs: either the population grows exponentially or it oscillates periodically with large amplitude.
Proceedings of the National Academy of Sciences of the United States of America | 2002
Inti Pelupessy; Marc J. M. Bonten; Odo Diekmann
The emergence of antibiotic resistance among nosocomial pathogens has reemphasized the need for effective infection control strategies. The spread of resistant pathogens within hospital settings proceeds along various routes of transmission and is characterized by large fluctuations in prevalence, which are typical for small populations. Identification of the most important route of colonization (exogenous by cross-transmission or endogenous caused by the selective pressure of antibiotics) is important for the design of optimal infection control strategies. Such identification can be based on a combination of epidemiological surveillance and costly and laborious as well as time-consuming methods of genotyping. Furthermore, analysis of the effects of interventions is hampered by the natural fluctuations in prevalence. To overcome these problems, we introduce a mathematical algorithm based on a Markov chain description. The input is longitudinal prevalence data only. The output is estimates of the key parameters characterizing the two colonization routes. The algorithm is tested on two longitudinal surveillance data sets of intensive care patients. The quality of the estimates is determined by comparing them to accurate estimates based on additional information obtained by genotyping. The results warrant optimism that this algorithm may help to quantify transmission dynamics and can be used to evaluate the effects of infection control interventions more carefully.
Siam Journal on Mathematical Analysis | 2008
Odo Diekmann; Philipp Getto; Mats Gyllenberg
We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new.