1,721,183 research outputs found
Mathematical structures of epidemic systems
No full textMathematical modelling of communicable diseases has in the past decades been the subject of intense research activity, on the part of both epidemiologists and biomathematicians; nonlinear forces of infection, spatial structure, age structure and other relevant features have been integrated to make the models more and more realistic and useful in prediction and control. The author's perspective in this book is that there is a concrete possibility of classifying most of the available models according to their mathematical structure, so to obtain a solid framework for analysing the behaviour of the modelled epidemic systems. This monograph suggests a possible classification of a large amount of models (bilinear, nonlinear, with or without structure), based on the Lyapunov stability theory and the theory of order preserving dynamical systems. The volume contains an original presentation of many worked out examples and case studies, mainly based on the author's experience, fully integrated with the exposition of the theory. It also contains a revisit of the most recent advances in the modelling of epidemics, including HIV/AIDS. Two appendices have been added for the ease of non-mathematicians. This monograph may be viewed as a research monograph for mathematically-oriented epidemiologists and for applied mathematicians. However, the detailed presentation of the methods make it a self-contained introduction to the mathematical modelling of infectious diseases so that it may also be used as a textbook in advanced courses of mathematical modelling in Biology and Medicine. The long and updated list of references makes this monograph a valuable survey of the subject.
In this second printing of the book the author has corrected all detected misprints, and updated the bibliography items
From individual dynamics at the microscopic scale to continuum dynamics at the macroscopic scale: The ant colony paradigm
No full textParticular attention is being paid these days to the mathematical modelling of the social behaviour of individuals in a biological population, for different reasons; on one hand there is an intrinsic interest in population dynamics of herds, on the other hand agent based models are being used in complex optimization problems (ACO's, i.e. Ant Colony Optimization). Further decentralized/parallel computing is exploiting the capabilities of discretization of nonlinear reaction-diffusion systems by means of stochastic interacting particle systems.
Among other interesting features, these systems lead to self organization phenomena, which exhibit interesting spatial patterns.
As a working example, an interacting particle system modelling the social behaviour of ants is proposed here, based on a system of stochastic differential equations, driven by social aggregating/repelling "forces". Specific reference to observed species in nature will be made.
Current interest concerns how properties on the macroscopic level depend on interactions at the microscopic level. Among the scopes of the seminar, a relevant one is to show how to bridge different scales at which biological processes evolve; in particular suitable "laws of large numbers" are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction-diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large.
In order to support a rigorous derivation of the asymptotic nonlinear integrodifferential equation, problems of existence of a weak/entropic solution will be analyzed.
Further the existence of a nontrivial invariant probability measure is analyzed for the stochastic system of interacting particles.
As a further application of the same paradigm, a multiscale model for tumour-driven angiogenesis will be presented.
REFERENCES
[1] Boi S., Capasso V., and Morale D., Modelling the aggregating behaviour of ants of the species Polyergus Rufescens. Spatial heterogeneity in ecological models. Nonlinear Analysis. Real World Appl. 1:163-176, 2000.
[2] Burger M., Capasso V., and Morale D., On an aggregating model with long and short range interactions. Nonlinear Analysis. Real World Appl. 2006.
[3] Morale D., Capasso V. and Oelschlaeger K., An interacting particle system modelling aggregation behaviour: from individuals to populations. J. Mathematical Biology. 50:49-66, 2005.
[4] Capasso V., and Morale D., Stochastic modelling of tumour-induced angiogenesi
Mathematical modelling for polymer processing: polymerization, crystallization, manufacturing
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Characterization of spatial Poisson along optional increasing paths---a problem of dimension's reduction
No full textA problem of reduction of dimension of a plane Poisson process is faced here by considering this process along optional increasing paths. A suitable reparametrization of an optional increasing path makes the process along the path a one-parameter Poisson process. A convenient characterization of the spatial Poisson process is so obtained. The key idea developed in this paper relates to the corresponding result which holds for one-parameter martingales stopped by a bounded stopping tim
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