P. Tautu – författare

On March 8/9, 1976 a workshop on "Mathematical Models in Medicine" was held at Mainz (German Federal Republic) by the group of "Mathe- matical Models" of the Deutsche Gesellschaft fUr Medizinische Doku- mentation, Informatik und Statistik. Purpose of this conference was to bring together experts from the GFR and neighbouring countries working in this field to evaluate possibilities and limits of this area of research in discussions with interested participants. This issue of Lecture Notes contains the invited contributions as well as the relevan~ remarks made by the discussants. Corresponding to the aims of the workshop the contributors had been encouraged to demon- strate their mathematical models in the light of actual applied examples. It had been our intention to restrict attention to a small number of specific areas in order to achieve a concentrated in depth treatment in these restricted areas. The areas chosen contain two - Epidemio- logy and Cell Models - which in the organisers feeling are not yet as well established in Continental Europe and one - Pharmacokinetic- with a more direct appeal to applied workers.While in the Epidemio- logy of infectious and parasitic diseases today strategies of control and eradication are gaining importance, the cell models are concerned with explaining the modes of genesis of cancerous growth and the kinetics and interactions within multi-cell structures.

These Proceedings have been assembled from papers presented at the Conference on Models of Biological Growth and Spread, held at the German Cancer Research Centre Heidelberg and at the Institute of Applied Mathematics of the University of Heidelberg, July 16-21, 1979. The main theme of the conference was the mathematical representation of biolog­ ical populations with an underlying spatial structure. An important feature of such populations is that they and/or their individual com­ ponents may interact with each other. Such interactions may be due to external disturbances, internal regulatory factors or a combination of both. Many biological phenomena and processes including embryogenesis, cell growth, chemotaxis, cell adhesion, carcinogenesis, and the spread of an epidemic or of an advantageous gene can be studied in this con­ text. Thus, problems of particular importance in medicine (human and veterinary), agriculture, ecology, etc. may be taken into consideration and a deeper insight gained by utilizing (more) realistic mathematical models. Since the intrinsic biological mechanisms may differ considerably from each other, a great variety of mathematical approaches, theories and techniques is required. The aims of the conference were (i) To provide an overview of the most important biological aspects. (ii) To survey and analyse possible stochastic and deterministic approaches. (iii) To encourage new research by bringing together mathematicians interested in problems of a biological nature and scientists actively engaged in developing mathematical models in biology.

Der vorliegende Band enthalt die auf der 28. GMDS-Jahrestagung gehal tenen Vortrage und ein gesondertes Kapitel mit Vortragen und Diskussionsbemerkungen des Workshops 'Sprachen und Grammatiken'. Die Reihenfolge und Einordnung der Arbeiten ist gegenUber der Vor- tragsfolge leicht verandert worden, um inhaLtlich einen noch besseren Zusammenhang der Themen herzustellen. Aile Obersichtsreferate und GrundLagenarbeiten wurden an den Anfang des Buches gestellt. Das ursprUngliche Vorhaben, den Satz fUr diesen Band auf einer com- putergesteuerten Lichtsatzanlage zu erstellen, muBten wir Leider nach zeitraubenden BemUhungen wieder aufgeben, da es nicht meglich war, die mit HiLle des Textsystems "SCRIPT" erstellten Texte auf einfache Art und Weise in eine fUr den Lichtsatz geeignete Form zu transferie- reno Die Zusage der damit betrauten Firma wurde von dieser wieder zurUckgezogen, da die Kosten fUr die Erstellung eines Interfacepro- gramms den moglichen Kostenrahmen bei weitem gesprengt hatten.FUr die Hilfe bei der Produktion des Bandes dank en wir Frau GabrieLe Acar, die den greBten TeiL der organisatorischen Arbeiten neben der Erfassung zu erLedigen hatte, sowie ihrer ebenfalls mit der Erfassung und Korrektur betrauten Kollegin Erika Kraus. SchlieBlich ist es uns eine angenehme Pflicht, dem Herausgeber der Reihe und dem Springer-VerLag fUr die UnterstUtzung bei der HersteL- Lung des Bandes zu danken.

These Proceedings have been assembled from papers presented at the Conference on Models of Biological Growth and Spread, held at the German Cancer Research Centre Heidelberg and at the Institute of Applied Mathematics of the University of Heidelberg, July 16-21, 1979. The main theme of the conference was the mathematical representation of biolog­ ical populations with an underlying spatial structure. An important feature of such populations is that they and/or their individual com­ ponents may interact with each other. Such interactions may be due to external disturbances, internal regulatory factors or a combination of both. Many biological phenomena and processes including embryogenesis, cell growth, chemotaxis, cell adhesion, carcinogenesis, and the spread of an epidemic or of an advantageous gene can be studied in this con­ text. Thus, problems of particular importance in medicine (human and veterinary), agriculture, ecology, etc. may be taken into consideration and a deeper insight gained by utilizing (more) realistic mathematical models. Since the intrinsic biological mechanisms may differ considerably from each other, a great variety of mathematical approaches, theories and techniques is required. The aims of the conference were (i) To provide an overview of the most important biological aspects. (ii) To survey and analyse possible stochastic and deterministic approaches. (iii) To encourage new research by bringing together mathematicians interested in problems of a biological nature and scientists actively engaged in developing mathematical models in biology.

This volume is a revised and enlarged version of Chapters 1 and 2 of a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two parts because of the large amount of new material. The present part is intended to introduce mathematicians and biologists with a strong mathematical and probabilistic background to the study of stochastic processes. We hope some readers will be able to discover by themselves the new features of our treatment such as the inclusion of some unusual topics, the special attention paid to some usual topics, and the grouping of the material. We draw the reader's attention to the numbering, because there are structural differences between the two parts. In Part I there are Chapters, Sections, Subsections, Paragraphs and Subparagraphs. Thus the numbering a. b. c. d. e refers to Subparagraph e of Paragraph d of Subsection c of Section b of Chapter a. Definitions, theorems lemmas and propositions are numbered a. b. n, n = 1,2, . . . , where a indicates the chapter and b the section. In Part II there are Sections, Subsections, Paragraphs, and SUbparagraphs. Thus the numbering a. b. c. d refers to Subparagraph d of Paragraph c of Subsection b of Section a. Theorems and lemmas are numbered a. n, n = 1, 2, . . . , where a indicates the section.

This volume is a revised and enlarged version of Chapter 3 of. a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two of the large amount of new material. The whole book parts because is intended to introduce mathematicians and biologists with a strong mathematical background to the study of stochastic processes and their applications in biological sciences. It is meant to serve both as a textbook and a survey of recent developments. Biology studies complex situations and therefore needs skilful methods of abstraction. Stochastic models, being both vigorous in their specification and flexible in their manipulation, are the most suitable tools for studying such situations. This circumstance deter­ mined the writing of this volume which represents a comprehensive cross section of modern biological problems on the theory of stochastic processes. Because of the way some specific problems have been treat­ ed, this volume may also be useful to research scientists in any other field of science, interested in the possibilities and results of stochastic modelling. To understand the material presented, the reader needs to be acquainted with probability theory, as given in a sound introductory course, and be capable of abstraction.