Part I [21] treated the case of a finite number of independent random uniform s-flats in an ‘admissible’ subset of Ed (s = 0, · · ·, d − 1). In this second part, the natural and fruitful ‘Poisson extension’ to a ‘countable number of independent random uniform s-flats in Ed itself” is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.

This Symposium was organized “to explore existing mathematical theories of disease and the possibilities offered to the medical sciences by further research in this field”. Because of the extreme breadth of these terms of reference, the discussion was limited to problems of communicable disease. The intention was to bring academic mathematical research into closer contact with applied problems in public health. It was thought that in this way, WHO would have greater access to theoretical development, while mathematical work in Universities and research institutes would achieve closer contact with practical problems.

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

This report is based mainly on the research conducted in 1969–70 in the Division of Research in Epidemiology and Communications Science of the World Health Organization, and gives a short résumé of methods elaborated for concise descriptions of what has been observed for prognosis of subsequent development of disease, and for the planning of new studies.

Many service systems exhibit variations of a random nature in the intensity of the arrival process or of the speed of service or of both. Changes in work shifts, rush hours, interruptions in the arrival process, server breakdowns, etc. all fall into this category.

The present study deals with a generalization of the classical M/G/1 queue by considering an extraneous process of phases which can be in one of the states {1, …, m}. During any interval spent in phase i, the arrivals are according to a homogeneous Poisson process of rate λi and any service initiated during such an interval has a duration distributed according to Hi(·). The process of phases is assumed to be an irreducible Markov chain in continuous time and is fully characterized by its initial conditions, by an irreducible stochastic matrix P and by the mean sojourn times σ1-1, …, σm-1 in each phase.

Independently of the queueing aspects, this arrival process is a generalization of the classical Poisson process which can be of interest in modelling simple point processes with randomly fluctuating “arrival” rate.

Two approaches to the time dependent study of this queue are presented; one generalizes the imbedded semi-Markov process obtained by considering the queue immediately following departure points; the other approach exploits the relationship between this queue and branching processes. The latter is more elegant from a purely theoretical viewpoint and involves iterates of a general type of matrix function introduced by the author. By making extensive use of the Perron-Frobenius theory of positive matrices the equilibrium condition of the queue is obtained. While retaining a similar intuitive interpretation the equilibrium condition is substantially more complicated than for the M/G/1 model.

The recurrence relations which yield the joint distribution of the phase state at time t, the queue length, the total number served and the virtual waiting time at t are exhibited in detail. Via transform techniques a number of limiting and marginal distributions are discussed. The discussion relies heavily on the theory of Markov renewal processes.

Throughout the paper and in a final section the author advocates the use of the structural properties of the queue and the resulting recurrence relations to organize the numerical analysis of complex queueing models such as the present one.

More explicit results for the case of two phases are given and are compared to results obtained by Yechiali and Naor for a closely related two-phase generalization of the M/M/1 queue.

The purpose of this paper is to bring to the attention of epidemiologists the possibility of setting up and utilizing structural models, more precisely, mathematical models based on graph theory. We shall present the first general results in this investigation using some statements and definitions.

The paper consists of three parts A, B and C. In part A, “A contribution to the methodology of biomathematics”, general considerations on mathematical analysis and automaton theory are given in order to show how and why new approaches for biomathematics are required from the standpoint of information science. In fact three requirements are mentioned: (1°) general system approach; (2°) historical approach; (3°) three indispensable constituent fields of biological science. Regarding (1°), the author refers to his work on the logic of information science explained in 1968 at the CAS Seminar in Dubrovnik and published in 1969 in a Japanese monograph. Regarding (2°), reference is made to his paper “A contribution to the methodology of futurology” published in International Conference on Futurology, April 1970, Kyoto. The three indispensable constitutent fields are given by (a) theoretical, (b) experimental and (c) engineering biologies. In view of these three requirements, characteristic features of biomathematics in connection with an information science approach are (1°) discrete, (2°) combinatorial, (3°) dynamical, (4°) evolutionary, and (5°) design mathematics.

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

Let v be a positive integer and for each k = 1, · · ·, v let mk and Nk be a positive and a non-negative integer, respectively. Denote by S'Nk,mk the set of (mk + 1) -tuples rk = (rk,mk, · · ·, rk,1, rk,0) having non-negative components summing to Nk, and by Xk(t) = (Xk,mk(t), · · ·, Xk,1(t), Xk,0(t)) an (mk + 1)-tuple random variable taking on values only from the set S′Nk,mk.

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinary M/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinary M/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of the M/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.

The effect of an epidemic of chickenpox on the course of an epidemic of yaws in a village of New Guinea was studied by Gart and deVries (1966) and Gart (1968). A deterministic continuous time model was used to confirm that the yaws epidemic accelerated dramatically in the month following the chickenpox epidemic. However, by the nature of the model, no statistical tests are possible.

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.

Recent research at Sheffield, which is briefly described below, has been concerned with chain-binomial models, and stochastic path integrals under infective curves.

In modern medicine it becomes more and more urgent every year to solve new epidemiological problems. One of these is the evolution of resistance to antibiotics of a large spectrum of microorganisms; this resistance is a consequence of the mass administration of antibiotics. In this problem it becomes necessary to apply the mathematical methods of epidemic theory as here the key lies in a purely epidemiological factor, namely the mass character of the use of antibiotics. A first model of the phenomenon encountered is presented.

There exists already a considerable literature about a mathematical approach to the description and control of malaria (see Ross (1911); Martini (1921); Lotka (1923); Moshkovskii (1950); Macdonald (1957), (1968)). This note gives an outline of the main assumptions of a simulation model which has been developed in close co-operation with epidemiologists, entomologists and immunologists in connection with a field project in Northern Nigeria where the necessary data for testing the model are being collected.

When a basal cell divides, both daughter cells remain in the basal layer of the epithelium, with one of the neighbouring cells being pushed out to make room. This fact opens the possibility that a cell with a heritable advantage over the normal cells may gradually produce a clone covering more and more of the basal layer. The advantage in question may consist in a faster rate of division than normal, or a more tenacious hold on the basement membrane; we shall limit consideration to the former situation.

We consider the usual stochastic model of a general epidemic, consisting of n + 1 homogeneously mixing individuals, assuming that initially when t = 0 there is one infective and n susceptibles (Bailey (1957)). At any time t we suppose that there are r susceptibles still uninfected, s infectives in circulation, and u individuals who have been removed (and are dead, isolated, or recovered and immune), where

This note is prompted by the papers of Weiss (this Symposium) and Bailey (1968). Weiss develops a technique for approximation to the moments of an epidemic process by regarding them as expandable in powers of N-1 where N is the size of the population, assumed constant. He first considers the simple stochastic epidemic with no removals and obtains explicit formulae for the terms of order N-1, the zero order terms being the deterministic values. Bailey is concerned with a similar type of approximation and he derives explicit results to the same order. Bailey uses an eigenfunction approach whereas Weiss's method is more direct and perhaps easier to generalise. However, in attempting to extend the method to the case of a closed epidemic with removals Weiss is led to intractable difference equations.