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.

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.

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.

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.

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.

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.

The theory of epidemics has been formulated as a deterministic theory and as a stochastic theory. There has been little work, except for a recent paper by Bailey [1], on the transition between the two formulations. In this paper we present an alternate formulation to that of Bailey that leads to a perturbation method more easily extendible to higher orders.

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Let n and m be the initial numbers of susceptibles and infectives respectively in the well-known model of Bailey. The successive states of a population will be described by (i,j), where i is the number infected since the beginning of the epidemic, and j is the number removed (dead, isolated or recovered and immune) during the same period. The transition probabilities obviously are the following

Bailey (1957) has considered the general stochastic epidemic with relative removal rate ρ, one initial infective and n susceptibles. His Equation (5.51) may be re-cast as where the fk satisfy the recurrence relationship

Influenza epidemics at a city or country level can be observed only through a daily officially registered morbidity, or briefly DORM. The influenza model for any country is defined as some set of equations containing the DORM for cities in that country as unknown functions of time, these being designed to approximate the cities' real DORM.

Three simulation models for stochastic discrete-time epidemics involving two competing and interfering viruses in highly structured communities were presented.

Gaver (1963) and Antelman and Savage (1965) have proposed models for the distribution of the time to failure of a simple device exposed to a randomly varying environment. Each model represents cumulative wear as a specified function of a non-negative stochastic process with independent increments, and assumes that the reliability of the device is conditioned upon realizations of this process. From these models are derived the corresponding unconditional joint distributions for the random failure time vector of n independent, identical devices exposed to the same realization of the wear process. It is shown that the identical failure time distribution for one component can arise from each model. In the Gaver model simultaneous failure times occur with positive probability. The probabilities of specific tie configurations are developed.

For an interesting class of Gaver models involving a time scale parameter, the maximum likelihood estimates from several devices in one environment are examined. In that case the tie configuration probability does not depend on the parameter. For the corresponding Antelman-Savage models a consistent sequence of estimators is obtained; the maximum likelihood theory did not appear tractable.

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.