## Albert Turner Bharucha-Reid's papers 1951-1955

Below we give details of some of Albert Turner Bharucha-Reid's papers published between 1951 and 1955. We give the Abstract of these papers and, in some cases, parts of the Introduction:

- A T Reid, On the diffusion of metabolic intermediates,
*Bull. Math. Biophysics***13**(1) (1951), 31-37.

A simple model of diffusible metabolic intermediates, produced in a spherical cell, is postulated. The rate of production of the intermediate is taken to be a linear function of its internal concentration. The intermediate is produced by the degradation of some precursor substance at a rate which is assumed to decrease with increasing internal concentration. Assuming the medium in which the cell is enclosed to be infinite in extent, and the external concentration of the intermediate constant, the time dependent and steady-state solutions of the diffusion problem involved are obtained. It is shown that production of the intermediate will not cease, for any value of*r*≤*r*_{0}where*r*_{0}is the radius of the cell. Possible application of this approach to the study of the occurrence of the lag phase in bacterial growth processes is indicated.

- A T Reid and H G Landau, A suggested chain process for radiation damage,
*Bull. Math. Biophysics***13**(3) (1951), 153-163.

It has been suggested that when an organism is exposed to ionizing radiation the initial damage results from the occurrence of ionization in a so-called sensitive volume due to absorption of radiation quanta. The initial radiation damage is then transmitted or amplified to a level of macroscopic perception. In this paper a mechanism by which this transmission may take place and a finite Markov chain model applicable to this transmission are postulated and discussed. This mechanism is assumed to be the depolymerization of essential chain molecules which are connected to some "central group" associated with the sensitive volume. The depolymerization of the macromolecules following a hit in the sensitive volume is postulated to be determined by a chain mechanism, which acts in a manner inverse to the mechanism controlling the polymerization process. A mathematical study of this problem is made using the theory of Markov chains. The probability of complete degradation of the chain macromolecule, and the probability of recombination of the units to give the intact chain were determined, assuming that the probability of successive steps in the degradation increase linearly from the intact state to that of complete breakdown.

- A T Reid, A Probability Model of Radiation Damage,
*Nature***169**(1952), 369-370.

In developing the mathematical theory of radiobiological phenomena it is necessary to consider two separate, yet related, problems. The first problem is concerned with the probability of occurrence of effective ionizing radiations within a so-called sensitive volume (or mass) of a living organism. This probability depends on the geometry of the target and the absorption of radiation quanta. The quantum hit theory which has been advanced to explain this initial effect has been discussed by D E Lea. The second problem, which is concerned with the probability that the initial damage to the system will cause a certain effect, has received very little theoretical consideration. In a paper to be published elsewhere I have considered a mathematical model of the transmission of primary radiation damage through a biological system. This model is based on the theory of Markov chains; It is the purpose of this communication to present the model developed and to suggest some possible biological interpretations of the results obtained. For the complete treatment of the problem the paper referred to above should be consulted.

- A T Reid, Note on Diffusion Controlled Reactions,
*J. Chem. Phys.***20**(1952), 915.

Two diffusion problems arising in an application of the Smoluchowski collision theory of reaction rates in solutions are considered. The diffusion equations are nonhomogeneous due to the introduction of an expression for the rate of production of the reacting particles. Two cases are considered: (1) The rate is constant, and (2) the rate is a linear function of the concentration. Solutions are obtained by the method of the Laplace transformation.

- A T Reid, Note on the growth of bacterial populations,
*Bull. Math. Biophysics***14**(4) (1952), 313-316.

In this note the explicit solution is given to an equation, suggested by C N Hinshelwood (1946), describing the growth of a bacterial population under the assumption that toxic products are a limiting factor. The behaviour of the culture as a function of time and the parameters (initial number, rate of growth, and rate of production of toxic substance) is discussed.

- A T Reid, On Stochastic Processes in Biology,
*Biometrics***9**(3) (1953), 275-289.

The purposes of this article are: (i) to discuss the role of the theory of stochastic processes in the methodology of mathematical biology, (ii) to present a review of some work dealing with the application of stochastic processes in biology, and (iii) to encourage, perhaps, other workers to utilise the theory of stochastic processes in formulating mathematical models of various biological phenomena. Mathematical biology is concerned with the development of a rationale for biological phenomena. Before the researches of N Rashevsky towards the development of a mathematical biology numerous studies were available in what might be termed mathematical ecology (Volterra, Kostitzin, Gause) and mathematical genetics (Fisher, Haldane, Wright). In the early days of mathematical biology the methods used by Rashevsky and his associates were mainly those of classical mathematical physics, with models of biological phenomena being postulated which could be treated by utilising these methods. It soon became evident, however, that the complex phenomena associated with the biological world could not be fully investigated by the mathematical methods of physics alone, and that new methods designed, perhaps, especially for biology would be required. That this situation might arise should come as no surprise when we reflect on the uniformity of physical and biological processes. In dealing with physical processes we have few variables between which to establish a functional relationship. Hence in physical experiments we need only to make a few determinations in order to predict the outcome of future experiments. This is an assumption on the uniformity of physical processes, and we find that it is deterministic. When we consider biological processes we find in the main a probabilistic or stochastic uniformity, a deterministic uniformity existing for only a few phenomena. With the above in mind it is perhaps clear why the formulation of abstract biological models should proceed more along stochastic rather than deterministic lines. Consider, for a moment, what we do in a biological experiment: We endeavour to measure the result of a series of interacting phenomena which vary in space as well as in time. We are unable, clue to our ignorance of biological organisation, to establish an exact relationship between the interacting elements which we have been able to ascertain as playing a role in the reactions leading to the measured result. Hence any attempts to give an exact relationship between biological observables are usually futile, or lead to complex models of the underlying mechanism which defy experimental verification or refutation. In developing mathematical theories in biology we should postulate models which answer the following sort of question: If observations on a biological system are taken as elements of the set of all possible observations, what type of model will enable us to predict observations that as a rule will be elements of a specified subset of the set of all possible observations on the system, while it is possible that the observations will not be elements of the subset? At present the theory of probability and stochastic processes is the only mathematical scheme suitable for answering such questions, and hence formulating the desired mathematical theories in biology.

- A T Reid, An age-dependent stochastic model of population growth,
*Bull. Math. Biophysics***15**(3) (1953), 361-365.

A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1 -*σ*), 0 ≤*σ*< 1, of the organism dividing into two at the end of its lifetime, and probability*σ*that division will not take place.

- A T Reid, Diffusion-controlled reactions in metabolizing systems,
*Archives of Biochemistry and Biophysics***43**(2) (1953), 416-423.

The Smoluchowski theory of diffusion-controlled reactions is applied to metabolizing systems. The following model is considered: A stationary spherical particle, surrounded by a reaction surface, is imbedded in a metabolizing system of infinite radius. Throughout the system reactant molecules are being produced autocatalytically. Reaction is said to occur when the diffusing molecules reach the reaction surface. Expressions are obtained for the concentration of the reactant molecules throughout the system and at the reaction surface, and the rate of reaction at the surface. The use of this approach in the study of lag phenomena in bacterial growth processes is discussed.

- A T Bharucha-Reid, Age-Dependent Branching Stochastic Processes in Cascade Theory,
*Physical Review***96**(1) (1954), 751-753.

A brief introduction to the recent Bellman-Harris theory of branching stochastic processes is given in the nomenclature of cascade theory; and a simple model in cascade theory formulated as an age-dependent branching process is given.

- A T Bharucha-Reid, On the Stochastic Theory of Epidemics, in
*J Neyman (ed.), Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, Vol.***1**(University of California Press, 1955), 111-120.

Early work in the mathematical theory of epidemics was mainly concerned with the development of deterministic models for the spread of disease through a population. An excellent review of the deterministic approach has been given by Serfling [Historical review of epidemic theory, 1852]. In this approach a functional equation (differential or integral equation, etc.) for*n*(*t*), the number of infected individuals in the population at time*t*, is derived on the basis of certain assumptions concerning the mechanism by which the disease is to be transmitted among members of the population. This equation, together with some initial condition (the number of infected individuals at time zero), is then solved to obtain n(t). In assuming a deterministic causal mechanism for the spread of an epidemic the number of infected individuals at some time*t*> 0 will always be the same if the initial conditions are identical. Because of the large number of random or chance factors which determine the manner in which an epidemic develops it became clear to workers in epidemic theory that probabilistic or stochastic models would have to be used to supplement or replace the existing deterministic ones. The development of the theory of stochastic processes has given the mathematical epidemiologist the proper theoretical framework within which his mathematical models can be constructed. Of particular interest are stochastic processes of the branching or multiplicative type. These processes can be described as mathematical models for the development of systems whose components can reproduce, be transformed, and die; the development being governed by probability laws [T E Harris, 1951]. A discussion of some stochastic models in epidemic theory has been given by Taylor [1956], and a detailed discussion of stochastic epidemic theory will be given in a monograph by the author [A T Bharucha-Reid, An Introduction to the Stochastic Theory of Epidemics and Related Statistical Problems, 1957]; hence, in this paper we will not give a review of previous work in this area. The purpose of the present paper is to consider the possible application of the Bellman-Harris theory of age-dependent branching processes [1952] to epidemics, and to discuss some statistical problems associated with stochastic epidemics.