Information Science Approach to Biomathematics and Its Implications to Ecology andQuantitative EpidemiologyAuthor(s): Tosio KitagawaSource: Advances in Applied Probability, Vol. 3, No. 2 (Autumn, 1971), pp. 198-200Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426154 .

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198

Statement 7. General epidemiologic situations-for instance, relations between the functional groups-may be carefully studied by means of the

theory of signed graphs and balancing processes.

Statement 8. The intersection of two digraphs is the indispensable opera- tion to detecting low epidemicity. We consider usually the space-time in- teraction (clustering), i.e., SD n TD, both digraphs having the same set of

points with the same ordering. Sometimes space-time interaction is not suf- ficient to emphasize the epidemic character of a disease, the two other types (iii) and (iv) of relations being also useful, and particularly the operation TD n XD.

Statement 9. The probabilistic approach to structural models can be made in two ways:

(a) if we consider a fixed structure: the percolation theory; (b) if we consider a random structure: statistical inference (for example,

estimating the actual structure of a graph which is not completely observable).

Selected references BARTON, D. E., DAVID, F. N., FIX, E. AND MERRINGTON, M. (1967) Tests for space-time

interaction and a power function. Proc. 5th Berkeley Symp. Math. Statist. Prob. Vol. IV, 217-227. Univ. California Press, Berekley.

CAPOBIANCO, M. F. (1970) Statistical inference in finite populations having structure. Trans. N. Y. Acad. Sci. 32, 401-413.

FLAMENT, C. (1963) Applications of Graph Theory to Group Structure. Prentice-Hall, Englewood Cliffs.

HAMMERSLEY, J. M. AND WELSH, D. J. A. (1965) First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In: Bernoulli, Bayes, Laplace. Ed. by J. Neyman and L. M. LeCam, 61-110. Springer-Verlag, Heidelberg.

HARARY, F., NORMAN, R. Z. AND CARTWRIGHT, D. (1965) Structural models. An Intro- duction to the Theory of Directed Graphs. Wiley, New York.

PIKE, M. C. AND SMITH, P. G. (1968) Disease clustering: A generalization of Knox's approach to the detection of space-time interactions. Biometrics 24, 541-556.

TOSIO KITAGAWA

Information science approach to biomathematics and its implications to ecology and quantitative epidemiology

TOSIO KITAGAWA, Kyushu University, Fukuoka, Japan

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

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Information science approach to biomathematics and its implications to ecology 199

approaches for biomathematics are required from the standpoint of infor- mation science. In fact three requirements are mentioned: (10) general system approach; (20) historical approach; (30) three indispensable constituent fields of biological science. Regarding (P1), 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 (10) discrete, (20) combinatorial, (3Y) dynamical, (40) evolutionary, and (50) design mathematics.

In part B, "Prolegomena to cell space approaches", which is the step to a realization of biomathematics, the author starts with the fundamental notions such as (i) unit cell, which may be either square, or triangular or hexagonal; (ii) basic cell space consisting of four unit cells; (iii) cell spaces such as m x n rectangular cell spaces and triangular cell spaces A("). Then the notion of local mapping transformation satisfying the principle of local majority, abbreviated by LMT, is introduced in each basic cell space. For instance for a 2 x 2 basic cell space with 4 square unit cells, a configuration is defined by an allocation of xi (i = 1, 2, 3, 4) where each xi is either 1 or 0. Let us introduce

X =xx X2 [I 1 5 0 0

x3 X4 1 1 ' 0

Now LMT is defined as

I, if S(X)?>3

LMT:'X-* = X, if S(X) =2 0, if S(X) ? 1,

where S(X) = =1x,.

Particular considerations are given to an m x n cell space consisting of

mn x n square unit cells. A configuration in the m x n cell space is defined as an m x n matrix (xij) (i = 1,2, ...,

m; j = 1,2, ...,

n), where each xij is either 1 or 0.

Regarding an application procedure of LMT's, various firing schemes are considered which involve transitions among configurations. A configuration which remains unchanged by any application of LMT is called a stable con- figuration.

The number of stable configurations in an m x n cell space is 2m+"n-1

Similarly for A)" cell space it is equal to 2at"-. The application of the Roma-

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200

nowsky-Gantmacher theorem in Markov chains gives an asymptotic behaviour of transitions of configurations under a certain firing scheme of LMT's.

The third part, C, "Guiding principles in cell space approaches", concerns our cell approach with numerical approximations in elliptic and hyperbolic partial differential equations. In this connection it is crucially important to refer to a new notion of determinative subspace introduced by the co-author Masako Yamaguchi.

These three parts have been extracted from three recent papers belonging to a series of contributions on "Information science approach to biomath- ematics", I (T. Kitagawa), II (T. Kitagawa and M. Yamaguchi) and III (T. Kitagawa), published as research reports of the Research Institute of Funda- mental Information Science, Kyushu University, Fukuoka, Japan. In Section 6 of Paper I, the author discusses biological control systems analysis and meth- odology of ecology to some extent. Some aspects of these cell approaches with reference to ecology are suggested.

However their implications to quantitative epidemiology are entirely due to the further developments and elaborations of cell space approaches sug- gested by the author.

NORMAN C. SEVERO

II. SOME PARTICULAR EPIDEMIC AND CELL MODELS

Multidimensional right-shift processes

NORMAN C. SEVERO, State University of New York at Buffalo

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',mk

the set of (mk + 1)-

tuples rk = (rkmk,, ... k,, rk,o) having non-negative components summing to

Nk, and by Xk(t) = (Xk,mk(t),-", Xk, (t),Xk,o(t)) an (k + 1)-tuple random variable taking on values only from the set S'•k,,. We now let r = (r ,

""., r,)

designate an element from the v-fold cross-product set

Sk, = S/,, x .-' x S., Sjm Nx**i NvSjm xS

thus the kth coordinate of the vector r is the vector rk, itself having mk + 1 coordinates. When N = ku, = 1,***...,V, we define ek(i), k = 1,..*,v and

i = 0, 1, ..-, mk, to be the vector r with components ru, = bku6i, U = 1,,

V, and v = 0, ..., m,. For example, when v = 2, mi = mi = 2, then

e1(2) = (1, 0, 0; 0, 0, 0), e2(0) = (0, 0, 0; 0, 0, 1).

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