Download NATURE

No_ 49SS
~October
17, 1%4
NATURE
225
GENERALIZATION OF EPIDEMIC THEORY
AN APPLICATION TO THE TRANSMISSION OF IDEAS
By DR. WILLIAM GOFFMAN
Center of Documentation and Communication Research, School of library Science,
Western Reserve University
AND
DR. VAUN A. NEWILL
School of Medicine, Western Reserve
NE of the most -fundamentsl problems in the field
O of infoI'JIUl.tion retrieval is that of determining the
circumstances under which it might be necessary to introduce an information retrieval system as an aid to a given
population of scientists. It is proposed. that this problem
be examined in terms of the tra.nsmission and development of ideas within a population. Specifically, the
transmission of ideas within a population will be treated
88 if it were the tre.nsmission of an infectious disease,
that is, in terms of an epidemic process. An attempt will
be made to indicate the role of information retrieval in
the development of such a process.
The EpidemiC Model
- Since the spread of disease in a population is to be our
model for the transmisaion of ideas, it is appropriate to
disoU88 the essential prinoiples pertinent to this issue.
These prinoiples are a part of epidemiology. The necess~- elements involved-in the process of the spread of
infectious dise..... are those of: (1) a .pecified population; (2) an exposure to infectious material. The
meinbera of the population belong to 1 of 3 bllSic 01"""".
at given instant in time: (a) infective8. those members
who are host to the infectious material; (b) 8U8ceptible8.
thoSe who can become infectives ·given conta.ct with in·
"tOOtious material; (0) removalB. those who have been
zreInoved for one of a variety of reasons suoh as death.
~unity, .hospitalization. etc.
These latter members
may have been either susoeptibles or infectives at the time
Of their removal_
n.e entire process is time-dependent, that is, a sequence
'events ocours which describes the process. An in~
'" uaI is exposed to infectious material either by direct
ilo.ntact with an infective or through some intermediary
i8Bt or vector. The exposed. person may either be resistant
to the incoming organism. in whioh case the organism is
d~, or may be infected by it, in which case the
inVading organism proceeds on a course of development.
·TJie time interval in whioh the development takes place
ii ea.lled the 'latenoy period' (or incubation period), that
, Ii; the interval of ,time. necessary for a susceptible 1p be
~transfonned into an infective. In other words, the latency
Plriod is the time between the receipt of infectious material
/'~. 8 rJUBCeptible and the time at which he is in position
16-tlansmit infectious material to another susceptible, thus
'~ting the process. An epidemio ocours when the
prooess of susceptibles being transformed to infectives
crOsses a certain threshold_ (The American Public Health
~cia.tion (1960) defined an epidemic as "the occurrence
. . in' 8 oommwlity ... of a group of illnesses of similar
nature. olearly in excess of normal expectation.")
The process already described here does not take into
(lOQ()unt the almost endless number of complexities which
aotusUy arise. - It is a model, that is, an idealization of
the real situation in which the complex process is reduced
<to ita essential properties.
an
a
University~
Cleveland. Ohio
Transmission of Ideas as an 'Epidemic' Process
In general. the 'epidemic' process oan be characterized
as one of transition from one state (susceptible) to another
(infective) where the transition is caused by eA-posure to
some phenomenon (infectious material). The process
need not be restricted to infectious diBeaBe but is a more
general abstract process that might be applied to many
situations. All that is needed is the appropriate interpretation of the process elements, that is, susceptibles,
infectiv6S, removals, infectious material, intermediary
host. latency period, disease, etc.
People are susceptible to certain ideas and resistant to
others. Once an individual is infeoted with an idea he
may in turn, after some period of time, tra.nBID.it it to others.
Such a process can result in an intellectual 'epidemio'
(Table 1). For example, consider the development of
psychoanalysis in the early part of this century. Freud
was no less host to the infootioUB material of the 'disease'
of psychoanalysis than the person carrying the organism
capa.ble of transmitting a cold, nor is his writing leas of a
'vector' carrying the 'infectious material' than the mos·
quito as a carrier of malaria.
",
Moreover. Abraham, Ferenozi, Jung and Jones were no
less 'susceptibles' who were infected by the ideas of Freud
and who. after a certain latency period, themselves became
'infectivea t than are those iadividua1s infected by the
droplete expressed by a cold oarrier. J ung might represent an example of acquired resistance to the disease
while the resistance of the medical community of Vienna
could represent innate immunity. The development of
the psychoanalytic movement in the early part of the
twentieth century was in its way no less an 'epidemio'
than WllS the outbreak of influenza in 1917 and 1918.
One can argue similarly that Darwin and evolution,
Cantor and set theory. Newton and mechanics. and so on.
were examples of 'epidemics' in the world o~ scientifio
thought which were instigated by the introduotion of a
single infective into a population. 'I'h6 analogy is J?-ot
restricted to science; for examples suoh as Christ,
Buddha, Moses and Mohammed can be cited. in the
Table 1.
ANALOGY BETWE.EN INnCTlOUS DISUSE JJfD INTELLECTUJ.L
El'mEliliCS
Elements of
the epidemic
proc~al5
HOlt
Agent
Infective
..
EJemenu Interpreted 1n tenna of
Infectioull dlaeat.e epidemic
Intellectual epidemic
Infectious material
Caae of disease
Susceptible Person who will be Infected
ginn ef[octh-o contact
Removal
Vector
Agent
Death or lmmunlty
Idea
Author of paper
Rellder of paper who will be
infecred given effective oontact
Death or lou oflnterellt
Infectious material (as for Idea (al for boat.)
host)
Infective Vector harbouring the agent Paper containing usefulldenl
Susceptible VectOr not harbouring the All papefll cClntalnInG' poten·
8{l:ent
tlAlIy naefuJ Ideas
Remo\-o.l
Death
Deletion or JOY
226
NATURE
October 17, 1964
YOL. 204
religious field and other examples can be given almost be 88 broad 88 medicine. but might be epidemiology, and
endlessly.
mathematical epidemiology might be the suhfield (D).
Although biological and intellectual epidemics can be N must be specified because D might develop into an
considered ns special cases of a general process. there are 'epidemio' in some popula.tion N 1 but not in some other
important differences. Intellectual epidemics are often N.. For oxample, the 'disease' of mathema.tical epidemi.
desirable while biological ones usually are not. This ology will probably follow different oourses in the fields of
difference. however. does not pertain to the structure of epidemiology, where it might become an 'epidemio'. and in
the proceasea hut only to external factors whioh might the field of mathematics, where it probably will never
be introduced. In one oase, one usually wishes to enhance become an 'epidemio'. An additional example from the
the process, while in tho other to eliminate it.
field of medicine might be the present epidemio of cancer
Another important differonoe relates to the infectious of the lung among smokers that is certainly different from
material. In the biological case the infective individual the pattern of development of the disease among nonproduces infectious material that closely resembles that smokers.
.
which started the process; there baa been little mutation
Let N I be the set of a.uthol"8 who have written papors
or change. Such is not the case with the intellectual in F a.t time t l , whore t. represents the starting time of the
epidemio, for here BOme mutation or ohange of the original proC688. Let I. be the set of a.uthors who have written
idea (infectious material) is prerequisite to placing it in papers in D at time teo Then I D represents the infective
print. •
popnlation at the beginning of the proceaa. Ifit is assumed
Ideas are transmitted by personal communication or that there are n() removals a.t time t l • then 8.=N.-I I
by puhlication. An epidemio cannot develop within a repreoent.. the suaeeptible popnlation at time,.. (Romovgiven population unless there is effective contact betw.!*Ul ·ala oonstitute .those membere of N who for BOme reason
the susoeptibles and infectious material. The purpose of or other are no longer produoing papers in F.). .
'an infonnation retrieval system is to provide such effective
Let I'. be the Bet of artioles produced in D .hy the
oontaot whore it does not already exist. If such is the 088e, members of I. at time 10 and 8'. be the set of artic1eo an information retrieval sy&.em might be introduced even produced by the mefnbers of S. at I .. Than N'.= S'. + I',
though such aystema are limited to the traosmission of oonatituteo the vector popnlation at time ,.. .
ideas contained in published mate1'ial.
.
In general as the proceaa developB we have N =8 +I +B
'. The essential question is : when is it desirable to intro· and N' = S' + I' + J!'. where B and B' repreoent removals.
duce a formal information retrieval system into the At time t l , RI=O and B'.=O.
-:
intellectual activity of a population of scientists' Here
The proceaa might be described as followa : A member
the epidemic model may be of value for it permits the • of 8 is infected hy a meniber of I'. After a certain
rephrasing of the queetion in terms' of epidemiological length of time (latency period) • becomes. an infective
problema Buch as: (I) the tranamiasion and spread of and giVOB birth to new infections in N' which in turn may
the disease; (2) the prediction of the epidemio COUl'B& ; infect other members of 8 and BO forth. In other words
(3) the diacovery ofthe threBhold densities ofthepopnlation 8 produces a paper in D in which he oites a paper in I'.
that might be paaead before an epidemio can develop. (The member of I' might contain more than one idea.
The answer to these epidemiological questions might help (more than one type of 'infectious material') with reapect
. to decide when an information retrieval system should to D. This same member of I' may contain ideas tha~
oould also infect members of a Bub·field other than D.
be introduced.
Our only interest in the member of I' is as a carrier of the
'infectious material' of D.) By producing this pBper
Mathematical Models
he has introduced a new infective in N'. He may also
There ha.a been extensive development of abstract have cited papers in 8' 88~well,·so that he has infected or
mathematical models in the theory of epidemics which associated members of 8' with the 'disease' D. The
can. be put to general use. There are two types of models, appearance of the pa.per written by 8 in D indicates that
namely, deterministic and stochastio. The deterministic he has made the transition from the susceptible to the
approach represents the process as a system of differential infective state.
equations while the stochastic approach describes· the
Let ~ be the rate at which the members ofB become
process as a finite state Markov process of either a discrete members of I. HODC6. 13 is the rate of infection in the
or continuous parameter (the discrete parameter is popnlation N. Let~' be the rate of infection in the
applicable where the latency poriod is constant, in which population N'. Then~' represent.. the rate at which
case the infectives occur in generations) depending on the papers in S' are being oited by membel"8 of N who are
physical situ'!otion. The stochastic representation, though producing papers in D.
generally moro realistic, is mathematically more sophisAs the process develops with time, a. certain number of
ticated. An excellent detailed treatment of this subject 8usccptibles and infectives in the host and vector popu·
as related to infectious disease can be found in Baileyl.
lations a.ro removed. In the case of the host population
this removal might be loss of interest, death, etc. In
the case of tho vector population it might be lack of citation
A Deterministic Model
due to deletion, inaccessibility. etc. Let y (y') be the
The most natural avenue for exploring the process of rate of removal of infectives I (I') and let 8 (8') he the
the tro.nsm.i::.sion of idOO8 within a population would seem rate of removal of susceptibles 8 (8').
Another aspect of the process as it develops is that new
'to be by way of the literature produced by the members
of that population. Although the published article is supplies of susceptiblcs and infectiv08 are introduced into
not the only way in which ideas can be transmitted, it the populations Nand N'. Let I' (1'.) be the rate at which
does play an important part. We are interested in studying new susC6ptibles are introduced and let u (u') be the rate
this role, for it is only in this respect that an information a.t which new infectivee are introduced. That is. ~
retrieval system can be justified at present. In terms of represents the rate at which new a.uthol"8 a.re introduced
the epidemic process we arc therefore concerned with the in F but not in D, while 11' is the rate a.t which new papers
tra.nsference of infectious material, that is, ideas. between are introduood in F but not in D. u represente the
human hosts by means of an intermediary host or vector, rate at which new authors are introduced in D and u' the
rate at which new papers are introduced into D.
namely, the written article.
The mathematical model to be presented includes the
We wish to study the spread of the 'disease' D within
the papulation N of the discipline F. Consider a. certA.in following states of transition of the host and vector
field. (F) such as medicine and a certain Bubfield (D of F) populations within a short time interval de: the sus·.
llUCh as medical epidemiology. The field (F) need not ceptible can rema.in a susceptible or become an infective .~
NO.4905
Table t.
NATURE
October 17, 1%4
BurE8 01' Tuxamo!f OJ' mJl B09T .4.lm Vscrolt POPtrL.UIONS·
8t.a£e loO which an lndlTldnal bdoDgl
At time,
At time t+ At
ll/SaICt!pUble SUlIOCptible.1nfec::tlve or removal
(2 Infeet:lvtl
(3) Removal
Infective or removal
or removal, the infective can remain an infective or become
a removal, and tho romoval must remain a. removal
(Table 2). (To keep the mathematical model simple in
this presentation. we are not permitting immunes to
lose their immunity and to return to the susceptible
population.) The latency period is asswned to be zero
80 that infective& occur continuously with time. If we
assume homogeneous mixing among the members of the
populations. the total process can be detenninistically
described by the follo,....ing systems of differential
equations :
dS
dt = -IlSI'- BS+{L
IlSI'-yI +u
dS'
<It
= -WS'I-BS'+{L'
dI'
dt =f)'S'I-y']' +u'
dR'
dR
dt =yI+BS
d:l =y'1'+8'8'
- A threshold density of susceptibles can be obtained if
we 8BSUID.e initial conditions of a single infectivo introe,luced into the host (N,l and vector (N',l populations
at time t., that is, No=(S., I, O) and N'.=(8'o. I, O}.
For an epidemic to develop from time to, the change in
the number of infeotives in both the host and vector
populations must be positive. Hence, from the above
differential equations, we obtain the threshold pp' as
follows :
IlSI', > yI - u
..
~'S'I'
> y'I' - u'
"or ~'8.s'o > rr-yu'-yu+uu'
.' and_, S,8', >
.
y-u
(y;,U») (y' WU'»)
where· - - = p; and
.
~.
-r =
y' -u'
= pp'
p'
,!thf;The epidemic curve whioh traoee the development of
~~ epidemio in time is given by the differential equation
Ti."-
other of epidemiologists. The 'disease' could be statisticel
epidemiology. Then one might study the development
of the field in terms of the interacting host and vector
populations rother t.han in terms of single populations.
Removal
• In ll'llDeral the bOllt and vector populatlonll need not follow the &arne
tran.s!Uonal patterna III In thlB ease.
.~ =
227
P(ll with peake at the points iIi time where :~
=O'
88.1
~ to infinity. that is, the size of the inf~tive population
!J; after a long period of time. This discU88ion indicates
~ ~ epidemic can only develop from time t. provided
~ the product of the hoat end vector susceptible popalation exceeds the threshold pp'_ When a potential
epidemic condition is present, the quantity of soientiste
~ information is suffioiently large that given effective
oonteot en epidemic can develop. It is at this time that
au information retrieval system might be introduced.
'If the population is already in an epidemic state, an
~ lDtormation retrieval system can be introduced in order
• inaintain or inorease the frequency of effective contacte
1ietween susceptiblee end infeotivee_ Hence, the points
.. at whioh an information retrieval system might be usefully
..introduced into a population are either at the initial
....pgint or the peak of the epidemio process or at a time
l'then intensification of the epidemio process is desired.
Other models different from the one we have presented
could be constrncted that would interpret the epidemic
hi~a different manner. For example, a model can be
elaborated by introducing two int.Precting populations
with rates of migration from one population to the other.
One population might consist of biostatistioians and the
:lII!i; size of the epidemio is given by the limit of I
The Stochastic Approach
In a stochastic model of the process described above tho
actual number of now infectives occurring in a sbort time·
interval ,,""ould be replaced with the probability of a new
case occurring in that interval. This type of mathematical
treatment is partioularly applicable when dealing with
small populations.
An epidemio process within a large population can be
characterized as a collection of epidemic processes within
sub.populationB. For example, the outbreak of an
infectious di.sease in a community might more specifically
be studied in terms of the household populations rather
than the community as a whole. ~imilarly the intelleotual
epidemio process within the population of a ce~in
discipline could be studied in terms of its sub-populatiOns,
that is, the 'disease' D of field F might be studied in t;.erma
of the sub-fields of F, indioating, from the point of view
of information retrieval, where intellectual epidemics are
developing in a given discipline.
Thua. a stochastic treatment of the epidemic procoases
would in general be more desirable because the size of
the populations being studied hae-been reduced. Unfortunately a stochastic model of an epidemic proceea
with a vector population does not exist as far as we lmow..
Such a model is in need of development and attention is
being given to this problem by the authors.
Deterministic models, however. can be very usoful,
partioularly in dealing with largo popuJetions, end experiments have been designed for the purpose of testing the
model presented above.
Discussion
The construotion of 0. matheIIl&tical model of a oomplex
physical situation" involves a certain amount of simplification. This is because the purpose of·the mathematicel
model is the description and prediotion 'of the essential
patterne of the phyeicel prooese end not the 80hievement
of its complete analysis.
A total understanding of tile procees requiree studies
complementary to tho mathematicel ones. In the C880
of the infectious diseaeee this means biologicel, eoologioaJ;
and cliniceJ investigations of the process as wen as mathematical; in the transmiMion of ideas, psychological,
sociological, and linguistic studies.
MathematiCB oannot indicate why exposure to a certain:
virus will lead to infection or why a certain article will
infect the reader with an idea. These important qU08tions
must be answered elsewhere. However, mathematics
can makeimportantcontn'butionstoward an understanding
of the epidemio prooess if one keeps its limitations in mind.
Of COUl'88, the usefulness of a mathematical model
depende on: (I) boing able to solve the model; (2)
obtaining reliable me88Ul'ements in order to test it.
These are by no means negligible problems. since even
the most simple representations present difficulties in
obtaining alg~braio solutions. This is particularly true
of the stochastic 0888. Additional problems are encountered in relation to defining popuJetions end obtaining
reliable estimates of the parameters of the proC668.
Although implementation of the epidemio model
presents certain difficulties, much useful information
concerning the transmission of ideas within a population
might be obtained from the mathematicel appro80b.
It could help in answering several questions that are
basio for the design and operation of information re·
trieval systems. These questions include: (l) Where
and when is aCtivity within a given discipline doveloping
into 'epidemio' proportions I (2) What is the expected
228
NATURE
duration of this 'epidemic' activity T (3) 'What is the
intensity of the 'epidemio"
(4) What are the clo.ssio
papers of a certain discipline, that is, the initial infecti vea
of lUl outbreak 1
.
Infonnation retrieval systems must be dynamic and
should reflect the interactions between the researcher
and the literature. It seems that the 'epidemic' approach
is a workable method for describing and predicting cert.ain
fundamental aspects of this interaction.
Since the process of t~tting ideaa, 88 in the case
of an infectious disease, is not a Single process within a
population but a collection of interacting processes ,vithin
October 17, 1964
VOL 204
Bub.populations, it would seem that the notion of an all.
encompassing information retrieval system spanning the
totality of knowledge should be replaced by the notion
of small dynamio interrelated. systems that appear when
needed and disappear when not needed.
'Epidemio' theory could have a wide range of applio.
stions other thnn to the transmission of ideas. for example:
to the study of certain ohronic diseases, divorces. accidenta,
etc.• but these will not be elaborated on in this report.
This work was supported by AFOSR grant 403-64 and
U.S. Public Healtb Service grlUlt AM 063911--03.
.
I
BalIey, X. T. I., PM Malktm4tWll Tht.orv 01 Epitkmiu (Grt.mD, 1980).
NEWS and VIEWS
Director of the Rowett Research Institute, Aberdeen:
Dr. D. P. Cuthbertson, C.B.E.
DR'-D. P. CtrnmERTSON, who will be retiring from the
directorship of the Rowett Research Institute, Bucksburn,
Aberdeen, on September 30, 1965, obtained his medical
'qualification in Glasgow and was for nearly twenty years
leoturer in pathological chemistry 6nd physiological
chemistry in the Royal Infinnary, Glasgow, lUld in the
University of Glasgow. ·.... He beeame an authority on
the biochemical effects of injury. and particularly on the
acute depletion of protein following injury. His work on
the prevention, and treatment of these changes became
of espeoial importance during the Seoond World War.
Cuthbertson worked for two years on the headquarters
staif of the Medical Research Council before he was
appointed director of the Rowett Research Institute,
Aberdeen. in 1945. Under his influence the Institute grew
rapidly in size and importance. and it is remarkable how
Cuthbertson was able to direct so much work and produce
an interesting report each year, summarizing work that
had been published in 100 papers or more. He managed.
however. to keep his own laboratory going and to give a
number of lectures, which were published. He is an
accomplished artist in water coloU1"8 and a keen golfer,
and will clearly be able to find something to do when he
retires.
Dr. K. L. Blaxter
DR. K. L. BLAXTER, at present head of the Nutrition
Department, Hannah Dairy Research Institute, has been
appointed to succeed Dr. Cuthbertson 88 director of the
Rowett Research Institute. Dr. Blaxter. who is fortyfive, is a graduate of the University of Reading. He has
been with the Hannah Dairy Research Institute since
1948. Before that he was a research officer at the
Veterinsry Laboratory of the Ministry of Agriculture,
Weybridge. During 1946-4-7 he was a Commonwealth
Fund Fellow at the University of Illinois, Division of
Nutrition. under Prof. H. H. Mitchell. With the exception
of B period of war service v.-ith the Royal Artillery. he
was a research assistant at the National Institute for
Research in Dairying from his first graduation until going
to Weybridge. Dr. Blaxter was Clive Behrens Lecturer in
the Faculty of Agriculture, University of Leeds, during
1958-61; 1960 Thomas Baxter Prizemsn lUld Gold
Medallist (for research on the nutrition of dairy cattle);
National Research Council of Canada visiting lecturer to
the Universities of British Columbia, Manitoba, SaskatchewlUl and Alberta in 1964; 1964 Royal Agricultural
Society Gold Medallist (for research on the nutrition of
farm livestock). He also gave the Fernhurst Lecture to
the Royal Society of Arts in 1962, lectures to the Pro·
fessors Council of the University, Uppsala, and to the
Berlin Acndemy of Sciences, D.D.R., in 1962, and the
third Samuel Brody Memorial Lecturer in·the University
of :Missouri in 1964. Dr. Blaxter's work has been concerned with the nutritive value of home.grown foods for
dairy cattle, food intake lUld food utilization, protein
requirements and protein metabolism of dairy cattle and
calves, hyperthyroidism in the ruminlUlt lUld tho use of
iodinated protein, hypomagnes::e.mic tetany. vitamin E
metabolism and muscular dy~trophy, energy metabolism
and the energy requirements of cattle. and their relation
to the environment. In 1962 his monograph entitled Tho
Energy Metabolism oj Ruminants was puhlished. Dr.
BJ.a.xter is a member of numerous learned societies and is
a member of the joint Agricultural Research CouncilMedical Research Council Development Commission sub.
committee on the monitoring of radioactive fall-out and
of tho Working Party on Ruminants of the Agricultural
Research Council's Technical Committee on Nutrient
Requirements of Farm Livestock.
Ministry of Aviation:
Director of Engine Research and Development (I):
Mr. -Dennis H. Mallinson
MR. D. H. MALLINSON has been appointed director of
engine research and development (I), Ministry of Aviation,
in succession to Dr. J. Remfry, who has retired from the
Public Service. Born in 1921, Mr. Mallinson was educated
at Prince Henry's Grammar School, Otley, before going
to the University of Leeds in 1939. He graduated with
honours in applied mathematics in 1942 and was posted.
88 a junior scientifio officer to the Engine Department of
the Royal Aircraft Establishment. As a member of the
small but growing turbine division under Hayne Constant
and W. R. Hawthorne, he was employed on engine performance investigations and particularly on the analysis
of results from the flight tests of the pioneer jet aircraft.
Transferring with the Royal Aircraft Establishment's
Turbine Division to Power Jets, Reeearch and Development, Ltd.• he remained on engine performance work and
was in charge of the Performance Section when the
National Gas Turbine Establishment came into being. In
1949, following a period of about a year in which he acted.
as technical and administrative assistant to the deputy
director (Mr. Peter Lloyd, the present directA:?r-general of
engine research and development), Mr. Mallinson became
a member of the National Gns Turbine Establishment's
ramjet research team Wlder Mr. R. P. Probert, concentrating first on thermodynamics but later encompassing
fuel control systems and combustion development and
finally assuming responsibility for the full-scale ramjet
test vehicle programme. This programme was successfully completed in 1957 lUld from then until ssrly 1963
he was superintendent of the Engine Research Division
at the 'National Gas Turbine Establishment in charge of
work on propelling nozzles, control systems and highspeed ramjets. He transferred to Ministry of Aviation
Headquarters, London, in· February 1963 88 888ist&nt
director responsible for engine project assessment and
continued in this post until receiving his present appointment. Dennis Mallinson is 0. district commissioner of the