466 



NATURE 



[October 5, 191 1 



SOME OV .IMITATIVE STUDIES IN 

 EPIDEMIOLOGY. 



A N account of some quantitative studies in epidemiology 

 "^ has recently been published in the second edition of 

 my book on the " Prevention of Malaria " (Murray), and 

 the Editor of Nature has asked me to give a general 

 description of them here. The attempts originated in the 

 following manner. Shortly after Anophelines were shown 

 to carry malaria, it was often observed that little apparent 

 correlation could be found between their numbers and the 

 numbers of infected persons in a locality. The observa- 

 tions were always far too scanty to establish any real 

 absence of correlation ; but they were used, nevertheless, to 

 support the thesis that the amount of malaria does not 

 depend upon the number of the Anophelines, and that 

 therefore the proposed anti-malarial measure of mosquito 

 reduction (then very unpopular) was useless. For many 

 reasons a trustworthy experimental investigation would 

 have been very difficult and costly, and it was therefore 

 all the more necessary to examine the subject by a care- 

 fully reasoned analysis of the relations which must hold 

 between the amount of the disease and the various factors 

 which influence it. My first attempt in this direction was 

 made in an official report on the " Prevention of Malaria 

 in Mauritius " (Waterlow and Sons, 1908), and fell into 

 the form of a simple difference equation. This was further 

 developed in the first edition of my book already mentioned, 

 and the subject was at the same time ably attacked by Mr. 

 H. Waite, at the instance of Prof. Karl Pearson, in 

 Biometrika, October, 1910. 



The attempt now referred to aims at extending the 

 reasoning to infectious diseases in general. The object is 

 as follows. Suppose that a given proportion of a popula- 

 tion in a given locality at a given moment are infected 

 with some disease. Then we know from experience that 

 the number will not remain fixed, but will vary from time 

 to time and from place to place. The problem is to calcu- 

 late these variations on the supposition that all the 

 coefficients are known, which, of course, is by no means 

 always the case. The use of the calculation will be (1) to 

 obtain more light regarding the coefficients by comparing 

 calculated with observed results ; (2) to obtain quantitative 

 estimates as to how far each coefficient should affect the 

 result ; and (3) to improve preventive measures by showing 

 which factors they should be directed against. My studies 

 have been hitherto concerned only with time-to-time varia- 

 tions, and the reader will understand that they require 

 verification and completion by better mathematicians than 

 myself. So far as I can ascertain, the subject has been 

 little dealt with hitherto. 



We must first obtain clear ideas on some points. 

 Infectedness is not the same thing as sickness. Infected- 

 ness begins when the infecting organisms first enter the 

 body of the host (man, animal, or plant), and ceases only 

 when the last of them die out of him or leave him, or 

 when he himself dies. Sickness may be quite absent 

 during the whole of this period, or may begin after an 

 " incubation period"; may cease long before or long 

 after infectedness ceases, or may be intermittent. It is 

 therefore merely an episode of infectedness, and one which 

 does not concern us greatly just now. Another episode, 

 and a more important one at the moment, is infective- 

 ness, that is, the state of the infected person dining which 

 the infecting organisms are able to pass from him to 

 others. The period or periods of infectiveness are always 

 contained within the period of infectedness, but do not 

 necessarily coincide with the periods of sickness. Thus 

 typhoid or diphtheria carriers may be ill for only a week 

 or so, or not at all, but may remain infective for months. 

 In yellow fever, according to good researches, sickness 

 and infectiveness begin together a few days after the com- 

 mencement <il infectedness at inoculation; but infectiveness 

 ceases three days later, often long before the sickness is 

 over. In malaria, sickness and infectiveness are inter- 

 mittent and not coincident episodes, and may recur for 

 years. Infectedness itself is only the preliminary stage 

 of affectedness, which begins at inoculation and does not 

 end until the last trace of the resulting sickness or 

 acquired immunity has vanished. Reinfection often occurs 

 during existing affectedness, and may increase its dura- 



tion and that of the episodes. Medical treatment may 

 have the opposite effect, and natural immunity and pre- 

 vention may reduce susceptibility to infection. Lastly, the 

 natural fluctuations of population, due to births, death-, 

 immigration, and emigration, must be considered, and 

 these may vary in consequence of the epidemic. 



Hence many coefficients have to be taken into account ; 

 and the principal difficulty lies, I fancy, in arranging for 

 all of them in the equations. The course which I have 

 adopted as being perhaps the best for a beginning is to 

 conceive the matter in the most general terms possible 

 by taking the act of infection as being one of any kind 

 of event, such as accident, death, marriage, bankruptcy, 

 receipt of bequests, insect-bite, &c, which may occur to 

 a population, the various coefficients being at present taken 

 as constant during the period considered. If such an 

 event occurs to a given constant proportion of the popula- 

 tion in unit of time, how many affected people will there 

 be in the locality on a given date, on a most probable 

 estimate, and how many of these have been affected once, 

 twice, thrice, &c. ? This simple form may be called the 

 problem of happenings, and its solution will often be 

 useful in epidemiology, as, for instance, in estimating the 

 most probable frequency of reinfections or of insect-bites. 

 But for some kinds of events, such as marriage, wealth, 

 and infectedness, we must contemplate a continuance of 

 the event in the individual, with a possible reversion to 

 the unaffected class after the cessation of affectedness. 

 Such events may be called becomings ; and we have now 

 to find the proportion of the population in this condition 

 on a given date. 



I will treat the equations as briefly as possible. 

 Consider the following : — 



„, +1 = (l-/ ; V,^ HVr, 



:,+,= h rw, + (l-H'V:, 



f t+l = va,+ Vs, (0 



Here a, and s, are respectively the numbers of unaffected 

 and affected individuals, and p, is the total population at 

 the end of t units of time ; v and V are respectively the 

 variations in number of the unaffected and the affected 

 due to births, deaths, immigrations, and emigration in 

 unit of time ; h is the proportion of the unaffected which 

 become affected, and H the proportion of the affected 

 which become unaffected (to be better defined presently) 

 in unit of time. Thus i-h and i-H are respective ly 

 the proportions which remain unaffected and which re- 

 main affected, and a (+] and z, +} are the numbers of the 

 groups after the lapse of one unit of time. The gain of 

 one group is the loss of the other group, and the total 

 population is the sum of the two groups, the factors h 

 and H disappearing in the summation. 



If n, m. i. e denote the (constant) nativity, mortality, 

 immigration, and emigration rates among the in 

 and \\ M, I. E the similar rates among the affected, it 

 is correct, I believe, to write ii=(i + b)(i-i«)(i + 0Ii-«)i 

 and a similar equation for V. Different symbols _ are 

 necessary for the two groups, because all the quantities, 

 even the immigration, may differ. We now take the 

 equations in mon lil, but omitting v and V for 



the moment. Thus 



<J, + 1 = (I -A)d, + (l -/i)mi, + (\ -/;)X:, + (l 

 z, +1 = // a, + h ua, + h X :, + // >-., f ( 1 - r)s, 



pt+l= «t+ ""-+ X:,+ .... (2 



Here n and N are the birth-rates of the two groups. 

 The second and third columns give the happenings ameng 

 the births; rz, is the proportion of the affected which 

 revert to the unaffected group in unit of time, and 

 the (very small) proportion of these which immediateH 



I m reaffected ; (] r)s, is the proportion of tie 



which Jo not revert, and (1- h)iz, the proportion of the 

 reverted which are not immediately reaffected. Obviously 

 /V +1 is men I3 the sum of the two groups a, and z, plus 

 the births that have occurred to both in the unit of time, 

 and the symbols h and r disappear in the summation. 

 rhi ei tion are not symmetrical, because, though the 

 progeny nf the unaffected are born in this group and belong 

 to it, the progeny of the affected are not born affected, 

 and therefore do not belong to the latter group. I think 

 that this is the better arrangement ; but it would be possible 



XO. 2l88, VOL. 87] 



