October 5, igi 1] 



NATURE 



4&7 



to add a term for affected births, as in syphilis. The first 

 two of the above equations may be written 



a l+1 = {l-A){l + n)a t + (i - /,)^±£(i + N).-, 



.= //(i 



+ »K+{i-(i--«)^ + ^}(i N) S< (31 



If, now, we restore the mortality, immigration, and 

 emigration rates, that is, affix to o, in both equations the 

 coefficient (i— m)(i + »')(i — e) and to z, the coefficient 

 <:-N)(i + I)(i-E), we have 



Ot+i=(l-h)va t +(\ - ''\~x V: ' 



=, +1 = &va t +{i-(i-A)?L±l}Vz t . . . (4) 



which are obviously the same as equations (i) if H is 

 now defined as the value of (i — fc)(N+r)/(i + N). 



The complete solution of these difference equations is 



(X -Y)a t =(a 1 -a Y)X'-(a 1 -a X)Y' 

 (X-YW=( Z] - :„Y)X'-(~-, -8 X)Y< 

 (X-V)/ ; = (/ 1 -/. 1 ,V)X'-(/ 1 -/ ll X)\- 



(5) 



«! = (I -h)va a ~. II V „ 





and X and Y are the roots of the auxiliary algebraic 

 quadratic equation 



x- \(i-h)v + (i -H)V}at + (i-/4-H)wV=o. 



These roots are rational for several particular values of 

 the constants. The most important instance is when 

 v=V, that is, when the happening does not affect the 

 normal fluctuations of the population. Here X = v and 

 Y = (i-/»)(i-r)/(i + N), and 



-**=*££**-**>■ 



(6) 



As Y is in this case less than unity, Y' diminishes with- 

 out limit as t increases, and therefore z t , the number of 

 affected individuals, asymptotes to a fixed proportion of 

 the total population, provided that all the elements remain 

 constant. I call this proportion the static value. In 

 disease it gives what is called the endemic index, or 

 ratio. 



In epidemiological applications the symbol s refers, not 

 to sickness or even infectedness, but to affectedness as 

 defined above ; and the symbol r does not mean recovery 

 from sickness or infectedness, but reversion to a suscepti- 

 bility to a fresh happening (inoculation), that is, to loss 

 of acquired immunity. Thus in drawing curves of 

 epidemics we must remember that this last factor may 

 not come into play until long after the commencement of 

 the epidemic, or not at all. 



In my book the above equations are treated also in the 

 infinitesimal form, when the integrals become exponential. 

 Thus the second of equations (2) becomes 



*,-.) + * 



where q = V-i-r-N. If the total population p remains 

 constant, this is easily integrable if h is also constant, or 

 (what more probably happens in epidemics) is a linear 

 function of z, say cz. 



Numerous applications are possible ; but I have space 

 to refer only to the important case of " metaxenous 

 diseases," that is, to infections common to two species of 

 animals or plants. The same equations apply to both 

 species, but the happening-factor h in one equation must 

 be a function of ; in the other equation. We thus have 

 two simultaneous equations to solve, namelv, 



where the marked symbols apply to one species of animals 

 (say, mosquitoes) and the unmarked ones to the other 



NO. 2 188, VOL. 87] 



species (say, man), and k and fc' are constants composed 

 of the most probable frequencies of communication 

 between the two species, of infectivity and of natural 

 immunity. Prof. F. S. Carey has referred these equa- 

 tions to Prof. A. R. Forsyth, who thinks that they are 

 not likely to be easily integrable in finite terms ; but the 

 most important case is where both s and z' have reached 

 static values, when the differential coefficients vanish. 

 We then obtain at once 



„ _ kpltf - qq' 

 kk'p'-kq 



with the similar equation for s'. In the case of some 

 insect-borne diseases this becomes (reduced) 



= 1( 1 " r)fb'f'b'A + r/6'\ =p {/d'/'i'A - rH ') , 



where z is the ratio of affectedness among men (say), 

 / and /' the proportion of infective men and insects, b' the 

 frequency of bites, r the reversion rate among the human 

 patients, N' the birth-rate of the insects, and A the ratio 

 of the number of the insects to head of human popula- 

 tion. Numerical estimates of the constants in malaria are 

 attempted in the book, and a table of calculated values 

 of A for various values of z and b' are given (as already 

 partly done by Mr. Waite). 



The following important laws seem to be established : — ■ 

 (1) the disease (a) will not maintain itself unless the pro- 

 portion of Anophelines (A) is sufficiently large ; (2) a small 

 increase of A above this figure will cause a large increase 

 of z ; and (3) s will tend to reach a fixed value, depending 

 on A and the other constants. I doubt whether these 

 laws could have been reached except by such mathe- 

 matical attempts. The second one is especially important. 

 If A is just at the critical value, s will be zero, or only 

 just above it; but if A is only about double this critical 

 value, a serious epidemic, amounting to about half the 

 whole population, may follow. Yet such a small increase 

 in the number of Anophelines will scarcely be detectable 

 except after very careful study, a fact which easily explains 

 why marked correlation has not always been observed. 

 The same equation shows that, if certain experiments are 

 to be trusted, yellow fever can scarcely be considered an 

 endemic disease of men at all ; and it also explains the 

 absence of certain diseases in the presence of capable 

 carriers, and the general phenomena of smouldering 

 epidemics. 



The most probable numbers of individuals to which a 

 happening has occurred never, once, twice, &c, can 

 easily be obtained, and are equal to the successive terms 

 in the expansion of |(i - /')->■ //|'V'/ in ascending powers 

 of h. This enables us to estimate the number of persons 

 who have been bitten, or the number of insects which 

 have succeeded in biting never, once, twice, &c, in a 

 given period, and to calculate the average number of bites 

 received or inflicted by each individual. It also enables 

 us to calculate (what I think has not been done before) 

 the frequency of reinfections. At present such reinfec- 

 tions are not much considered during the course of an 

 already existing infection, but I estimate that in a locality 

 where half the people are statically affected with malaria 

 no fewer than about 63 per cent, will be infected or re- 

 infected every four months (under constant conditions). 

 In 1898 I showed that birds reinoculated with malaria 

 could exhibit renewed and severe infections. 



Lastly, to complete the study, it is necessary to estimate 

 the most probable proportion of affected individuals who 

 are also infected, or infective, or sick at a given moment. 

 This will be the same as the proportion of the average 

 number of days lived during these " episodes " to the 

 average number lived during the whole period of " affected- 

 ness," which can be calculated from the special patho- 

 logical data. 



These studies require to be developed much further ; 

 but they will already be useful if they help to suggest a 

 more precise and quantitative consideration of the 

 numerous factors concerned in epidemics. At present 

 medical ideas regarding these factors are generally so 

 nebulous that almost any statements about them pass 

 muster, and often retard or misdirect important preventive 

 measures for vears. Ronald Ross. 



