definition is more correct.
27] CONDITIONS OF INFECTION 153
(6). By the malarial case mortality we mean, of course, the
proportion of infected persons who die of the disease.
(7). By the malarial mortality we mean the proportion of
the total population who die of the disease.
(8). I use the word ratio to denote a proportion, and the
word rate to denote a percentage. The latter is
obtained by multiplying the former by one hundred.
27. Conditions required for the Production of New
Infections in a Locality. — From what has been written already
it will appear that new infections can occur in a locality only
if all the following conditions exist : —
(i). That a person whose blood contains a sufiflcient number
of gametids (sexual forms) is living in or near the
locality.
(2). That an Anopheline capable of carrying the parasites
sucks enough of that person's blood.
(3). That this Anopheline lives for a week or more after-
wards under suitable conditions — long enough to
allow the parasites to mature in it.
(4). That it next succeeds in biting another person who is
not immune against the disease or is not protected
by quinine.
To these we must now add the following principle : —
(5). That few or no new infections will occur in a com-
munity unless the persons with gametids in their
blood and the carrying Anophelines are sufficiently
numerous.
This last proposition is the basis of the public prevention
of malaria. Although tacitly admitted, it is never properly
discussed in the monographs, and the reader must therefore
examine it here in detail and as precisely as possible.
Let us suppose that we have to do with a population of
1,000 people living over an area in which indigenous malaria
does not exist ; and suppose that one of these people is an
154 MALARIA IN THE COMMUNITY [Sect.
imported case with suitable gametids in his blood. Next,
suppose that a single suitable Anopheline is liberated within
the area. What are the chances that this insect will ever cause
a new infection ?
First, we observe that not every mosquito can succeed in
biting human beings at all : suppose that the chances are
4 to I against this happening. Next, we observe that as there
is only one infected person among the i,ooo people in the place,
and as the particular Anopheline liberated in the area may
bite any one of these people, the chances are i,ooo to i against
its happening to bite the patient, even if it succeeds in biting
at all. That is, altogether, the chances against its biting the
patient are 4,000 to i. But suppose that this has happened.
It must now live for a week or more afterwards, and not all
mosquitos live so long. Suppose that the chances are 3 to i
against this particular Anopheline living long enough to mature
the parasites in it — so that the chances are 12,000 to i against
the Anopheline reaching the infective stage. But it must now
bite a second person. Suppose that the chances against this
are, again, 4 to i. Thus the total chances against this
Anopheline inoculating another person will be something like
48,000 to I. In other words, if instead of liberating a single
Anopheline within the area of observation we had liberated
48,000, the chances are that only one of these would succeed
in biting the patient, and would also live long enough to become
infective and to bite a second person. Of course, by bad luck,
so to speak, a larger proportion of the insects might succeed
in this ; but, on the other hand, by good luck, none at all might
succeed. Such calculations, though obviously based on con-
jectural data, yet suffice to show the absurdity of supposing
that the presence of a few Anophelines must cause an epidemic
of malaria.
Or we may consider the subject as follows : — Suppose that
48 Anophelines on the average live near the infected person.
Then, if the conjectural data are sound, 1/4 of these, or 12,
27] CONDITIONS OF INFECTION 155
will succeed in biting him ; 1/3 of these, or 4, will live for a
week or more ; and only 1/4 of these, or i, will succeed in biting
another person. If 48 Anophelines live near each one of the
999 healthy people, there will be 47,952 Anophelines" which
can bite only healthy people, and which cannot therefore cause
new infections. Thus, on the average, only i out of 48,000
Anophelines will succeed in infecting another individual.
If instead of only one person with gametids in the blood
there are 2, 3, 4 ... or more in the locality, then if there are
48 Anophelines on the average to each person, 2, 3, 4 ... or
more new infections may be caused. Thus, obviously, the
number of new infections in a locality, that is, the inoculation
rate, depends on two factors (other things being equal), namely,
the number of Anophelines and the number of previously
infected persons in the locality.
It is useful to put these thoughts into simple symbolic
language. Let/ denote the human population of the locality ;
nip the number of malaria - infected persons ; and imp the
number of these with gametids in the blood. Here vi and /
are fractions, since 7np is less than /, and imp less than mp.
Also m may vary from o to i, since the number of infected
persons may be anything from zero to the whole population.
The fraction i, being the proportion of infected persons with
gametids in their blood at the moment of enquiry, may be
put at 1/4, or may be much less.
Again, let a denote the number of Anophelines (of some
malaria - bearing species) to each human being — so that ap
denotes the total number of Anophelines in the locality, and
aimp the number of Anophelines compared with the number
of persons with gametids. Let b be the proportion of these
(say 1/4) which succeed in biting ; s the proportion (say 1/3)
which succeed in maturing the parasites ; and b the proportion
which succeed in biting another person. Then bsbaimp gives
the number of Anophelines which succeed in infecting persons.
156 MALARIA IN THE COMMUNITY [Sect.
We have assumed that b^ij^, •$'=i/3. 2= 1/4; so that if each
of these bites a different person, we shall have —
No. of \noc\x\2A.\ons = b'^saiinp = amp I ig2
Inoculation rate % = — ^- • — = about h am.
'° 192 p
That is to say, the inoculation rate per icx) of population equals
about half the malaria ratio (w) multiplied by the mosquito
ratio {a).
For example, in a village containing 1,250 people, 750
infected people, and 3,000 Anophelines,/'= 1,250, m = o-b, rt = 2"4;
and we calculate roughly that the number of infecting Anophe-
lines, and also of inoculations, will be about 9*4. The inoculation
rate per cent, will be about 1/2 (2*4XO"6) = o72 ; that is, the
chances of being inoculated in the village will be as 72 is
to 10,000.
Or suppose that in another village half the people are in-
fected, and there are about twenty Anophelines to each person.
Then the chances of becoming infected there will be about 5/100.
Such calculations may appear far - fetched to many ; but
they are useful, not so much for the numerical estimates yielded
by them, but because they give more precision to our ideas,
and a guide for future investigations.
28. Laws which Regfulate the Amount of Malaria in a
Locality. — The number of infecting mosquitos which succeed
in biting again is b^saiinp. If all of these bite different people,
and all these people are healthy, and all become infected, this
expression will also denote the number oinew infections occurring
in the locality. But, of course, the infecting mosquitos may
often happen to bite on the second occasion, not healthy
persons, but persons already infected, especially if the pro-
portion of the latter is large.
If, as before, mp denotes the number of persons already
infected at the beginning of the enquiry, then/ — ?;// or (i —m)p
denotes the number of healthy people. Hence by proportion
28] LAWS OF DIFFUSION 157
the number of infecting mosquitos which bite healthy persons
(on the second biting) will be x, where
X : U^saimp :: {i—m)p : p
or X = b'^sai {i — m) mp . . . . (i)
and if each bites a different person and each person becomes
infected, the same expression will denote the number of new
infections which occur in the locality — that is, will denote the
addition to the number of malaria cases there.
But this is not the whole change which may occur. While,
during the period of observation, new infections are being pro-
duced, it may happen that some of the old cases may have
recovered. The number of those old cases was originally mp :
let rmp denote the number of those who have recovered during
the period of observation — so that r is a fraction. Hence the
whole number of cases in the locality will have increased or
decreased at the end of the period of observation, according
to whether b'^sai (i — ;;/) mp, the number of new cases, is greater
or less than rmp, the number of recoveries. Thus (neglecting
common factors) the change depends upon whether b-sai {\ —in)
is greater or less than r.
Suppose that no change occurs — that the recoveries exactly
equal the new infections. Then
b'^saz {i—in) = r .... (2)
From this equation we can calculate the values of a and m
when the amount of malaria in the locality remains constant —
that is, if we know the values of b, s, z, and r.
Suppose that the period of observation is one month. Now,
in section 21, I estimated roughly that only about half the cases
of malaria remain infected after three months. If this rate
holds for smaller periods, we may suppose that the ratio of
people who remain infected after only one month will be given
by the cube root of 1/2 — that is, by 07937 ; so that the pro-
portion of those who recover in one month will be 1—07937.
That is, we may write r = 0*2062. Let us take the values of
b, s, t as suggested in the previous section — so that b-si= 1/192.
158 MALARIA IN THE COMMUNITY [Sect.
Hence the new infections will equal the recoveries if
a{i- m) = 192 X 0-2062 = 39-59 = 40 {say) ... (3)
and the malaria will increase or diminish in the locality accord-
ing to whether a{\ —in) is greater or less than 40. Thus, if in,
the original malaria rate, is very small, the malaria will not
increase unless a, the monthly number of Anophelines per head
of human population, is greater than 40. On the other hand,
if ;;; is a larger fraction, say 1/2, the malaria will not increase
unless a is greater than 80. If in is still larger, say 3/4, the
malaria will not increase unless a is greater than 160. This
suggests that the malaria rate is not likely to increase
indefinitely unless the number of mosquitos is enormous. On
the other hand, if the number of Anophelines is below the
figure given by the equation, the malaria rate ought to begin
to fall, because the new infections can no longer keep pace with
the recoveries.
It should be noted that by the number of Anophelines we
here mean the number of different Anophelines which may bite
each person during a whole month, and not the insects which
may be, so to speak, allotted to each person at any one moment.
We say one month because this is the period we have selected
for observation — the period during which we suppose that
0*2 or I /5th of the cases recover. If we had selected one week
for the period of observation, the proportion of recoveries would,
of course, be lower (about 0*056 142), and the number of different
Anophelines to each person, required to compensate for the
recoveries during the week, would be correspondingly less (as
shown by the equation).
Now, what will happen if the malaria rate, instead of
remaining constant, increases or decreases ? On the one hand,
will it increase until every one becomes infected ; or, on the
other hand, will it decrease until it vanishes?
We have supposed that inp denotes the number of cases
at the beginning of the enquiry, which lasts, let us say, for a
month. Let ni^, tn^ip, w^p , . . denote the number of cases
28] LAWS OF DIFFUSION 159
at the end of i, 2, 3 . . . months respectively. Then if the
other figures remain constant, we have,
m^p = original cases -\- new infections — recoveries
= vip + b'^sai ( I — w) mp — rmp . . . . . (4)
and as the same process repeats itself month after month we
continue to have,
m^p = Wj/ + d'^sai ( i — m■^) m^p — rni^p
m^p = m^J) + b'^sai ( i — vi^ w^p — rm^J)
and so on. We can calculate m^ from the first equation.
Substituting its value in the second equation we calculate m^ ;
and substituting this in the third equation we calculate m.^ ;
and so on indefinitely.
If r=o'2 and b^si='00% the value of the 7ith term of the
series m^p, m^ . . . m„..,p, m„p may be written
in „p = {160 -\- a — am „..,) m„..,px o-oo^ • • • (S)
Let us now consider for examples the case of a village
containing 1,000 people of whom half are infected at the
beginning of the enquiry. By equation (3) of this section, and
also by this equation, if the number of different Anophelines
per person during one month {a) is 80, the malaria will neither
increase nor decrease, so that w„ will always be the same as w,
namely, 0*5. But if a is greater or less than this, the malaria
will increase or decrease accordingly.
(i). First, suppose that there are 100 different Anophelines
per person during a month — so that the malaria should increase.
Thus we calculate,
m^p = {i6o+ 100— iooxo"5)x 5x0*5 = 525
W2^ = (i6o+ 100— iooxo'525)x5xo-525 = 544'5.
Proceeding in this way we find that the number of cases
should increase every month as follows : —
Increase of Cases with 100 Anophelines
Months
I
2
3
4
. finally
Cases
500
525
544
560
571
600
i6o MALARIA IN THE COMMUNITY [Sect.
But how do we reach the last figure ? As we calculate the
number of cases month after month we observe that they always
increase, but by a constantly decreasing increment. Finally,
this increment becomes very small, so that the number of
cases approaches a fixed limit. It is easy to calculate this
limit mathematically (as will be done presently), but we can
also calculate it very simply as follows. If the number of cases
does ever arrive at a fixed limit, so that it no longer increases,
then equation (3) of this section must hold ; that is, the number
of cases and the number of mosquitos will be exactly balanced
according to the formula a (i— ;//) = 40. From this we have,
when this exact balance is reached,
m=\— Afi\a (6)
Here
multiplied by the population (1,000), gives 600 cases as the
final limit.
(2). Next, suppose that there are only 60 Anophelines per
person. Then we calculate as before,
Decrease of Cases with 60 Anophelines
Months
I
2
3
4
-. finally
Cases
500
475
455
438
424
â– â– 333
The final result is obtained as before from equation (6). Here
then, although the malaria ratio diminishes rapidly at first, it
never disappears altogether, but ultimately stands at 1/3 of the
population.
(3). Next suppose that there are 40 Anophelines per person.
Decrease of Cases with 40 A nophelines
Months o I 2 3 4 . . . finally
Cases 500 450 409 376 348 ... o
Here the fall is quicker and the malaria finally vanishes,
because ?«= i —40/40 = ; but very many months will elapse
before this result is approached.
28] THE LIMIT OF MALARIA i6i
(4). We also have
Decrease of Cases with i Anopheline per head.
Months
I 2 3 4 . .
. finally
Cases
500 401 322 259 207
(5) Decrease of Cases with no Anophelines,
Months
I 2 3 4 . . .
finally
Cases
500 400 320 256 205
Thus the presence of only a small number of Anophelines does
not affect the result very much. In fact we can see this from
the form of the equation (5), namely,
because i — 7;/^^ is always a fraction, and a must therefore be
a considerable number if it is to have any marked effect
compared with the first term 160.
Returning now to the subject of the limit mentioned in
example (i). It is known in mathematics that if a series such
as Wj, w^, W3 . . . here considered tends to a limit, that limit
can often be easily found. Let M denote the limit — that is,
M=in^ when Jt, the number of terms, is indefinitely increased.
In other words, M is the malaria ratio which is finally arrived
at after the lapse of many months.
Now we had in equation (4),
By the mathematical rule referred to, the value of w , or M.
can be easily found, when n is large, by solving the equation
Mp={i+bhai{\-M)-r]Mp
that is,
M=i-rlb'^sai (7)
If ;- = o-2 and b'^si= 1/200, this gives 7J/= i —40/di, which is the
same as equations (2) and (3), which have already been used to
find the limit shown in example (i). There we calculated the
limit by another line of reasoning, but now we see that it can
be obtained also by the ordinary mathematical rule. (This
rule is that if ^";ir is a repeated function of x, its value when n
L
i62 MALARIA IN THE COMMUNITY [Sect.
is indefinitely large will often be given by the roots of the
equation ,r= (px).
The reader should make a careful study of those ideas,
and will, I think, have little difficulty in understanding them,
though he may have forgotten most of his mathematics. If
our reasoning has been correct and complete enough, the main
principles involved may be stated as follows : —
(i). Whatever the original number of malaria cases in the
locality may have been, the ultimate endemic
malaria ratio will tend to settle down to a fixed
figure, dependent on the number of Anophelines
and the other factors — that is, if these factors
remain constant all the time,^
(2). If the number of Anophelines is sufficiently high, the
ultimate malaria ratio (M) will become fixed at
some figure between o and i (that is, between
0% and 100%). If the number of Anophelines is
sufficiently low (say below 40 per person), the
ultimate malaria rate will tend to zero — that is, the
disease will tend to die out. (In this calculation a
negative malaria ratio, that is, one which is less than
nothing, must be interpreted as meaning zero).
Consider, for example, the case of a village with 1,000
inhabitants and 60 different Anophelines for each person
during one month. If, to begin with, every person starts with
being infected, then the malaria ratio will fall month by month
until it reaches the value i1/= 1—40/60; that is, until 333
persons are left infected. And this occurs because there are
not enough Anophelines to maintain the original high rate of
100% infection. But now suppose that with the same number
of people and of Anophelines, the epidemic began with only
one infected person from outside : the number of Anophelines
1 This is a well-known peculiarity of repeated functions. See, for instance, my
paper "A Method of solving Algebraic Equations," Nature, 29th October 1908,
p. 663.
28] THE LIMIT OF MALARIA 163
will now be too large to allow this original low rate of 0'i%
infection to continue, and the rate will consequently rise until
the value M= 1—40/60, giving 333 infected persons, is reached.
Thus, whether every one, or only one person, is infected to start
with, the ultimate result will be the same — the number of cases
will be 333, and will continue at that figure so long as all the
factors remain the same. That is, an exact balance between the
number of cases and the number of Anophelines will be arrived
al Many months may, however, elapse before this happens.
Suppose that all the factors d, s, z, a, r do not remain the
same. Then if any of them is temporarily altered a correspond-
ing change will be made in the rate of increase or decrease ;
but after the temporary disturbance has ceased, things will
again tend to return to the normal state — just as water always
seeks to find its own level. Thus, if the recovery rate r is
increased by a temporary administration of quinine, or the
number of mosquitos a is temporarily diminished by winter or
by active " petrolage," a change for the better will be made,
but it will cease soon after these alterations cease. If, however,
the alteration is permanent, then the value of M will, of course,
be permanently affected by it.
We have often assumed certain conjectural values for some
of the factors; does this fact invalidate the reasoning? We
will examine these values more closely in the next section,
but the exact figures do not affect the general law. We have
supposed that, in equation (7), r\b'^si=\o\ from which we
argued that the ultimate malaria ratio will vanish if the number
of different Anophelines to each person is less than about
40 per month. The actual number may be more or less, but
the fact that a limit must be reached remains. As I have said,
the whole calculation is useful, not so much for its numerical
results, but because it gives precision to our ideas.
A more serious objection is that, in framing the funda-
mental equation (4), we have disregarded certain factors which
would modify it. For example, the population will be subject
i64 MALARIA IN THE COMMUNITY [Sect.
to constant changes owing to immigration, emigration, births
and deaths — all of which operate both on the healthy and
on the infected. Thus the birth-rate will give a continuous
supply of healthy individuals, while the death-rate will remove
many of the infected ones. Also many of the recovered cases,
and many of the mosquitos, may be at least partially immune ;
gametids may appear in the blood more during some seasons
than during others, and infected mosquitos may bite many
more than one person each ; but the introduction of all these
factors would give a much more complicated equation than
we have need for here. It will still be of the form
m-J> = oi'iginal cases + new infections — recoveries {or deaths),
and from this we shall still be able to argue that a limit
must be reached when the new infections exactly balance
the recoveries. We may therefore conclude,
(i) That the amount of malaria in a locality tends towards
a fixed limit determined by the number of malaria-
bearing mosquitos and by other factors.
(2) That if the number of malaria-bearing Anophelines
is below a certain figure, that limit will be zero.
It is often thought and said that malaria should exist
wherever susceptible Anophelines exist, and that anti-mosquito
measures will therefore be useless so long as any of these
insects remain. But more careful reasoning will convince
us that malaria cannot persist in a community unless the
Anophelines are so numerous that the number of new
infections compensates for the number of recoveries.^
29. Laws which Regfulate the Number of Anophelines in
a Locality. — We have seen, then, that the amount of malaria
in a locality depends (among other factors) upon the number
of suitable Anophelines, and upon the proportion of them
which succeed in biting human beings, in living long enough
to mature the parasites, and in biting human beings again.
^ See section 65 (10).
29] OUTPUT OF MOSQUITOS 165
We must now study these points in greater detail. Unfor-
tunately, although much entomological work has recently been
done on mosquitos, such important subjects as these have
received little attention.
(i). The otitpiit of mosquitos from marshes. — I know of no
adequate studies on this point. In 1908, in Mauritius, I
stretched a mosquito-net over 9 square yards (7"5249 square
metres), and counted every day the mosquitos hatched within
it. The selected spot (Clairfond Marsh) was covered with
rank grass, the roots of which were submerged by an inch or
two (2"5-5 cm.) of water. It was sheltered by trees, and was
an ideal spot for the breeding of Myzorhynchus mauritianus
Daruty and D'Emmerez, 1900 (an Anopheline which does
not carry malaria). The observations were continued for
sixteen days in January (the warm rainy season), and only
this kind of mosquito was found. Altogether thirty males
and thirty-one females were obtained,^ giving an average of
0*423 per square yard per diem, or 5,062 per 10,000 square
metres. Numbers continued to hatch out on the sixteenth
day. During the daytime the adults took refuge in the
grass, from which it was necessary to expel them. Clairfond
Marsh is about 1,400 feet (427 metres) above sea - level
(section 30 (21) ).
This output (about 5,000 mosquitos per diem for 100 yards
square of marsh) seems to have been rather large, as when
the net was placed in another position the yield was much
smaller. Taking twenty days as the average life of an
Anopheline (conjecture), 100,000 of them should thus be in