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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

Online LibraryRonald RossThe prevention of malaria → online text (page 14 of 55)