Purim: a rapid method with reduced cost for massive detection of CoVid-19
Benjamin Isac Fargion, Daniele Fargiona,b,1 , Pier Giorgio De Sanctis Lucentinic,2 , Emanuele Habibd
a Physics
Department & INFN Rome1, Rome University 1, P.le A. Moro 2, 00185, Rome, Italy
Institute of Fundamental Physics, Via Appia Nuova 31, 00040 Marino (Rome), Italy
c Physics Department, Gubkin Russian State University (National Research University), 65 Leninsky Prospekt, Moscow, 119991, Russian Federation
d DIAEE, Department Of Astronautical, Electrical And Energy Engineering, University of Rome ”La Sapienza”, Via Eudossiana 18, Rome, Italy
b Mediterranean
arXiv:2003.11975v1 [q-bio.PE] 26 Mar 2020
Abstract
The CoVid-19 is spreading pandemically all over the world. A rapid defeat of the pandemic requires carrying out on the
population a mass screening, able to separate positive from negative cases. Such a cleaning will free a flow of productive population.
The current rate and cost of testing, performed with the common PCR (polymerase chain reaction) method and with the available
resources, is forcing a selection of the subjects to be tested. Indeed, each one must be examined individually at the cost of precious
time. Moreover, the exclusion of potentially positive individuals from screening induces health risks, a broad slowdown in the effort
to curb the viral spread, and the consequent mortality rates. We present a new procedure, the Purified by Unified Resampling of
Infected Multitudes, in short Purim, able to untangle any massive candidate sample with inexpensive screening, through the crosscorrelated analysis of the joint speciments. This procedure can reveal and detect most negative patients and in most cases discover
the identity of the few positives already in the first or few secondary tests. We investigate the the two-dimensional correlation case
in function of the infection probability. The multi-dimensional topology, the scaled Purim procedure are also considered. Extensive
Purim tests may measure and weight the degree of epidemic: their outcome may identify focal regions in the early stages. Assuming
hundreds or thousand subjects, the saving both in time and in cost will be remarkable. Purim may be able to filter scheduled flights,
scholar acceptance, popular international event participants. The optimal extension of Purim outcome is growing as the inverse of
the epidemia expansion. Therefore, the earlier, the better.
Keywords: Virus, CoVid-2019, Detection
1. Introduction
The CoVid-19 hidden pandemia, born recently [1–6] requires urgent detection for larger group sample, to disentangle
positive and negative cases. The urgency of the screening of the
population cannot face easily the exponential infection growth.
Any test examined individually, may take precious time, causing bottlenecks in the waiting list and reducing our ability to
win the virus exponential diffusion growth.
Because the possible geographical relation to the climate,
[7], selection test among countries and continents might be request for seasons and even for years. The mobility among nation depends on ability to recognize quickly the health state of
travelers. At the present time the rapid tests, as the ones based
on the saliva (gingival exudate)[8] probe may offer, one by one,
a validation. There is no other known cheap and fast methods to test a large population at once. Here we suggest a joint
and unified unique test or a mixed sub-groups test array (for
instance by gingival exudate), united into several independent
ones, whose cross checked results might discover in an rapid
and economic way the main infected cases. The mathematical
procedure might be exact or it might just lead to an overestimate
Email address: daniele.fargion@uniroma1.it (Daniele Fargion)
author, ORCiD:0000-0003-3146-3932
2 ORCiD:0000-0001-7503-2064
1 Corresponding
Preprint submitted to International Journal of Infectious Disease
of the infected cases. Therefore, because of overestimation one
could require a few additional individual tests with marginal
cost.
The Purified by Unified Resampling of Infected Multitudes,
Purim3 , may become the key strategy to separate, to trace the
virus diffusion leading to a sooner suppression.
Moreover, the screening procedure of a large population to avoid
the spreading of the infection should detect any suspect case.
The highlighting of suspect cases that are not actually positive
to nCoV would increase the cost of the screening but won’t
reduce its effectiveness. Most virus tests are individual, often
testing in a blind and costly way wide numbers of the population. We believe that such single individual test politics in
present pandemic stage is time consuming, costly and unable
to face the fast virus growth and the largest population screening. Moreover, the foreseen return of secondary waves of infection might require iterative or permanent filter of travelers from
abroad. Quick and massive tests are needed. The procedure remind somehow the procedure to disentangle a scrambled card
deck that is not well shuffled [9].
3 Let us remind that the word Purim, in Hebrews, means ”fate”; the name
of the Queen, Ester, means, ”hidden”, as the Covid-19 pandemia behavior: the
proposal was indeed conceived recently, on the Purim 5780. During Purim, like
Carnival, all wear a mask ; the fate of all the people is turn suddenly from the
disgrace to the life.
March 27, 2020
K = (N + 1)2 − N 2 − M. Finally any way we imagine to locate
all the NALL patients into a square matrix array.
Therefore, as described as an example, by Fig.1 ( NALL =
N 2 , N = 9), we may study several basic situations. On the
left, we assume an unique positive patient; in the center, a second with two True positive infected; the right figure shows a
final case with three positive infected candidates. As shown
in Fig.1 a corresponding cross-check sub-group (intersection of
sub-group) may disentangle immediately the unique infected
one, obviously a True positive shown by a label T . Let us define the infected total number by NT as the whole number of
true positive patients. Let us also define by NF the number of
false positive.
As shown in the same figure in multiple cases arise also mirror intersection pointing to a False F infected subject. A larger
number of infected samples will cause, generally, a larger number of intersection and of False F infected. From the analysis of
their multiple intersections there will be individuated both the
infected T ones but also the additional parasite false F positive.
Two False and two True in the second figure in Fig.1 show the
first of such situations. Three True and five False in the last
figure side show a more complex geometry. Of course the overestimation NT + NF ≥ NT depends on the geometrical disposal
and the total number of infected in the matrix array. As obvious, in general, more True presences, more False ones. However, some rare multiple infected presences may be lead also to
no False positive suspects. We will discuss the overestimation
in next section. We describe now in details the practical procedure in the Purim selective tests, as the few examples shown
in Fig.1, to figure out the infected case by the intersection of
sub-group tests.
Figure 1: CrossCheck The different cross-checked situations for a sample of
N = 9, N 2 = 81 subjects, where one, or two or three different infected True
positives candidates are present. The sub-groups are just made by the horizontal
and by the vertical column array = 2 · N = 18 sample of patients. Any subgroup test (composed by N = 9 individuals each) is mixed and united with the
all N = 9 individual saliva. Their 2 · N array results are later cross-checked.
False positive cases might also arise as a ”mirror or shadows” effect. In a first
case on the left, the single positive case may be immediately identify as an
unique True positive case. Remaining candidates are negative. Al least ((81 −
(18 + 1)/81) = 79% doubled checked negative. In the second and in the third
arrays the presence of two or three True positive infect subjects produce more
False positive born by the ”mirror or shadows” SG False intersections. Such
overabundance may be disentangled by additional, individual tests for the few
suspect candidates.
2. The United Group test
Let us consider as the simplest experiment, the Purim of a
sample of suspect infected ones as large as NALL . The simplest
case might be a passenger group waiting for a flight. Let us
assume, as this first simplest Purim procedure to combine and
mix all the NALL test in an unified one. This whole (NALL ) mixed
test may be either positive to CoVid-19 or not. The absence of
any infected, the negative case, will free the NALL suspect candidates. In particular, in an airport, they may fly abroad. On
the contrary, if the test offer a positive result, at least one or
even more components of the group must be infected. An additional inspection may be either individual (costly and time consuming) or, as shown below, by a quick and inexpensive Purim
procedure based on partial sub-group cross-checked tests.
3.1. The bi-dimensional Sub Group Purim procedure
Imagine that all along the vertical side of the square we label each horizontal arrow places for the candidates by a letter:
a,b,c..i. In analogy at the top of the square let us label each vertical column by the number 1,2,3–9. See Fig.1. Now we should
call each sub group of patients S G contained in an arrow or in a
column, each composed by N = 9 tagged candidates (in column
or in arrows) respectively as shown in Fig.1:
3. The bi-dimensional (2D) Purim
Let us therefore assume that the suspect group to disentangle and to analyze by Purim may be located, one by one, within
an ideal square matrix array whose components are NALL = N 2 .
In general NALL number is not a perfect square, but it may always be reduced to the sum of two (or more) comparable sub
square subgroups. It may be also reduced to a quadratic group
with a few remnants candidates. This secondary procedure for
an optimal fragmentation of NALL into two comparable squared
and to a few remnants is irrelevant here and it might be discussed elsewhere. Anyway, one may also find an approximation NALL = N 2 (±M) where M is the smallest integer number needed to fill the N 2 square matrix. In such a case one
may build such an square array either with NALL = N 2 and
with a few empty (M) places, in both rows and in columns.
In other cases one may arrange the patients in a larger square
where: NALL = (N + 1)2 ; consequently, as before, few K empty
places will be added in the new arrow and new column, where:
S Ga , S Gb , ..S Gi
S G1 , S G2 , ..S G9
The cross sections of their cross checked test results imply,
for instance, for the first case, the coexistence of two positive
sub-groups:
S Gc , S G2
The two group intersection identify an unique infect place
subject sitting both in S Gc and in S G2 . He got a double positive
test. Therefore, the patient sitting in the c2 matrix coordinate
is the infected one. No other is positive nor any false suspect
arise. The the remaining persons are in majority negative. Even
77% of them, are double checked.
The next situation, describing 2 positive T True case, will
arise by the finding of the following four subgroup positive
tests:
2
1
S Gc , S G2
1
100
400
900
2500
4900
10000
40000
250000
1000000
0.8
0.4
0.2
0
-0.2
S Gc , S G6
64
81
100
121
144
169
S
S
S
S
S
S
361
400
441
484
529
576
625
S
S
S
S
S
S
S
[1 iteration+ residues]
0.6
Saved Tests Fraction
Saved Tests Fraction
However as shown in Fig.1 there are also suddenly two
additional (false or ”shadows”) F candidates, due to different
”mirror” intersections:
[1 iteration+ residues]
0.95
S Gg , S G6
0.9
0.85
0.8
0.75
-0.4
0.7
0.001
0.01
0.1
0.0001
0.001
Infection Probability
S Gg , S G2
In conclusion: in present two real infected T there are also
two virtual false F (not infected) cases. A first over-estimation
of the Purim procedures. In the final third situation described
by the right side in Fig.1, the disposal, with the three True positive candidate induced a total of 6 false positive candidates. In
general the suspects (real or True ones NT ) are less than the
suspected ones P s ≥ NT + NF ones.
The last case in the Fig. 1 shows the eventual three True positions and the presence of six suspected positive tests; the True
infected ones are located by the sub-group intersection below:
0.01
Infection Probability
Figure 2: Saved Tests The figures show the trend with respect to the probability
of infection Pi of the fraction of tests that is saved with the Purim procedure.
On the left the overall trend, on the right a zoom on a specific area. The usual
procedure involves a single test for each individual: if the individuals are NAll ,
then NAll tests are performed. The Purim procedure, as explained in the text,
provides for the clustering of individual tests in groups with N = NAll 1/2 individuals each. It also prescribes to put the sample of each individual in D (here
equal to two) groups with the additional condition that each pair of individuals
can be present simultaneously in only one group. We define T s the number of
tests saved as T s = N 2 − 2N − N p , N p is the average number of individuals
who tested positive. The figure shows, by a MonteCarlo the best trend of the
saved test fraction T s /NAll as a function of the probability of infection Pi and
for different values of NAll . The figure point out that the best efficency is obtained with a number of subgroups N which depends in a good approximation
(inversely) on the probability of the infection.
S Gc , S G2
S Gg , S G6
S Ge , S G8
.
Multiple Negative
Ratio R--
1
S Gc , S G6
S Gg , S G2
S Ge , S G2
0.8
0.6
0.4
S
S
S
S
S
S
S
S
S
S
S
S
S
[R =N /NAll=0.5]
1
Infection Probability
Fraction of Multiple Negative
However, there are additional 6 independent sub-group intersections defining six more possible or virtual positive, False,
candidates:
64
81
100
121
144
169
361
400
441
484
529
576
625
0.1
0.2
0.01
0
0.001
0.01
0.1
Infection Probability
.
S Ge , S G6
10
100
NAll
Figure 3: Multiple Negative Tests. From the Purim procedure some of the
sample can result negative in all subgroup tests, hence multiple negatives, N −− .
In the left pane, the curves for different values of NAll , the subgroups dimensions growing with the arrow, with the ratio N −− /NAll , as a function of the
probability of infection p. In the right panel the intersections of these curves
with the value of 0.5.
S Gc , S G8
S Gg , S G8
.
All of these 9 possible intersections or infected cases must
be tested individually or by similar mixed and cross-checked
procedure. Therefore, the Purim procedure has a quite good accuracy for diluted sample of infect cases. Naturally the present
pandemia is, hopefully, below 1% of the population and in such
a group (nearly hundred) sample, as in Fig. 1, Purim procedure will detect positive subjects with quite a good precision.
Therefore, we may already conclude that in general the single
tests versus the Purim procedure does cost, in economy and in
time, by a ratio R = N 2 /(2N). In a hundred sample, a Purim
procedure may be 5 times more inexpensive and faster than individual tests. In a thousand sample number (located in a square
array N of nearly 32 · 32 size) the cost in time and in expenses
for the ideal (just one positive) Purim case would be only 6.4%
of N 2 tests; if there are a few True infected, as < 5 candidates
or about 0.5%, the additional mirror or false candidates would
lead to a final total 8.9% of the cost respect to N 2 = 1024 single
tests expense. The whole saving will be above 91%.
3.2. Over-estimation in a crowded sample
We may wonder about the previous overestimation in Purim
procedure for a crowded source positive presence. There might
be not always a simple quadratic growth of the arrow and column intersections, as in the case described in Fig. 4. The general complexity is not obvious. For very fine tuned locations of
NT = N infected subjects, each of them with a different number
of row and column, they may produce, 2 · N, all columns and
rows positive, implying N 2 cross-checked intersections, suggesting that all of the sample is infected. Indeed, in the case
described below only N are True positive while N 2 − N are
just False positive; see for instance left side in Fig. 5. Such a
rare situation may occur once the sample of infected has a large
3
probability P to be present within the total group
P≥
In analogy the number of false negatives NF is analogously
bounded by
1
4 · NALL 1/2
(0 ≤ NF ≤ (NH · NV ) − (Max(NH , NV ))).
. For a square of 32 element and NALL = 1024, this occur for
nearly a large infected presence ≥ 0.75%, or 7 or 8 infected
patients. An extreme additional, fine tuned case occurs when
only any arrow of the sub-group (in the example, the first vertical one) is identical positive for all sub-groups while all the
others (horizontal) sub-groups are different ones: in that ”well
aligned” configuration, both horizontal or vertical, there are no
False, but just N True cases as shown in the right side in Fig. 5.
(2)
For example in the simple case as in Fig. 4 where NH = 3,
NV = 4 the previous bounds for NT becomes:(4 ≤ NT ≤ 12).
Fig. 4 shows indeed NT = 5 True and NF = 7 as in their
bounded limits.
Naturally, because NH and NV are in general much less N,
and if N 2 is a large number (hundreds or above) and the present
false cases are rare (a few unities). Therefore additional test are
not too costly. There are some cases where the True sample may
be aligned in an axis or in all diagonal places; rare situations
where False will be ruling or will be absent at all: see Fig.5.
These situations might be over estimating because the infect
presence is not negligible (N versus N 2 ).
The bidimensional cross-cheked sub-group procedure might
be extended to multi-dimensional Purim procedure, as it is shown
in a more rich but more complex correlations approach considered below.
4. The Multi-Dimensional Purim procedure
The natural extensions of the previous bi-dimensional Purim
procedure could be described for geometrical visibility by a cubic array where the sample to be tested would be located each
one to a mini-cell tagged by its 3D coordinate.The cubic matrix
may be imagined as in figure Fig. 6. The candidates are located
in 3D cell structures labeled by three array axis. Two quite different different Purim procedure may used to disentangle the
positive infect subjects; a planar one or a vector one, base on
different cross-checked procedure.
Figure 4: CrossCheck As in the previous Fig.1 multiple intersections may occur simultaneously, leading to a complex situation with the presence of false
positives. In this case with 5 True and 7 False positives. This redundancy is
limited and can be disentangled by further tests, only on the 12 positives.
4.1. The Planar (2D) cross-check inside a 3D array
Let us imagine to locate the NALL suspected patients inside a cube cell defined by its 3D coordinates. As in the bidimensional case, this may be reached by dividing the whole
number NALL in comparable sub cubic numbers or by to located
the surplus candidate into few empty cells in such a larger cubic
array location. The different alternative to optimize NALL into
cubic sizes are irrelevant here. Each person is tagged by its Hi ,
horizontal, Vi , vertical, Li , longitudinal position inside the ideal
cubic cell volume. For instance a sample of NALL = N 3 = 216
passengers are tagged by a 3D array made by 63 cubic side cell
structure as shown in Fig. 6. The easy way to act within a
planar Purim procedure in 3D cells (candidates) array is based
on the construction of 3 · N planar sub-groups associated to all
the three possible planes: HVi , HL j , LVk . (i,j,k are bounded
within N cube size). Each of the planar array is made by N 2
sub-group components united by their test. In present example
each planar sub-group contains N 2 = 36 patients tests united.
These sub groups (3 · 6) are organized and tested independently.
Therefore one needs only a total of 3 · 6 = 18 test to intersect
to discover potential (True and False) infected cases and not
NALL = N 3 = 216. In the presence of an unique positive person
in the whole cube sample its planar intersections may define just
Figure 5: Opposite configurations Two extreme fine tuned and antithetic configurations may be able to affect the outcome of the Purim procedure in an
opposite way. On the left side the N infected samples are each in a different
row or column. It is the equivalent of having them on the diagonal. The test result will not discriminate, indicating all N 2 subjects as possible positive. In the
right part instead, the N infected are all positioned on the same column (or row),
for instance the first subgroup S G1 . The result is exhaustive by identifying all
N True positive subjects without any False positive.
In general for a number of NH (horizontal) positive row and
for a number of NV (vertical) column sub-group positives, the
resulting number of True NT cases is bounded by a maximal
and minimal value:
(Max(NH , NV )) ≤ NT ≤ (NH · NV ).
(1)
4
a single cell candidate, as shown in Fig. 6 left side correlated to
the upper figure. Such a unique identification (one True) is well
defined. In the very diluted sample this procedure is extremely
effective and economic. The ratio between 3 · N and N 3 test is
3/N 2 . Therefore very inexpensive. In the present NALL = 216
case the cost is below 10%. There are of course mirror effects
due to many parasite F False intersections, even more than in
previous bi-dimensional case. As shown in the zoomed side in
Fig. 6 the 2 True positive cell location, defining a complete
inner mini-cube, they produce 6 additional planar intersections
whose corner are forming 6 False or virtual infected cases by
the other two T True infected presence. Indeed the total intersections of three independent planar sub-group are the exactly
the corners of a cube, that are in a total of 8.
number NT inside bounded by:
(Max(NH , NV , NL )) ≤ NT ≤ (NH · NV · NL ).
(3)
In analogy the False NF are also bounded by
(0 ≤ NF ≤ (NH · NV · NL ) − (Max(NH , NV , NL ))).
(4)
In the Fig. 6 these bounds are confirmed. The true events
are NT = 2, the False are NF = 6 = 8 − 2. Both are following
their extreme bound limits.
Figure 7: The candidates are tagged on the horizontal, vertical and longitudinal
axis. The construction of independent sub-vector-groups by their 3D intersection, may identify the True positive infect case with redundancy. This vector
procedure requires many more tests 3N 2 and cost but it imply less overestimation of False events respect to the previous planar procedure.
4.2. The Vector (1D) cross-check inside a cube array
The gain by planar inspection is obvious: to collect larger
planar sub-group number is a cheaper way to inspect a very
large sample. However the cost is paid in the over-estimation
(for polluted overcrowded positive populations). The need for
additional tests is reducing the planar Purim advantage. There
is an additional tool in the Purim procedure based on the vector
(column) sub-group in one dimension section. In this case the
cross-check is in general overabundant and redundant: this procedure may avoid many False presences. However, this accuracy is paid by the growth of test vector needed tests: see Fig.7.
Assuming as above NALL = N 3 one needs N 2 column or vector
sub groups, each one of N cell or patients, for each cube face
side. This imply 3 · N 2 tests respect to the individual N 3 . For
large numbers this approach may be interesting, but the gain or
the huge cost-saving (the ratio Rgain between 1 dimension procedure and individual one) is only Rgain1 = 3 · N 2 /(N 3 ) = 3/N,
while in previous one (for the 2D array structure) it was Rgain2 =
3/(N 2 ). For N = 10 and a thousand of the whole candidates,
this imply for the planar Purim just Rgain2 = 3% while for a
vector 1 dimensional procedure a less attractive: Rgain1 = 30%.
Figure 6: The 3D array distribution of the candidates in a cell array 63 tagged
by horizontal, Vertical and Longitudinal labels. For sake of simplicity in upward figure, the array cells are shown with the colored plane sections that are
crossing and selecting an inner central True infected case. The inner detection
by the independent 3 sub-planar-groups is shown on the left side. Moreover,
as shown in the right side, the presence of just 2 True infect cases creates six
plane intersection forming 23 cubic vertex: 2NT + 6NF = 8. Therefore a larger
growth of False suspected cases. The geometry is zoomed in the right corner
to show the cubic structure. This False growth imply a wider over-estimation
respect to the bi-dimensional cases.
The number of 3D planar intersections are very often leading to an over-estimation of True candidates whose number may
be nevertheless well bounded. For instance a simple generalization may occur for any three True location, all with different
coordinates. Their tests will tag 3·3 positive planes results. The
consequent whole possible intersections will be much larger
than the few three of the True cases: indeed the 33 planar intersections will suggest (at each corner), 27 suspected positive
cell candidates. In the suggested case, assuming non co-planar
3 True patient, there will be 33 − 3 False ones. A very large
polluted result. Naturally there may be also some coexistence,
in the same plane of few True positive subject, reducing the
False overabundance. The complexity may be restricted to the
extreme bounds for the NT ones. Let us assume that there are
NH horizontal positive planes, NV vertical and NL longitudinal
positive planes. Their total vertex (or intersections) number will
be: NVertex = (NH · NV · NL ). There might be in general a True
5. The Multi-Dimensional (3D,..MD) Purim
The generalization to more M dimension is obvious: the
same extension may be done from the third dimension case to
more, M, dimensions. The procedure is less intuitive but it
is very simple, not a necessary step to discuss here in detail.
5
The chain of sub structure (planar, vector) for Purim procedures might be extended to (Cubic, planar, vector) and so on
for each greater dimension stage. The most effective procedure
may have an ratio R (among Purim test versus single ones) as
R = M · N/(N 3 ) = M/(N M−1 ) = M/(NALL (M−1)/M ). Therefore
the larger M and the better the effective R (assuming a very
diluted sample of candidates).
Fig. 2. Another limit is due to the dilution of the biological
sample by mixing of the positive ones with the negative ones
in the same sub-group. This limit depends on both the concentration of the nCoV in the samples collected from infected
people and on the limit of detection (LoD) of the testing technology. No specific data on the actual LoD is available in the
open literature. The CDC reports [10] a limit for common RTPCR lies between 100.0 and 100.5 RNA copies µL. Yet no data
are available on the concentration of RNA copies in the specimens of infected people. On the other hand, authors from China
[11, 12] report relevant negative PCR tests in symptomatic patients, even with repeated tests. Nonetheless, a dilution between
10 to 100 would be probably acceptable leading to the opportunity for rapid testing of clusters between 100 and 10,000 people
with reduction of number of laboratory tests between 80% and
98%. In a limited time scenario, in which the number of people tested is limited by the availability of laboratory time, this
means that the application of the proposed method will allow
to increase the number of tested people between 5 times and 50
times in comparison to standard one-by-one testing.
5.1. The zoom scaled Purim filter
The procedure that we are suggesting might be seen as a
filter able to pick up, by its narrow orthogonal array net, the
single or just a few positive infect patients. The same filter may
applied in larger scale. Indeed the Purim procedure may be enlarged in wider scale changing the granularity of the sample.
Let us imagine a whole clustered group (hundreds or thousands
persons), whose mixed test are jointed and unified, as a single subject, whose collective test is either a positive or negative
one. Such a positive clustered group would contain at least an
infected person. Now, let us imagine a larger scale array, made
by such clustered groups, each individually located in a square
matrix, of such clustered groups. Each of those sub-group as a
whole may be tagged as positive or negative. For example, let
us imagine that the clustered groups are the school classes of
25 students each. Let us also suggest, that the total class number in the big school is: ( NclALL =25). Therefore we have a
625 total students in the whole school. The classes are identify, for instance, by their age (14th − 18th year old) and also
by their peculiar section (a,b,c,d,e), into a wider square matrix
classes-elements. Each class may be localized by its letter and
class age. Such an array 5x5 of clustered groups (= classes)
resemble, in full analogy, the previous bi-dimensional Purim
section. We may find by column and row intersection the (one
or few) infected (true or false) positive classes, with just 10 unified tests. We verify the 25 classes. If any is positive, Finally
we may further inquire, restricting as in a microscopic zoom,
the positive class, into a narrow scale tests, looking for the positive and negative or false individual students. Any positive class
will be at once again re-analyzed, by the inner zoomed Purim
procedure, to figure out each individual infected student. With
only a first 10 tests we may inspect 625 student classes. Like
from a telescope toward a microscope view, from the large to
the small scale, Purim procedure may offer rapid, inexpensive,
as well a high filter accuracy.
7. Applications and Conclusions
The present common project to defeat the CoVid-19 by a
severe quarantine of the whole national economy might be successful, but extremely costly and late. The ability to distinguish
and filtering positive from negative subjects may free a steady
component of the population into the economy life, while holding the rest of the positive or untested citizens in a restrictive
quarantine. The usual one-test to each patient in present international CoVid-19 pandemia is exhausting both in the economy
and in time prospective. A fast and economic is based on the
Purim procedure (Purified by Unified Resampling of Infected
Multitudes): the collective array sub-group tests whose crosschecked results may rapidly intersect and disentangle the real
and the few suspect positive patients. For a flight or a ship embark procedure, respectively of 102 −104 individuals passengers
the present Purim strategy may cost about (20%) up to (2%) of
the whole sample test, both in cost and in time. In a diluted
< 1% infected population stages these procedure might require
only a negligible additional tests to better identify the real True
positive infected passengers, from the few possible False ones.
In general such Purim test will free and guarantee the majority
of the passengers as being negative by a double-checked subgroup test. The Purim application in the present medical subjects it may avoid the presence of infected doctors at the hospital care. The same Purim purification may be applied to classes
or to schools leading, step by step, to a fast reopening of the education system. For an ideal national population, as large as the
Italian one of 6 · 107 candidates, the fragmentation of the whole
population into 6 · 103 sub group, each made by NALL = 104
citizens, may be achieved by 2 · N 2 , just 2 hundreds, sub test
each, within a realistic goal: minor city areas may be tested
step by step and they may be let free in life while positive identified subjects may be better guaranteed in their medical and
quarantine needs. This procedure would cost nearly 2 − 3% of
the individual national population test, with possibly dozens of
6. Discussion
The proposed method is suitable for detecting suspects, but
also identifies as possible positive subjects that would have been
clinically negative due to a ”shadow effect” (F). As far as the
method is used for screening large groups with low probability
of being infected, this reduction of accuracy won’t decrease its
ability to avoid the spreading of the infection.
The effectiveness of the method in comparison to one-byone testing is related to the increase in the number of people tested together as a unique sample. Yet, this is limited by
the probability of each individual being infected, as shown in
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billions of euro saving expenses. The procedure might be naturally extended in the international arena, where the pandemia
is just at the earliest stages, with more profit and success.
Acknowledgement
The authors wish to thank the Prof. Moshe Labi, of New
York A. Einstein Hospital, for very helpfull suggestions and for
letting us know today of a few days ago Jerusalem Post [13]
article, regarding a Technion proposal, analogous to our first
and simplest United Group Purim procedure [see section 2].
References
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