Purim: a rapid method with reduced cost for massive detection of CoVid-19
Benjamin Isac Fargion Daniele Fargion Pier Giorgio De Sanctis Lucentini Emanuele Habib
q-bio.PE physics.soc-ph










      ]Benjamin Isac Fargion
      sap,mifp]Daniele Fargioncor daniele.fargion@uniroma1.it
      RU]Pier Giorgio De Sanctis LucentiniPgDSLORCiD
      DIAEE]Emanuele Habib

      [cor]Corresponding author, ORCiD:0000-0003-3146-3932
      [PgDSLORCiD]ORCiD:0000-0001-7503-2064


      [sap]Physics Department & INFN Rome1, Rome University 1, P.le A. Moro 2, 00185, Rome, Italy
      [mifp]Mediterranean Institute of Fundamental Physics, Via Appia Nuova 31, 00040 Marino (Rome), Italy
      [RU]Physics Department, Gubkin Russian State University (National Research
      University), 65 Leninsky Prospekt, Moscow, 119991, Russian Federation
      [DIAEE]DIAEE, Department Of Astronautical, Electrical And Energy Engineering, University of Rome "La Sapienza",  Via Eudossiana 18, Rome, Italy





 
 
 
 
 
 
 
 









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 cross-correlated 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.



  VirusCoVid-2019Detection









§ INTRODUCTION



 The CoVid-19 hidden pandemia, born recently <cit.> 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, <cit.>,  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)<cit.> 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 of the infected cases. Therefore, because of overestimation one could require  a few additional individual tests with marginal cost.
      
  The , [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.], 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 <cit.>.













§ THE UNITED GROUP TEST


Let us consider as the simplest experiment, the of a sample of suspect infected ones as large as N_ALL. The simplest case might be a passenger group waiting for a flight. Let us assume, as this first simplest procedure to combine and mix all the
N_ALL test in an unified one. This whole (N_ALL) mixed test may be either positive to CoVid-19 or not. The absence of any infected, the negative case, will free the  N_ALL 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  procedure based on  partial sub-group cross-checked  tests.
    


§ THE BI-DIMENSIONAL (2D) 


Let us therefore assume that the suspect group to disentangle and to analyze by may be located, one by one, within an ideal square matrix array whose components are  N_ALL = N^2. In general  N_ALL 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 N_ALL  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  N_ALL = 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 N_ALL = 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:  N_ALL = (N+1)^2; consequently, as before,  few K empty places will be added in the  new arrow and new column, where:  K = (N+1)^2- N^2 -M. Finally any way we imagine to locate all the  N_ALL patients into a square matrix array.
    
Therefore, as described as an example, by  Fig.<ref> ( N_ALL= 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.<ref> 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  N_T as the whole number of true positive patients. Let us also define by N_F 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.<ref> 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  N_T +N_F ≥ N_T 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  selective tests, as the few examples shown in  Fig.<ref>, to figure out the infected case by the intersection of sub-group tests.
 
 

 §.§ The bi-dimensional Sub Group 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.<ref>.
   Now we should call each sub group of patients SG 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.<ref>:
       
    SG_a, SG_b,..SG_i

       
    SG_1, SG_2,..SG_9

       
       The cross sections  of their cross checked test results imply, for instance, for the first case, the coexistence of two positive sub-groups:
            
    SG_c, SG_2

        
            The two group  intersection identify an unique infect place subject sitting both in  SG_c and in SG_2. 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:
     
            
    SG_c, SG_2
 
            
    SG_g, SG_6

            
      However as shown in Fig.<ref> 
      there are  also suddenly two  additional (false or "shadows") F candidates, due to different "mirror" intersections:
         
    SG_c, SG_6

         
    SG_g, SG_2

         
        In conclusion: in present two real infected T there are also two virtual false F (not infected) cases. A first over-estimation of the procedures.
        In the final  third situation described by the right side in Fig.<ref>, the disposal, with the three True positive candidate induced  a total of 6 false positive candidates. In general the suspects  (real or True ones N_T) are less than the suspected  ones P_s ≥ N_T+ N_F ones. 
        
        
        
    The last case in the  Fig. <ref>  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:
            
    SG_c, SG_2

            
    SG_g, SG_6

            
    SG_e, SG_8
.
             
       However, there are additional 6 independent sub-group intersections  defining six more possible or virtual positive, False, candidates:
             
            
    SG_c, SG_6

            
    SG_g, SG_2

            
    SG_e, SG_2
.
            
    SG_e, SG_6

            
    SG_c, SG_8

            
    SG_g, SG_8
.
            
            All of these 9 possible intersections or infected cases must be tested individually or by similar mixed and cross-checked  procedure. Therefore, the 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. <ref>, procedure will detect positive subjects with quite a good precision. Therefore,  we may already conclude that in general the  single tests versus the  procedure  does cost, in economy and in time, by a ratio R= N^2/(2N). In a hundred sample, a 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) 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%.
        





        





                


 §.§ Over-estimation in a crowded sample
             
We may wonder about the previous overestimation in 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. <ref>.
   The general  complexity is not obvious.
   For very fine tuned locations of N_T=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. <ref>. Such a rare situation may occur once the sample of infected has a large probability P to be present within the total group 
    P ≥1/ 4·N_ALL^1/2
. For a square of 32 element and N_ALL = 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. <ref>.

        














 In general for a number of N_H (horizontal) positive row and for a number of N_V (vertical) column sub-group positives, the resulting number of True N_T cases is bounded by a maximal and minimal value:

    (Max (N_H, N_V)) ≤ N_T ≤ (N_H· N_V) .
 
In analogy the number of false negatives N_F is analogously bounded by

    (0 ≤ N_F ≤ (N_H· N_V)- (Max(N_H, N_V))).



 For example in the simple case as in Fig. <ref> where N_H =3, N_V =4 the previous bounds for  N_T becomes:( 4 ≤  N_T ≤ 12). Fig. <ref> shows indeed  N_T = 5 True and N_F = 7 as in their bounded limits.
                
   Naturally, because N_H and  N_V 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.<ref>. 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 procedure, as it is  shown in a more rich but more complex correlations approach considered below.
               
                


§ THE MULTI-DIMENSIONAL PROCEDURE
   

The natural extensions of the previous bi-dimensional 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. <ref>. The candidates are located in 3D cell structures labeled by three array axis. Two quite different  different procedure may used to disentangle the positive infect subjects; a planar one or a vector one, base on different
    cross-checked procedure.



 §.§ The Planar (2D) cross-check inside a 3D array 

Let us imagine to locate the N_ALL suspected patients   inside a cube cell defined by its 3D coordinates. As in the bi-dimensional case, this may be reached by dividing the whole number N_ALL 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 N_ALL into cubic sizes are irrelevant here.
  Each person is tagged by its H_i, horizontal,  V_i, vertical,  L_i, longitudinal position inside the  ideal cubic cell volume. For instance a sample of  N_ALL = N^3 = 216  passengers are tagged by a 3D array made by 6^3 cubic side cell structure as shown in Fig. <ref>. The easy way to act within a planar  procedure in 3D cells (candidates) array is based on  the construction of 3· N planar sub-groups associated to all the three possible planes: HV_i, HL_j, LV_k. (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  N_ALL = N^3 = 216.  In the presence of an unique positive person in the whole cube sample its planar intersections may define just a single cell candidate,  as shown in Fig. <ref> 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 N_ALL= 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. <ref> 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. 

 





 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 3^3 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  3^3- 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 N_T ones.
Let us assume that there are N_H horizontal positive planes, N_V vertical and N_L longitudinal positive planes. Their total vertex (or intersections) number will be: N_Vertex= (N_H · N_V · N_L). There might be in general a True number N_T inside bounded by:


    (Max (N_H, N_V, N_L)) ≤ N_T ≤ (N_H· N_V · N_L) .
 

In analogy the False N_F  are also bounded by

    (0 ≤ N_F ≤ (N_H· N_V · N_L)- (Max(N_H, N_V,N_L))).


In the Fig. <ref> these bounds are confirmed. 
The true events are N_T =2, the False are N_F = 6 = 8-2. Both are following their extreme  bound limits.









 §.§ 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 advantage.
    There  is an additional tool in the 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.<ref>. Assuming as above N_ALL= 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 R_gain between 1 dimension procedure and individual one)  is only R_gain1 = 3 · N^2/ (N^3) = 3/N, while in previous one (for the 2D array structure) it was  R_gain_2 = 3/(N^2). For N=10 and a thousand of the whole candidates, this imply for the planar just R_gain_2 = 3% while for a vector 1 dimensional procedure a less attractive: R_gain1 = 30%.




§ THE MULTI-DIMENSIONAL (3D,..MD)  

 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. The chain of sub structure (planar, vector) for 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 test versus single ones) as R= M ·  N/(N^3)= M/(N^M-1) = M/(N_ALL^ (M-1)/M). Therefore the larger M and the better the effective R (assuming a very diluted sample of candidates). 
 
 

 §.§ The zoom scaled   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 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: ( Ncl_ALL =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 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 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, procedure may  offer rapid, inexpensive, as well a high filter accuracy. 

 


§ 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-by-one 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 Fig. <ref>. 
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 <cit.> a limit for common RT-PCR lies between 10^0.0 and 10^0.5 RNA copies μ L. Yet no data are available on the concentration of RNA copies in the specimens of infected people. On the other hand, authors from China <cit.> 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.




§ 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 procedure (): the collective array sub-group tests whose cross-checked results may rapidly intersect and disentangle the real and the few suspect positive patients.    
  For a flight or a ship embark procedure, respectively of  10^2-10^4 individuals passengers the present 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 test will free and guarantee the majority  of the passengers as being negative  by a double-checked sub-group test. The application in the present medical subjects it may avoid the presence of infected doctors at the hospital care.  The same 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· 10^7 candidates, the fragmentation of the whole population into 6· 10^3 sub group, each made by  N_ALL =10^4 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 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 <cit.> article, regarding a Technion proposal,  analogous to our first and simplest United Group procedure [see section <ref>]. 




§ REFERENCES

*


