relatively low reported number is attributed to a number of factors, including but not limited to mismanagement of the epidemic at the political level, high ER visit costs, and a lack of resources that limit the number of tests drastically. For example in the UK 25,000 tests were carried out in the period since January 2020 and up until March 11, 2020, which is equivalent to the number of tests carried out in South Korea in two and half days according to the World Health Organization. Thus, while in some countries there is a (front-end) problem of sample collection, in other countries the main concern is in processing the collected samples (backend problem). Here we are concerned with the back-end problem, namely processing a large amounts of collected samples.
Specimens can be collected from the upper respiratory tract as nasopharyngeal (through the nose) and oropharyngeal (through the mouth) swab or wash in an ambulatory regime 2 .
The laboratory confirmation of the COVID-19 is based on the Nucleic Acid Amplification Tests (NAAT); the assay detects the genomic sequences of virus RNA by real-time reverse transcription polymerase chain reaction rRT-PCR.
The current outbreak has evoked researchers and experts from various fields to reevaluate the feasibility of multi-sample pools 3, 4 , where samples from a number of patients are mixed together, as opposed to testing individual samples. In this work, we propose an advanced testing method where samples from each patient are mixed in multiple tubes in a unique configuration, then variation of test "positiveness" of each tube are employed in order to calculate all possible positives. The first part of the method is by itself powerful when the percentage of infected patients is extremely low, as long as proper mixing is done -keeping in mind the dilution threshold (due to mixing) required for identifying the disease. However, as the percentage of the infected increases it becomes much more challenging to determine positives without performing new tests. To this end, an accurate quantitative approach (the second part of the proposed method) can be employed to determine all positives without risking having false negatives. For this part to be effective, an accurate method for quantification of PCR is required 5 . The proposed method takes into account the uncertainty (error) in the test. Even when the uncertainty increases (i.e. for less accurate tests), all positives are still obtained with no false negatives, though false positives start to arise as well. The mathematical model of the proposed method is presented in the following section.
Let n be the number of patients (sample size), m the size of the test tube set, and l the size of a subset of the tube test set. Each patient sample is distributed to a different configuration of l tubes. Thus, the maximum number of patients is given by
The test results in each tube j = 1..l can be described by
where r i j is the contribution of the i-th patient to the 'positiveness' of the j-th tube, and δ i j is the delta function, being zero or unity for negatively or positively tested patients, respectively.
Thus, we can now construct a set of n algebraic equations 
or for simplicity we write r T δ = R, where subscript T is the transpose operator. The solution vector R represents the test results, and thus known with some degree of test uncertainty, ∆R, which we shall account for. On the other hand, while the matrix r is unknown, the distributions of the samples in the tubes is our choice, and that is simply the matrix r with nonzero elements replaced by ones, 
which is the distribution matrix (n × l) that shows how samples from each patient (rows) are added to the tubes (columns). Finally, the vector δ contains the information we seek, on positive and negative patients. Thus, our objective is find δ. 
Phase 2: Identifying all positives. We rewire equation (3) 
To gain more quantitative understanding of the proposed method, we performed numerical tests with actual scenarios under various conditions. We considered group sizes of 28, 56, and 70 patients that were tested using 8 tubes only (figure 1), and group sizes of 120, 210 and 252 patients using 10 tubes -see figure 2 . Infected patients were selected randomly, with a percentage that ranged from 0.8 to 21.43. Each result point is an average of a hundred test repetitions, which is important when discussing uncertainties in the tests that were allowed to be between ∆R = 0.05%...30%. Without loss of generality, each of the positively tested patients were allocated a random number between 0-220 nano grams (ng), that mimics possible reading, e.g. using a PCR technique, though any other range could be equally implemented.
Uncertainty in the test results, ∆R, depends on a number of factors among which are the accuracy and precision of the test method. The less sensitive the test is the larger ∆R becomes, e.g. an uncertainty of 10% is equivalent to 20 ng so that we are unable to distinguish between two readings with difference that is less than 20 ng. Therefore, as ∆R increases false solutions may appear, and thus false positive results are expected. However, since the actual solution lies within any given uncertainty, we never obtain false negatives, which is extremely important for disease management and control. If the uncertainty is very small, we are always able • Samples from each patient are added to all tubes that correspond to digits with number '1'. For example, samples from patient (0 0 0 1 1 1) are added to tubes number 4, 5, and 6 (from left).
• All m tubes are sent for testing. It is important that testing is obtained at a cycle where no saturation has been reached, so that the value of each positive test represents a sum- 