Assuming that tests are perfect, we usually aim to correctly classify all *N* individuals as infected or noninfected. We often summarise the performance of algorithms through the *expected tests per individual*. If an algorithm uses a (possibly random) number of tests *T* to classify *N* individuals as either 'infected' or 'non-infected', then the expected tests per individual is *(*E*T )/N*, where E*T* denotes the expectation (or mean, or average) of the random variable *T* . Clearly, it is desirable for the expected tests per individual to be as small as possible. Note that individual testing clearly has *(*E*T )/N* = *N/N* = 1. The expected tests per individual is useful for comparing how much better (or worse) an algorithm is than individual testing.

A standard information-theoretic bound called the *counting bound* (see, for example, Aldridge (2019), Aldridge et al. (2019), Baldassini et al. (2013)) states that, for any successful pooled testing procedure, the expected tests per individual satisfies the lower bound

$$\frac{\mathbb{E}T}{N} \ge H(p). \tag{11.1}$$

Here, *p* is the prevalence of the disease, and *H(p)* is the *binary entropy* function, defined by

$$H(p) = p\log\_2\frac{1}{p} + (1-p)\log\_2\frac{1}{1-p}.$$

The bound (11.1) immediately implies that, when the prevalence is high enough, pooled testing cannot significantly outperform individual testing (under the model of perfect tests). For example, when the prevalence of infection in the population being tested is 20% (*p* = 0*.*2), we have *H(*0*.*2*)* ≈ 0*.*72, and therefore no pooled testing algorithm can use less than 72% of the number of tests per individual required by individual testing, at this prevalence level (under the model of perfect tests). In fact, a deeper mathematical result of Fischer et al. (1999) states that (under the model of perfect tests), individual testing is optimal among all pooled testing algorithms whenever the prevalence is at most 38.2% (*p* ≤ 0*.*382).

It is useful to consider, for different pooled testing algorithms, how close their 'expected tests per individual' is to the counting bound (11.1), at different prevalence levels, under the model of perfect tests, and this is what we shall do in this section.

In this section, we study the following four classes of pooled testing algorithms:

Figure 11.2 shows the performance of these algorithms (with the grid and *(r,s)* regular methods algorithms in what we will later call their 'conservative two-stage' variants—see Sect. <IP_ADDRESS>). The top subfigure shows the expected tests per item, and indicates the potential benefit of using pooled testing (compared to individual testing) when the prevalence is below 20%.

However, this top subfigure does not sufficiently convey the difference between the pooling algorithms at very low prevalence. The bottom subfigure shows the *rate*, defined by

$$\text{rate} = \frac{H(p)}{\mathbb{E}T/N} = \frac{H(p)N}{\mathbb{E}T}.$$

The rate measures how close an algorithm gets to the counting bound (11.1) higher rates are better. (We remark that the rate can also be interpreted informationtheoretically, as the number of bits of information learned per test. See for example Aldridge et al. (2019), Baldassini et al. (2013).) The rate illustrates better the comparison between the different pooling methods, and shows the advantage of the *(r,s)*-regular design at very low prevalences (e.g., below 2%).

#### *11.2.2 Individual Testing*

Clearly, the individual testing of *N* individuals requires *T* = *N* tests, yielding an expected number of tests per individual equal to *T/N* = *N/N* = 1.

An obvious advantage of individual testing is that one does not need an estimate of the prevalence. It is a one-stage algorithm, so has the fastest possible turnaround time.

**Fig. 11.2** Performance of one- and two-stage group testing algorithms under perfect testing, as measured by **a** the expected tests per item, and **b** the rate. 'Grid' and '*(r,s)*-regular' refer to the conservative two-stage variants. At all but the lowest prevalences, the *(r,s)* regular design has optimal parameter *r* = 1, so is equivalent to Dorfman's algorithm, or *r* = 2, and is equivalent to the grid algorithm

It is also the simplest testing algorithm to implement. However, we will see that other algorithms yield huge resource savings over individual testing, at low prevalencelevels, and are therefore well worth considering, when prevalence is fairly low and testing capacity is constrained.

#### *11.2.3 Dorfman's Algorithm*

In Sect. 11.1.2, we introduced Dorfman's original two-stage algorithm. In this section, we discuss it in more detail.

Suppose we receive samples from *N* individuals in a large population where the prevalence of infection is *p*, and we run Dorfman's algorithm on these *N* individuals, using pools of size *s* ≥ 2, where *N* is assumed to be a multiple of *s*. Clearly, one test for each pool is always used at the first pooled testing stage. A pool will also require an extra *s* individual tests in the second stage if the pooled test was positive. The pooled test will be positive unless all *s* items are noninfected, so the probability it will test positive is 1 − *(*1 − *p)s*. So for each of the *N/s* pools we definitely have 1 pooled test, then with probability 1 − *(*1 − *p)s* another*s* individual tests are required, giving an expected number of tests as

$$\mathbb{E}T = \frac{N}{s} \left( 1 + s \left( 1 - (1 - p)^s \right) \right) = \left( \frac{1}{s} + 1 - (1 - p)^s \right) N.$$

Hence the expected tests per individual is

$$\frac{\mathbb{E}T}{N} = \frac{1}{s} + 1 - (1 - p)^s.$$

It is easy to check that *s* = 2 is never the best choice of pool size *s*, but that *s* = 3 improves on individual testing for *p <* 1 − *(*1*/*3*)*1*/*3 = 0*.*307. That is, Dorfman's algorithm improves on individual testing for prevalences below roughly 30%.

If the prevalence *p* is known accurately beforehand, we should choose the pool size *s* so as to minimise the quantity 1*/s* + 1 − *(*1 − *p)s*, and thereby minimise the expected number of tests required. For fixed *p*, the function *s* → 1*/s* + 1 − *(*1 − *p)s* is very well-behaved, and it is very easy to numerically find the integer *s* that minimises it. But it is useful also to note that when *p* is small, we may use the approximation

$$\frac{1}{s} + 1 - (1 - p)^s \approx \frac{1}{s} + 1 - (1 - ps) = \frac{1}{s} + ps,$$

which is minimised over the reals at *s* = 1*/* √*p*. This gives an expected tests per individual of approximately 2√*p*. More formally, choosing *s* = 1*/* √*p* yields that the expected number of tests per individual is 1*/s* + 1 − *(*1 − *p)s* = 2 √*p* + *O(p)*,

**Table 11.1** Resource requirements of using Dorfman's algorithm to test *N* individuals, at different prevalence levels, with tests of perfect sensitivity and specificity

where the error term *O(p)* is small compared to √*p* when *p* is small. Hence, when *p* is small, a good estimate for *p* is known beforehand, and the pool-size *s* is chosen sensibly, our model predicts that Dorfman's algorithm uses, on average, approximately 2√*pN* tests to identify all the infected individuals among a population of size *N*.

Table 11.1 shows the predicted resource requirements (under the simple model above, i.e., perfect tests), of using Dorfman's algorithm to test *N* individuals, at different prevalence levels, when the prevalence level is known accurately beforehand, and the pool-size *s* is chosen optimally.

There are shortcomings in this analysis. We mention three here. Firstly, tests used in the real world do not have perfect sensitivity or perfect specificity; we will refine the model to take this into account, in the following section.

Secondly, an accurate estimate for the prevalence may not be known beforehand, so it may not be possible to choose the pool size *s* optimally beforehand. We return to this issue in Sect. 11.4.

Thirdly, the independence assumption may fail because infections of different individuals being tested by the laboratory are unlikely to be truly independent. But if the overall prevalence remains the same, then Dorfman's algorithm will not perform any worse than the above analysis predicts. In fact, it is advantageous in terms of resource requirements if there is a 'clustering' of infected individuals in the same pool. In terms of resource use, the worst case for Dorfman's algorithm is when infected individuals are spread between many pool, and the best-case is when infected individuals are concentrated in few pools.

#### *11.2.4 Grid Algorithms*

In Dorfman's algorithm, each individual was tested once in the first stage. In a family of algorithms called *grid algorithms*, each item is tested twice in the first stage. One variant attempts to classify samples as infected or noninfected just from this single stage, while other variants follow up with a second stage of individual testing.

#### **<IP_ADDRESS> Variants of Grid Algorithms**

The grid algorithms always begin as follows. Suppose we have *N* individuals to test. We split these into *N/s*2 groups each of *s*2. (We assume for simplicity that *N* is a multiple of *s*2.) Let us concentrate on a single group. Each of the *s*2 individuals is swabbed, and the sample from each swab is divided in two, so it can later be a part of two different sample pools. We now picture the *s*2 individuals as laid out on a *s* × *s* grid. In the first stage we conduct 2*s* pooled tests: we make one pool from each of the *s* rows of the grid, and one pool from each of the *s* columns. A PCR test is run on each of these sample pools.

We assume, again, that tests are perfectly accurate. What can we learn from these results of these pooled tests?

What should one do after receiving the pooled test results? Here, we briefly look at three possible choices, leading to three different variants of the grid algorithm:

**Table 11.2** Resource requirements of using two variants of the grid algorithm at different prevalence levels, with tests of perfect sensitivity and specificity. In the one-stage variant, parameters are chosen such that each grid of *s*2 individuals is correctly classified with probability at least 0*.*99. A dash—means that the method is worse than individual testing

• *Conservative two-stage grid algorithm*: Alternatively, we can run a second stage where all individuals that appeared in two positive-testing pools are given an individual test—even in Case 1 or Case 2A, where the infected individuals can be logically determined after the first stage. This has the advantage that every infected individual can be definitely confirmed as infected by the 'gold standard' of an individual (non-pooled) PCR test, which might be personally reassuring for the individual or their employer, beneficial when tests are imperfect, or required by regulators. Two-stage algorithms where infection must be confirmed with an individual test are often known in the literature as *trivial two-stage algorithms*, but in this chapter we will use the term *conservative two-stage algorithm*, as it is more descriptive. Conservative two-stage algorithms can be easier to analyse mathematically, than the standard two-stage ones.)

In real-life situations, when using the one-stage variant of the grid algorithm, one must make a decision on what to do with individuals with an inconclusive result. One option would be to inform all those individuals they should self-isolate as a precaution; another option would be to individually re-test each of them at a later date (effectively running a two-stage algorithm with a delayed second stage); another option would be to restart the testing from scratch with different grids; a more reckless option would be to inform all the individuals being tested that the test results were inconclusive and that they should continue their lives as if they had tested negative. Which option the laboratory or the regulatory authorities choose may depend on the impact of letting an infection go undetected: if the individual in question is a school pupil or a member of the general community in a mass-testing programme, the impact is likely to be much less than if the individual is a healthcare worker working with highly vulnerable patients or a resident-facing social care worker, for example.

In the next subsection, we give a brief analysis of the conservative two-stage variant of the grid algorithm, under our convenient assumption that the tests are perfectly accurate. We summarise these results in Table 11.2.

#### **<IP_ADDRESS> An Analysis of the Conservative Two-Stage Grid Algorithm**

Recall that in the conservative two-stage variant of the grid algorithm, every individual in two positive pooled tests receives an individual test. Our analysis here is similar to that in Aldridge (2020), Broder and Kumar (2020).

For each of the *N/s*2 grids of *s*2 individuals, there are 2*s* pooled tests in the first stage. An individual can then receive an individual test for one of two reasons. One is that the individual is infected, which occurs with probability *p*. The second is that the individual is uninfected (which occurs with probability *q* = 1 − *p*), and that further, its row contains an infected individual (this happens with probability 1 − *qs*−1), and its column also contains an infected individual (this also happens with probability 1 − *qs*−1).

Hence, using the linearity of expectation, the expected number of tests is

$$\mathbb{E}T = \frac{N}{s^2} \left( 2s + s^2 (p + q(1 - q^{s-1})^2) \right) = N \left( \frac{2}{s} + p + q(1 - q^{s-1})^2 \right),$$

and the expected tests per individual is *(*E*T )/N* = 2 *s* + *p* + *q(*1 − *qs*−1*)*2.

As before, given the prevalence *p*, this is a well-behaved function of*s*, so it is easy to numerically choose the optimal *s*. Results were shown in Table 11.2 above, and far outperform the one-stage variant when a second stage is available. Comparing with Table 11.1, we see that the conservative two-stage grid algorithm also outperforms Dorfman's algorithm for the values of *p* under consideration, though not by more than a factor of approximately two, for these values of *p*. We note, however, that the optimal choice of *s* (the pool-size in the conservative two-stage grid algorithm) for *p* = 0*.*1%, is 106, and large pool-sizes do have practical down-sides (usually requiring automation, reducing sensitivity, and posing regulatory problems—see Sect. 11.4).

From a more mathematical perspective, a similar approximation to that for Dorfman's algorithm shows that, for small *p*, the optimal *s* is of the order 1*/p*2*/*3; this yields an expected number of tests per individual of the order *p*2*/*3. For *p* sufficiently small, this is an improvement on Dorfman's algorithm (where the expected number of tests per individual is of the order *p*1*/*2).

#### *11.2.5 Pooling Algorithms Based on (r, s)-Regular Designs*

The *(r,s)*-regular designs are a family of algorithms that generalise the algorithms we have seen so far. These algorithms have a first stage where each individual's sample appears in exactly *r* different pools, and each pool contains samples from exactly *s* individuals. The second stage (if there is a second stage) consists of individual testing—as with the grid algorithm discussed in the previous section, in the 'standard' variant, an individual is given an individual test at the second stage only when their infection status cannot be determined from the first stage, whereas in the 'trivial' variant, an individual is given an individual test at the second stage whenever all the pools they appear in test positive at the first stage.

Individual testing is a special case of an *(r,s)*-regular design, with *r* = 1, *s* = 1 and a single stage. Dorfman's algorithm is a special case with *r* = 1, *s >* 1, with two stages. The grid algorithms we discussed above are special cases with *r* = 2.

There are a number of ways to construct a pooling design that is *(r,s)* regular.

We note that, by counting the number of times a sample appears in a pool in two different ways, the number of pooled tests *T*1 used by the first stage of an *(r,s)* regular design satisfies *Nr* = *T*1*s*, and therefore the number of tests per individual used by the first-stage of an *(r,s)*-regular design is *T*1*/N* = *r/s*.

As stated above, an *(r,s)*-regular pooled testing algorithm can be used in the form of a one-stage, a 'standard' two-stage, or a conservative two-stage algorithm, just as with the grid algorithm.

We briefly present a summary of an analysis of the conservative two-stage variant, following Aldridge (2020), Broder and Kumar (2020). With this variant, any individual whose *r* stage-one pooled tests are all positive receives and individual test in the second stage. For large *N*, the expected tests per individual in the random *(r,s)*-design (described above) satisfies the following with high probability:

**Table 11.3** Resource requirements of using an *(r,s)*-regular design in a conservative two-stage algorithm to test *N* individuals, at different prevalence levels, with tests of perfect sensitivity and specificity

$$\frac{\mathbb{E}T}{N} \sim \frac{r}{s} + p + q(1 - q^{s-1})^r,\tag{11.2}$$

where *p* is the prevalence and *q* = 1 − *p*. Here, *r/s* is the number of tests per individual in the first stage. An individual requires retesting in the second stage either if it is infected, with probability *p*, or if it is noninfected, with probability *q*, but all *r* of its tests are positive, each of which happens with probability 1 − *qs*−1. Thus if the results of the tests containing a given noninfected individual were independent, then (11.2) would hold exactly. It turns out that a randomly sampled *(r,s)*-design satisfies this independence condition for most individuals (with high probability); in fact, with positive probability, it satisfies the condition for all individuals.

Further, a lower bound is given in Aldridge (2020), which shows that, among all conservative two-stage algorithms, the random *(r,s)*-regular design is extremely close to optimal for all *p <* 0*.*3.

Table 11.3 shows the performance of the *(r,s)*-regular design according to (11.2) with an optimal choice of *r* and *s*. We note that the *(r,s)*-regular algorithm outperforms individual testing, Dorfman's algorithm, and the grid algorithms for all values of *p* in the table.

Table 11.3 shows results with the mathematically optimal choice of *(r,s)*, but as the prevalence gets small, these parameter choices can get quite large. This could be unwieldy or even infeasible for a laboratory to carry out, and large values of *s* (the pool-size) provoke worries about sensitivity (with imperfectly sensitive tests). However, typically *r* can be reduced somewhat and *s* reduced quite a lot with only a marginal reduction in performance. For example, at *p* = 0*.*5%, the optimal choice is *r* = 7, *s* = 147, giving an expected tests per individual of 0*.*063. But reducing the parameters to the much more manageable *r* = 3, *s* = 62 still gives an expected tests per individual of 0*.*072, which is only slightly worse. The practically best choice of parameters will depend on a laboratory's capability for carrying out complicated procedures, and worries about the impact of dilution on test sensitivity (see Sect. 11.4).

#### **11.3 Pooled Testing Algorithms for Imperfect Tests**

#### *11.3.1 The Model*

In this section, we refine the model of the previous section to take into account the fact that the tests we are dealing with do not always give the correct answer. Recall that the PCR test has very high sensitivity, typically higher than 99%, meaning false positive test results are extremely rare, and has moderate sensitivity, typically between 70 and 90%, meaning that false negative results are not uncommon.

Here, we use a very simple model for such tests. We assume that each test on a pool containing at least one infected sample has a fixed probability *u* of correctly returning a positive result, and that each test on a pool of entirely noninfected samples has a fixed probability *v* of correctly returning a negative result, independently of the outcomes of all other tests (including of tests on overlapping pools), and independently of the size of the pool (that is, with no 'dilution' effect).

Whether or not this is a realistic model will depend upon the main sources of false negatives and of false positives, and therefore on the precise protocol being used and the practical situation. If the main source of insensitivity or nonspecificity is a shortage of reagents, faulty equipment, or faulty lab-procedures, then it is probably quite realistic. If individuals are frequently swabbed incorrectly (as can happen when individuals are asked to self-swab), then incorrectly taken swabs will be an important source of insensitivity, and in this case, unless individuals are re-swabbed at each successive stage of a group-testing algorithm and there are no overlapping pools at any single step, the independence assumption will not be valid. Moreover, the assumption that dilution (where a small number of positive samples are diluted by a large number of negative samples), does not affect sensitivity, is likely to be fairly realistic with pool-sizes of 10 or less and with typical viral loads, but will be less realistic with pool-sizes of 100 or more. See Sect. 11.4 for a further discussion of this issue.

In the rest of this section, we look at some results regarding two-stage and onestage algorithms, under this model for noisy tests.

# *11.3.2 Analysis of Individual Testing and Dorfman's Algorithm*

For a given algorithm, there are (at least) three things we want to know: First, how many tests do we expect to use? Second, how many false negative declarations do we expect to make? Third, how many false positive declarations do we expect to make? A useful quantity for comparing algorithms is the *expected number of tests per isolated individual* (ETI): that is, the expected number of tests used, divided by the expected number of infected individuals correctly discovered and instructed to isolate. Since the isolation of infected individuals is the main public-health goal of a screening programme, ETI is a good measure of how much benefit we are getting per test used (though it does not take into account turnaround time).

Let us start with individual testing. To test *N* individuals, this requires exactly *N* tests. There are *pN* infected individuals on average, and we find each one if its test correctly gives a positive result, which happens with probability *u*. So on average we correctly find *upN* infected individuals but falsely miss *(*1 − *u)pN* of them; so the ETI is *up*. Similarly, also on average, of the *(*1 − *p)N* noninfected individuals, we correctly identify *v(*1 − *p)N* of them, but falsely declare *(*1 − *v)(*1 − *p)N* of them to be infected.

Now consider using Dorfman's algorithm to test *N* individuals with pools of size *s*, with *N* a multiple of *s*, and suppose we use the protocol of declaring an individual to be infected only if both their pooled test and their individual test are positive. (If a pool tests positive in the first stage but all the corresponding individual tests in the second stage are negative, the pooled test is assumed to be a false positive.)

First, an individual will make it through to the second stage if either the pool is infected, and correctly gives a positive result; or if the pool is noninfected, but incorrectly gives a positive result. Thus the expected tests per individual is

$$\frac{\mathbb{E}T}{N} = \frac{1}{s} + \mu(1 - q^s) + (1 - v)q^s,$$

where *q* = 1 − *p*. Here, 1*/s* represents the requirements of the first stage (i.e., the pooled tests), *u(*1 − *qs)*represents the requirements of the second stage in the case of a true positive pool result, and *(*1 − *v)qs* represents the requirements of the second stage in the case of a false positive pool result. For small *p*, it turns out that an essentially optimal choice for minimising this quantity is *s* = 1*/* √*up* ; this can be shown in a similar way to in the previous section. This yields an expected number of tests per individual which is approximately 2√*up* + *(*1 − *v)*, compared to 1 for individual testing. For *p* = 0*.*02, *u* = 0*.*8, *v* = 0*.*995, we get an improvement from 1 test per individual to 0*.*25 tests per individual.

Second, the expected number of infected individuals found is *u*2 *pN*, as there are *pN* infected individuals on average, and they are found if both their pooled test and their individual test are correctly positive. The other (on average) *(*1 − *u*2*)pN* infected individuals get false negative declarations. Compared to individual testing, where the total expected number of false negatives is simply *(*1 − *u)pN*, we have

$$\frac{(1-\mu^2)pN}{(1-\mu)pN} = 1+\mu \le 2,$$

and therefore the expected number of false negatives under Dorfman's algorithm can never be more than twice that when individual testing is used.

Third, a noninfected individual is falsely declared infected if both their pooled test is positive—either due to a false positive test or the presence of an infected individual and a true positive test—and their individual test is a false positive. This event has probability

**Table 11.4** Expected resource requirements and impact of using individual testing or Dorfman's algorithm to test *N* individuals, at different prevalence levels, with tests of sensitivity 80% and specificity 99.5%

$$((1 - q^{s-1})u + q^{s-1}(1 - v))(1 - v).$$

Hence, the expected number of false positives is

$$\left( (1 - q^{s-1})u + q^{s-1}(1 - v) \right) (1 - v) qN.$$

Again, for small *p* with *s* = 1*/* √*up* , this is approximately *(* √*pu* + 1 − *v)(*1 − *v)q N*. Compared to the expected number of false positives under individual testing, which is*(*1 − *v)q N*, Dorfman gives an improvement by a factor of √*up* + 1 − *v*. For *p* = 0*.*02, *u* = 0*.*8, *v* = 0*.*995, Dorfman gives an expected number of false positives which is approximately 0*.*12 times its value under individual testing. Thus Dorfman produces far fewer false positives than individual testing, a feature which is common to many other pooled testing algorithms.

Under the assumption that the sensitivity *u* of each test is 0.8 and the specificity *v* of each test is 0.995, Table 11.4 summarises, for different prevalences, the expected number of tests, false negatives, and false positives for individual testing and for Dorfman's algorithm. Note that, compared to individual testing, Dorfman's algorithm dramatically decreases the number of tests required and the number of false positives, but roughly doubles the number of false negatives.

The ETI of Dorfman's algorithm (used in the way above), is *(*1*/s* + *u(*1 − *qs)* + *(*1 − *v)qs)/(u*2 *p)*; if *p* is small and we choose *s* = 1*/* √*up* , as described above, then this is approximately *(*2 √*up* + 1 − *v)/(u*2 *p)*.

#### *11.3.3 One-Stage Testing*

We now briefly consider general one-stage (nonadaptive) pooling algorithms, under our simple model of imperfect tests. Here, we see the results of the tests, and we must try to come up with a 'best guess', from those results, as to which individuals were infected. The precise meaning of 'best guess' depends (for example) on the down-sides of missing infected cases (false negatives), and of false positives. This kind of problem is known as an *inference problem*.

We suppose the pooling design is chosen according to a *pooling matrix* A = *(ati)* ∈ {0*,* 1}*T*×*N* , a matrix of zeros and ones, where *ati* = 1 if the sample from individual *i* is included in the *t*th pooled test, and *ati* = 0 otherwise. The *T* rows of the matrix A represent the *T* pooled tests, and the *N* columns represent the *N* individuals being tested.

Some further notation is useful. Let **x** = *(xi)* ∈ {0*,* 1}*N* be the vector of zeros and ones where *xi* = 1 if individual *i* is infected, and *xi* = 0 if individual *i* is uninfected; the vector **x** represents which individuals are truly infected and which are not, and it is what we really want to guess; we refer to it as the 'infection vector'.

We write **y** = *(yt)* ∈ {0*,* 1}*T* for the actual outcomes of the tests, i.e., *yt* = 1 if the *t*th pool tests positive, and *yt* = 0 otherwise. Finally, we write **y**˜ = **y**˜*(*A*,* **x***)* for what the *T* outcomes of the pooled tests on the infection vector **x** would be, under perfect testing, i.e., *y*˜*t* = 1 if the *t*th pool would test positive under perfect testing, and *y*˜*t* = 0 otherwise. Explicitly, *y*˜*t* = 1 if *(*A**x***)t* ≥ 1, and *y*˜*t* = 0 if *(*A**x***)t* = 0.

Given **y** and A, we must come up with an estimate **x**ˆ for **x**. If we are only interested in estimating the most likely set of infected individuals (i.e., we do not want to err on the side of caution when we report to the individuals whether they are infected or not), then it makes sense to report a maximum a posteriori (MAP) estimate (though there are other reasonable alternatives, e.g. minimising the expected number of false positives plus false negatives). It can be shown using the standard techniques that the MAP choice for **x**ˆ is one where **y**˜*(A,* **x**ˆ*)* is chosen to minimize the 'penalty function'

$$f(\hat{\mathbf{x}}) = a \times \#\{i : \hat{\mathbf{x}}\_i = 1\} + b \times \#\{t : \mathbf{y}\_t = 1, \tilde{\mathbf{y}}\_t = 0\} + c \times \#\{t : \mathbf{y}\_t = 0, \tilde{\mathbf{y}}\_t = 1\},$$

for some constants *a, b, c* ≥ 0—these constants depend on the assumed prevalence, and the assumed sensitivity/specificity of the (pooled) tests. As a simple example, the P-BEST algorithm (see Sect. 11.5.2) minimises the penalty function with *a* = 0 and *b* = *c* = 1. We note that, if the estimated cost of declaring an infected person uninfected is much greater than the estimated cost of declaring an uninfected person infected, then the estimate with minimum expected cost may be different from the MAP estimate, and the penalty function to be minimized, should be changed, but Bayesian analysis can still be applied. We do not here get into the question of how to efficiently perform such an analysis.

#### **11.4 Practical Challenges for Pooled Testing**

The practical challenges and the downsides of implementing a pooled testing algorithm for COVID-19 testing—either as part of a national or local testing programme or within an autonomous institution or company—depend to some extent on the algorithm in question. A cost-benefit analysis is of course desirable, in each setting where pooled testing may be considered.

**Small benefit at high prevalence**. As mentioned earlier, when prevalence is greater than 38.2%, no pooled testing algorithm can outperform individual testing (Fischer et al. 1999) (under the assumption of perfect tests). Even when prevalence is significantly less than 38.2%, it will often be judged that the down-sides of pooled testing outweigh the advantage of resource-savings. For example, pooled testing has been approved in India only for use in areas where the population prevalence is 2% or less (see Sect. 11.5.4).

**Increased turnaround time**. As mentioned above, a single PCR test can typically be performed in four to six hours. If pooling can be automated, using e.g. a pipetting robot, then one-stage (non-adaptive) pooled testing algorithms will not have a significantly longer turnaround time than individual testing. For multistage pooled testing algorithms, such as Dorfman's algorithm, conservative two-stage pooled testing algorithms based on *(r,s)*-regular designs, or the multistage algorithm piloted in Rwanda (see Sect. 11.5.3), the increase in turnaround time (relative to individual testing) will depend partly on the laboratory set-up. The impact of increased turnaround times clearly depends upon the damage done by letting infections go undetected for longer; in the case of screening healthcare workers or social care workers, who work with individuals highly vulnerable to severe illness if exposed to SARS-CoV-2, this impact is likely to be much greater than in the case of screening university students or factory workers (for example).

**Laboratory infrastructure**. Dorfman's algorithm does not require any sophisticated equipment to implement: the pooling of the samples can be done by hand. The pooling of the samples can even be done, as at the University of Cambridge (see Sect. 11.5.1), by the individuals to be tested, thus imposing no extra workload on laboratory staff. It only requires the laboratory to keep track of which individuals correspond to which pooled samples, and a capability to perform individual follow-up tests on the individuals whose pools test positive (or alternatively, for part-samples from each individual to be kept back during the first step, in case follow-up testing is needed on that individual in the second step). Some laboratories, e.g., those suffering from a shortage of well-trained personnel, or of equipment, would struggle to implement even Dorfman's algorithm (the simplest to implement): even keeping track of which samples belong to which individuals, once these have been divided in half, may prove challenging under conditions of extreme pressure and consequent disorganisation.

The grid-algorithm is most efficiently implemented using a pipetting robot with an arm that can move in two dimensions; this piece of equipment, while commercially available, may be too expensive for organisations operating with a low budget, and for poorer countries. An alternative is to do the pooling manually, if sufficient manpower is available.

**The impact of dilution on test sensitivity**. When one infected sample is pooled with several others that are uninfected, the viral RNA is diluted, and this dilution leads to a decrease in the sensitivity of the PCR test on the pooled sample. The precise impact on sensitivity depends upon the laboratory protocol used, and also upon the distribution of viral loads in the samples being tested (which, in turn, depends upon the stage of the illness in the individuals being tested, as well as on individual biological factors). The following, however, gives a rough idea of the impact of dilution on sensitivity. Using a common protocol, and a set of 838 SARS-CoV-2 positive specimens, Bateman et al. (2021) found that dilution by a factor of five led (on average) to a 7% reduction in sensitivity, dilution by a factor of ten led (on average) to a 9% reduction in sensitivity, and dilution by a factor of 50 led (on average) to a 19% reduction in sensitivity. By contrast, a systematic review of the accuracy of individual PCR tests found false negative rates ranging from 2 to 29%, using repeated PCR-testing as the gold standard (for true positivity). Using repeated PCR testing as the gold standard for positivity is likely to underestimate the true rate of false negatives.

More importantly, for tests taken in the field, is the fact that swabs are sometimes taken incorrectly (see the previous chapter of this book, by Dunbar and Tang, for further discussion of this issue). One community-based study of close contacts of confirmed COVID-19 cases in China, found an overall sensitivity of 71% for upperthroat swabbing (by trained medical personnel) followed by an individual PCR test, using repeated PCR tests as the gold standard (for true positivity). When individuals self-swab, sensitivity is likely to be lower, unless the individuals themselves are appropriately trained (e.g., healthcare workers). Compared to these factors, overall, the impact of dilution on sensitivity, is relatively minor.

In the theoretical results we have described here, the number of samples per test *s* can get extremely large when the prevalence *p* is very small. In real applications, laboratories would be unwilling (due to dilution concerns) or unable (due to equipment capacity) to pool together very high numbers of samples. Thus for extremely low prevalences, the gains of pooled testing are unlikely to be as high as the theory suggests.

**How to deal with inconsistent test results**. When the sensitivity or the specificity of the tests being used is less than 100%, a pooled testing algorithm can yield inconsistent results. For example, when Dorfman's algorithm is implemented, a particular pool can test positive due to the presence of one infected individual (a true positive), but then in the second stage of follow-up (individual) testing, all the individuals in that pool can test negative, due to the infected individual testing negative (a false negative). In this case, the testing authorities are faced with a dilemma: they could assume (wrongly, in this case), that the first (pooled test) was a false positive and that the follow-up tests were true negatives, or they could simply choose to declare the individuals in question free from infection (to avoid having to recall them for further testing), but they could also decide to repeat the second round of individual tests on that group (pool) of individuals, to hedge against the possibility that there was a false negative in the second round of individual tests. This could be regarded as a third, 'confirmatory', testing-step in the algorithm. (If extra samples cannot be taken from the individuals in question, a third, confirmatory, testing-step may in fact be impossible.)

Which option the authorities choose, may depend on the impact of letting an infection go undetected; if the individual in question is a school pupil or a member of the general community (in a mass-testing programme), the impact will be less than if the individual is a healthcare worker working with patients highly vulnerable to severe illness or death in the case of COVID-19 infection, or if the individual is a resident-facing social care worker. The extra resource-requirements, and the delay, of a third (confirmatory) round of testing, should it be judged necessary, would have to be taken into account when deciding whether to adopt the pooled testing strategy.

**Prior prevalence estimates**. In situations where surveillance is poor, or infection levels are changing very rapidly, a laboratory may have very little idea of the prevalence of infection in an incoming batch of samples to be tested for SARS-CoV-2 infection. In such circumstances, a suboptimal choice of the parameters in a pooled testing algorithm algorithm (due to an underestimate of the prevalence) can lead to an inefficient second step—less efficient, in fact, than individual testing. For example, if the prevalence is 25%, then using Dorfman's algorithm with pools of size 10 requires on average approximately 1.04 tests per individual, which is slightly worse than individual testing (in addition to having a longer turnaround time). However, if an upper bound on the prevalence is known with a reasonably high degree of certainty (and this upper bound is not too high), then the parameters in a pooled testing algorithm can still be chosen so as to achieve a significant resource-saving over individual testing, even though the parameters cannot be fined-tuned to the exact prevalence. For example, the number of tests per individual for Dorfman's algorithm with fixed pool size *s*, is increasing in *p*, so the 'worst case' is when *p* is maximal. Hence, if the prevalence is known to be at most 1%, then Dorfman's algorithm with pools of size 10, will require at most 1 10 + 1 − *(*1 − *p)*10 ≤ 1 10 + 1 − *(*1 − 0*.*01*)*10 ≈ 0*.*196 tests per individual (on average). Thus we get at least a five-fold improvement on individual testing, as long as the prevalence does not rise above 1%.

**Regulatory approval**. All of the issues listed above may be obstacles to regulatory approval. An additional obstacle to regulatory approval is the complexity of pooled testing algorithms, compared to individual testing. Often, policymakers will need to give their approval, bearing in mind public opinion, and a procedure which cannot be understood by a large percentage of the public (or by policymakers without the requisite quantitative training), may be less likely to gain such approval. On the other hand, the simpler pooled testing algorithms, such as Dorfman's algorithm and the grid algorithm, almost certainly can be explained in such a way that policymakers (and the majority of the public) can understand them—and this may well be part of the reason why these two algorithms have seen the widest use, of all pooled testing algorithms, during the COVID-19 pandemic so far.

## **11.5 Uses of Pooled Testing in the COVID-19 Pandemic**

Hitherto in the COVID-19 pandemic, Dorfman's algorithm has been the most widelyused pooled testing strategy. This is almost certainly because it is (i) easy to implement without necessarily needing large changes in laboratory equipment or infrastructure, (ii) relatively robust to changes in prevalence, (iii) simple and transparent enough for non-scientific decision makers or the public to understand and for regulators to approve, and (iv) of easily predictable and controllable sensitivity. At the same time, it can yield large efficiency gains over individual testing, as outlined above. In this section, we give some examples of places where pooled testing has been applied during the COVID-19 pandemic, focussing on examples where detailed information is available.

#### *11.5.1 Dorfman's Algorithm at the University of Cambridge*

During Autumn Term of 2020 (6 October–4 December 2020), University of Cambridge students in college accommodation were asked to participate in the University's asymptomatic COVID-19 screening programme (University of Cambridge 2020).

Students were divided into 'bubbles' of average size 8 and maximum size 10, with each bubble consisting of students sharing facilities (e.g., a bathroom, kitchen or living room). In a typical week, half of the students in each 'bubble' were requested to provide nasal swabs, which were then collected together into a single container by one of the students. This container was then sent to a local laboratory, where a pooled PCR test was performed on the pooled samples. If a pooled sample tested positive, each student in the corresponding bubble was informed, and instructed to take an individual PCR test at one of the National Testing sites. A simple rota determined which students were asked to provide swabs on which weeks; on average, a student was asked to provide a swab approximately once per fortnight. Students who were symptomatic did not take part on weeks when they were symptomatic, as all students experiencing symptoms were instructed to seek an individual test. Students who had recently tested PCR-positive did not take part either. Participation was voluntary, but consent rates were high, starting at 75% of all 15,479 eligible students during the first week of term, and steadily increasing to 82% of all 15,310 eligible students in the last week.

Students were requested not to socialise outside their bubbles. Assuming a high level of compliance, this would mean that positive cases were more likely to cluster within bubbles, leading to a resource-saving at the second stage (of follow-up individual testing): this is the 'best case' in Dorfman's algorithm, in terms of resource-use at the second step, when the infected individuals are distributed among as few pools as possible. In the sixth week of term, for example, 80 students individually tested positive across 59 positive-testing pools (Warne 2020), giving an average of approximately 1*.*3 infected students per positive testing pool. Had the infected individuals been more evenly distributed, there would have been only one positive-testing student per positive-testing pool.

#### *11.5.2 The Grid and P-BEST Algorithms in Israel*

In August 2020, Israel's Ministry of Health approved two single-step pooled testing protocols for use in clinical laboratories in the country: one based upon the 'grid algorithm', and one based upon the 'P-BEST' algorithm of Shental et al. (2020). The grid algorithm has been described earlier.

P-BEST uses an *(r* = 6*,s* = 48*)*-regular design with a code-based construction. This deals with *N* = 384 individuals in *T* = *Nr/s* = 48 pooled tests. A 'best guess' for the identifies of the infected individuals (from the results of the pooled tests), is obtained using the method described in Sect. 11.3.3, above.

It should be noted that the P-BEST algorithm does require software and computing resources to implement, in addition to a pipetting robot with an arm that can move in two dimensions. However, this equipment is affordable by most countries.

In trials where at most 5 out of 384 individuals are PCR-positive under individual PCR testing (corresponding to a prevalence of 1.3% or lower), the P-BEST algorithm usually correctly identified all infected and uninfected individuals. Problems can arise when a batch of samples is received with a much higher prevalence than 1.3%; in this case, even if there are no false positive and or false negative test results, it is often not possible to use P-BEST to determine which individuals are infected and which are not.

In Israel, in clinical laboratories where P-BEST (or the grid algorithm) has been employed, data analysis and machine-learning has been employed also, to predict which batches of samples are likely to have much higher prevalence rates than the national average (based on origin); such batches were typically dealt with using individual testing.

#### *11.5.3 A Multi-stage (r, s)-Regular Algorithm in Rwanda*

Starting in August 2020, a multi-stage algorithm was piloted in Rwanda, where infection prevalence was low but the supply of PCR tests was limited (Mutesa et al. 2021).

The stages are as follows. It is required to pick two integer parameters, *a* and *r*2.

1. A Dorfman-like stage with *r*1 = 1 pooled test for each individual and *s*1 = *ar*2 samples in each test. A negative result shows that all*s*1 individuals are noninfected; individuals in a positive pool go through to stage two.

The parameters are chosen to be as efficient as possible while still ensuring that the chance of more than two stages being required is very small. One common choice is *a* = 3*,r*2 = 3, so that the first stage is a *(*1*,* 81*)*-regular Dorfman-like design, and the second stage is a *(*3*,* 9*)*-regular hypercube design of 9 tests for 81 individuals.

## *11.5.4 Other Uses of Pooled Testing*

Here is a brief (and far from exhaustive) list of some other examples of the use of pooled testing in the COVID-19 pandemic.

Initially 10,000 people tested per day, from April 2020 onwards (World Health Organization 2020).

• PCR testing using Dorfman's algorithm with pools of size 5, by the states of Uttar Pradesh (Sharda 2020) and West Bengal (Yengkhom 2020), India, in areas with estimated prevalence of 2% or lower.

# **11.6 Applications of Pooled Testing for COVID-19: Some Conclusions**

In this section, we conclude, by drawing some of the above analysis together and discussing our own personal perspective on the practical settings where pooled testing for COVID-19 is likely to be useful, or at least may merit serious consideration.

#### *11.6.1 Pooled Testing for Asymptomatic Subpopulations*

As stated in the introduction, we believe that pooled testing is most likely to be useful for the screening of asymptomatic people, for surveillance, and possibly for the testing of contacts of confirmed cases, provided the prevalence of infection among the group to be tested is sufficiently low. On the other hand, we believe that in most countries, pooled testing is unlikely to be useful for the testing of symptomatic people. (Here, we use the term 'asymptomatic' to denote someone who is not experiencing the recognised symptoms at the time of their test. This includes 'true asymptomatics', who never experience symptoms, and 'pre-symptomatics', who go on to develop symptoms after their test.)

There are two main reasons why we believe that, in most countries, pooled testing will be of limited use for the testing of symptomatic people. First, the prevalence of COVID-19 infection among those presenting symptoms is usually sufficiently high that the resource savings of pooled testing are modest compared to individual testing, and may be outweighed by the down-sides of pooled testing, such as increased turnaround time. Among those presenting COVID-like symptoms, prevalences of between 4 and 33% are realistic, depending upon the setting, the location, the symptoms used in the definition of 'symptomatic', and the prevalence of other respiratory viruses (which in turn depends on the time of year) (Menni 2020; Pueyo 2020). Second, many countries already have well-established testing programmes using individual testing for those presenting symptoms. In many countries, including the UK, an individual test on symptomatic people is mandated by the regulatory authorities to confirm infection, even if they can be proved positive solely via pooled tests.

On the other hand, in many countries, the prevalence of COVID-19 infection among the general population has for quite long periods been at levels low enough that pooled testing of asymptomatic people can yield large efficiency gains over individual testing. For example, the estimated prevalence of current SARS-CoV-2 infection in the general community in England, as estimated by the ONS Infection Survey (Office for National Statistics 2021), has ranged from 0.026% in early July, to 2.1% in early January (and 3.6% in London in early January). Hence, for the prevalence among asymptomatics in England, values of *p* between 0*.*02 and 1*.*6% are good estimates. Countries that have adopted more stringent non-pharmaceutical interventions (such as stricter lockdowns or strongly enforced quarantines for international arrivals) have experienced lower prevalence rates; for example, New Zealand probably eliminated COVID-19 infections in the general community between early May and mid-August 2020, except for international arrivals, who were quarantined (Baker et al. 2020).

It is often desirable to screen subpopulations where the prevalence of infection is likely to be significantly higher than in the general population: for example, patientfacing healthcare workers, resident-facing social care workers, and factory workers in high-risk environments such as meat-processing plants. But it may also be desirable to screen subpopulations where the prevalence of infection is likely to be similar to the general population: see some of the examples below.

We list here some of the settings where we believe pooled testing for COVID-19 is most likely to be useful. Of course, a careful cost-benefit analysis should be carried out for each potential application, with the decision to adopt or not depending on certain factors, including the prevalence level, laboratory resources and capabilities, the impact of increasing the turnaround time, and regulatory constraints.

particularly primary-school aged children—may not tolerate regular swabbing, although saliva tests would not have this problem.

One potential application of pooled testing that many authors are circumspect about is the screening of asymptomatic healthcare workers and social care workers. Given the major risks to vulnerable patients and social care residents associated with any additional delay in finding positive cases among these workers, screening using individual testing is often thought to be preferable, if there are sufficient resources. Even in the presence of very severe resource constraints, a careful cost-benefit analysis should be performed to compare the impact of screening based on individual testing with that based on pooled testing, taking into account the increased turnaround time. Even Mutesa et al. (2021), who are in general strong advocates of the use of pooled testing for COVID-19, state explicitly that they do not advocate its use for the screening of healthcare workers. We do note, however, that pooled testing was used by Saarland University Hospital for this purpose (see Sect. 11.5.4).

#### *11.6.2 Pooled Testing and Vaccination Programmes*

At the time of writing (March 2021), vaccination programmes are proceeding rapidly in many developed countries and are having a large effect in reducing the number of COVID-19 cases (Aran 2021). It might be thought that, in such countries, there will soon be no need to consider pooled testing. We believe that such an assumption may be premature at this stage, mainly because we do not yet have reliable data on the extent to which the vaccines currently being distributed reduce the number of cases (symptomatic and asymptomatic) of new variants of COVID-19, particularly the South African and Brazilian variants (Mahase 2021). (See Aldridge and Ellis (2021) for a discussion of the evidence base on this to date.) Bearing in mind this uncertainty, and the risk of further new variants arising that are resistant to available vaccines, we believe it would be wise for decision-makers to bear in mind the possibility that it may be desirable to rapidly increase testing-capacity using pooled testing in the medium term. In poorer countries, vaccination programmes are likely to be long delayed. In such countries, pooled testing may still be a valuable tool to consider for the foreseeable future.

## *11.6.3 Pooled Testing for Surveillance*

At low prevalence levels, pooled testing has the potential for very large resourcesaving in national COVID-19 surveillance programmes. In some countries with a very large testing capacity, this may not be necessary—for example, the UK's ONS ( Coronavirus) Infection Survey currently tests a random sample of approximately 400,000 members of the community population in England, once per fortnight, using individual testing (Office for National Statistics 2020); compared to the UK's Pillar II testing capacity of more than 200,000 tests per day, this is not too great a resource requirement. However, if it is desired to reduce the resource requirements of a nationwide surveillance programme, pooled testing provides a way of doing so. Using pools of size up to 100, the hypercube-based algorithm piloted in Rwanda by Mutesa et al. (2021) can estimate the prevalence fairly accurately, while achieving an approximately 100-fold reduction in number the tests used when the prevalence level is at 0*.*05%. The main concern here is the reduction in sensitivity caused by dilution, but Mutesa et al. (2021) report proof-of-concept experiments which suggest that, using an appropriate protocol, sensitivities of 98% or 92% (depending on the gene targeted), can be achieved at a 100-fold level of dilution. This suggests that their scheme can be used reliably to monitor prevalence. The extra turnaround time compared to individual testing is likely to be much less of an issue with surveillance than with case identification, particularly at low prevalence levels.

It is also plausible that pooled testing could be used to monitor the prevalence of new variants. This may become particularly important if new variants begin to seriously hinder the success of vaccination programmes. A new PCR-testing method (involving only a minor update to existing PCR tests) that can detect which variant of SARS-CoV-2 a patient is carrying is currently undergoing clinical trials by the biotechnology firm Novozymes (Merrifield 2021).

**Acknowledgements** The authors were supported in part by UKRI Research Grant EP/W000032/1.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# **Chapter 12 Outlier Detection for Pandemic-Related Data Using Compositional Functional Data Analysis**

**Christopher Rieser and Peter Filzmoser**

**Abstract** With accurate data, governments can make the most informed decisions to keep people safer through pandemics such as the COVID-19 coronavirus. In such events, data reliability is crucial and therefore outlier detection is an important and even unavoidable issue. Outliers are often considered as the most interesting observations, because the fact that they differ from the data majority may lead to relevant findings in the subject area. Outlier detection has also been addressed in the context of multivariate functional data, thus smooth functions of several characteristics, often derived from measurements at different time points (Hubert et al. in Stat Methods Appl 24(2):177–202, 2015b). Here the underlying data are regarded as compositions, with the compositional parts forming the multivariate information, and thus only relative information in terms of log-ratios between these parts is considered as relevant for the analysis. The multivariate functional data thus have to be derived as smooth functions by utilising this relative information. Subsequently, already established multivariate functional outlier detection procedures can be used, but for interpretation purposes, the functional data need to be presented in an appropriate space. The methodology is illustrated with publicly available data around the COVID-19 pandemic to find countries displaying outlying trends.

## **12.1 Introduction**

The crisis caused by COVID-19 in almost all areas of life has also revealed that an accurate data collection is a challenge that cannot be easily resolved due to political or logistic problems. However, the availability of clean and reliable data is a key step in fighting a pandemic. On the one hand, knowing the real number of tested, newly infected and dead people allows to investigate the causes of the observed

C. Rieser · P. Filzmoser (B)

Institute of Statistics and Mathematical Methods in Economics, TU Wien, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria

e-mail<EMAIL_ADDRESS>
C. Rieser e-mail<EMAIL_ADDRESS>
developments and to take appropriate measures to stop the spread of an infection. On the other hand, insurance companies offering a protection linked to some specific events during a pandemic would like to have reliable data to avoid the possibility of moral hazard.

Many countries report the number of cases, deaths, tests, and further parameters (variables) related to the COVID-19 pandemic regularly over time, and the data are accessible in public data repositories. Rather than treating the data with tools from time series analysis, it is common to consider them as functional data, so that the measurements are represented by smooth functions over time. One could then analyse the multivariate information contained in the functions for the different variables, and compare the countries with respect to this information. Thus, countries for which the multivariate information differs from the main trend given by the majority of the countries are possible outliers. Instead of directly considering the reported number (represented by the functions), one could also focus on analysing relative information. This can be done by taking (log-)ratios between the variables. Thus, the source of information for the analysis would not consist in the number of cases, death, tests, etc., for a particular day in a particular country, but in the (log-)ratios between these numbers. This is what is done in compositional data analysis, and outlier detection in this context will focus on atypical behaviour in the multivariate information of such (log-)ratios. For example, if the development of the number of cases over time is similar in some countries, but in one country the number of deaths develops more rapidly, this could be much better visible in a (log-ratio) than in the reported values. Thus, treating COVID-19 data as compositional data and analysing relative rather than absolute information can be very beneficial for outlier detection.

In this paper we consider a new method for the detection of outliers in the compositional functional data setting. The detection of outliers in the *p*-dimensional multivariate data case has been intensively investigated throughout the years and many methods have been developed. Denote by **x***k* ∈ R*p*, for *k* = 1, ..., *K*, the observed samples. A popular approach considers an outlier of these samples as a point **x***k*0 for which the robustified version of the Mahalanobis distance, - (**x***k*0−**m**) **C**−1(**x***k*0−**m**), where **m** respectively **C** are robust estimators for the mean and the covariance matrix, is above a certain threshold and thus far away from the centre **m** with respect to the covariance structure **C**; see Rousseeuw (1985), Rousseeuw and Driessen (1999) and Hubert and Debruyne (2010). The idea of defining an outlier as a point being far away from the centre has been extended to more general measures related to statistical depth, see Tukey (1975), Serfling (2006) and Mosler (2012).

In recent years, many methods of multivariate statistics have been generalised to Functional Data Analysis (FDA). In FDA one considers data points to be whole functions, i.e. in the notation above, data points **x***k* : *I* → R*p* are multivariate functions; for an overview of FDA we refer to Ramsay (2004), Ferraty and Vieu (2006) or Kokoszka and Reimherr (2017). Accordingly, the concept of outliers has been extended from the multivariate to the FDA setting, see Fraiman and Muniz (2001), Febrero et al. (2008), Sun and Genton (2011) and Hubert et al. (2015b).

In this paper we consider extending the ideas of outlyingness to functional data with image in the compositional data space. Thus, Sects. 12.1.1 and 12.1.2 provide a short introduction to the concepts of compositional data analysis and functional data, respectively. Further, in Sect. 12.2 we consider smoothing for functional data with image in the compositional space. In Sect. 12.3 we look at how one can detect outliers for the latter setting. That is, we extend the methods of detecting outliers from the non-compositional FDA case to the compositional one. Furthermore, Sect. 12.4 contains an application of the method presented. The data is comprised of COVID-19 data of different countries over time. Each country represents a functional data point. We finish in Sect. 12.5 with a summary and some conclusions.

#### *12.1.1 Compositional Data Analysis Concepts*

Assume we have given a *D*-dimensional random vector **x** for which each entry is strictly positive, i.e. **x** ∈ R*D* +, where R*D* + denotes the *D*-dimensional real number space with strictly positive entries. In the framework of compositional data analysis (CODA) it is assumed that the ratios *x j xk* , for any *j*, *k* ∈ {1, ..., *D*}, *j* = *k*, carry the relevant information, and thus only relative information is essential. As ratios do not change when multiplying **x** with a strictly positive scalar λ > 0, it holds that λ**x** =: **y** carries the same information as **x**. This motivates defining the equivalence relation

$$\mathbf{x} \sim \mathbf{y} \qquad \Longleftrightarrow \qquad \exists \lambda > 0 \quad \lambda \mathbf{x} = \mathbf{y} \text{ for any } \mathbf{x}, \mathbf{y} \in \mathbb{R}\_+^D$$

which partitions the space R*D* + into equivalence classes. Choosing for each equivalence class the representative **x** = (*x*1, ..., *xD*) satisfying *D j*=1 *x j* = 1, leads to the set of equivalence classes called the *D*-part simplex

$$\mathcal{S}^D := \left\{ \mathbf{x} = (\mathbf{x}\_1, \dots, \mathbf{x}\_D)' \in \mathbb{R}\_+^D, \ \sum\_{j=1}^D \mathbf{x}\_j = 1 \right\}.$$

The space *S D* is turned into a Hilbert space—called the Aitchison geometry on the simplex, see Aitchison (1982)—by defining addition (perturbation), multiplication with a scalar (powering), an inner product and a norm for **x** = (*x*1, ..., *xD*) , **y** = (*y*1, ..., *yD*) ∈ *S D* and α ∈ R:

$$\langle \mathbf{x}, \mathbf{y} \rangle\_A := \frac{1}{2D} \sum\_{j=1}^D \sum\_{k=1}^D \log \left( \frac{x\_j}{x\_k} \right) \log \left( \frac{y\_j}{y\_k} \right)$$

$$\begin{array}{ll} \bullet \text{ Norm:} & \qquad \|\mathbf{x}\|\_{A} := \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle\_{A}}. \end{array}$$

Furthermore, the Aitchison geometry is (bijectively) isometric to R*D*−1. To show this, firstly define the centred log-ratio (clr)

$$\text{clr}: \mathcal{S}^D \to \mathbb{R}^D, \quad \text{clr}\left(\mathbf{x}\right) := \left(\log\left(\frac{\mathbf{x}\_1}{\sqrt[0]{\prod\_{j=1}^D \mathbf{x}\_j}}\right), \dots, \log\left(\frac{\mathbf{x}\_D}{\sqrt[0]{\prod\_{j=1}^D \mathbf{x}\_j}}\right)\right)' \tag{12.1}$$

which satisfies the properties of being invariant under the above operations and the norm, i.e.

$$\text{clr}(\mathbf{x} \oplus \mathbf{y}) = \text{clr}(\mathbf{x}) + \text{clr}(\mathbf{y}) \tag{12.2}$$

$$\operatorname{clr}(\alpha \odot \mathbf{x}) = \alpha \operatorname{clr}(\mathbf{x}) \tag{12.3}$$

$$
\langle \mathbf{x}, \mathbf{y} \rangle\_A = \langle \text{clr}(\mathbf{x}), \text{clr}(\mathbf{y}) \rangle\_E,\tag{12.4}
$$

see Filzmoser et al. (2018). However, as for any **x** ∈ *S D*, the entries of clr(**x**) sum up to zero, *D i*=1 clr(**x**)*i* = 0, it follows that the clr mapping does not satisfy the property of being one-to-one onto R*D*. To obtain a bijective mapping, choose a *D* − 1 dimensional basis **V** = (**v**1, ..., **v***D*−1), where **v***j* ∈ R*D*, for *j* = 1,..., *D* − 1, are clr coefficients, and define the isometric log-ratio (ilr) mapping as

$$\text{ilr}\_{\mathbf{V}} : \mathcal{S}^{D} \to \mathbb{R}^{D-1}, \quad \text{ilr}\_{\mathbf{V}}(\mathbf{x}) := \mathbf{V}' \text{clr}(\mathbf{x}).\tag{12.5}$$

The latter is a one-to-one mapping fulfilling (12.2), (12.3) and (12.4), see Filzmoser et al. (2018). As there are infinitely many possibilities to choose a basis **V**, ilr coefficients are frequently considered to express all relative information of a composition appropriately in the usual Euclidean geometry, for which the common statistical tools have been designed. If an interpretation is desirable, the relative information is often re-expressed in terms of clr coefficients by clr(**x**) = **V** ilr**V**(**x**), because they relate to the original compositional parts in terms of relative information of the part to an "average" (geometric mean), see (12.1).

#### *12.1.2 Functional Data*

In FDA we consider observations to be multivariate smooth functions **f** : [*t*1, *tN* ] → R*D*. In practice, such observations often originate as time series, measured at certain time points *ti* , with *i* = 1,..., *N*, and thus they are not necessarily forming smooth functions. In this case, a preprocessing step is needed to find an estimate ˆ**f** for **f** given (*ti*, **y***i*), with **y***i* ∈ R*D*,*i* = 1,..., *N*, being noisy samples of **f**(*ti*). We assume in the following Gaussian centred uncorrelated noise with equal variance. Although many methods exist to recover smooth functions, it is common that ˆ**f** is estimated by smoothing spline methods. The literature on spline methods is vast and we refer to Reinsch (1967), Wood (2017) and Yee (2015) for a good overview. The main idea is that given multivariate data (*ti*, **y***i*) we find an estimate ˆ**f** which is, on the one hand, sufficiently smooth but, on the other, also a good approximation to the data. It is common to look at the following vector valued smoothing problem

$$\hat{\mathbf{f}} := \operatorname\*{arg\,min}\_{\mathbf{f}} \sum\_{i=1}^{N} \|\mathbf{y}\_i - \mathbf{f}(t\_i)\|\_E^2 + \lambda \int\_{t\_1}^{t\_N} \left\|\mathbf{f}''(t)\right\|\_E^2 dt,\tag{12.6}$$

where λ > 0 is a fixed smoothing parameter, and ·*E* denotes the Euclidean norm. The idea is that with increasing λ, the second derivative **f** is forced to zero, i.e. towards a linear function. From Problem (12.6) it can be deduced that the solution is of the form **f**(*t*) := *N i*=1 **a***ibi*(*t*), see Yee (2015), with *bi* being basis functions of the cubic spline space, and **a***i* being fixed vectors in R*D*. Plugging this basis expansion into (12.6) shows that the penalty function acts as regularisation penalty on **a***i* restraining the flexibility of the latter. In reality, one never uses the full basis expansion as given above, but rather a different and equally flexible expansion with less basis functions to save coefficients and avoid unnecessary computation in the case of a lot of data, for example a B-spline basis. Plugging in a specific basis expansion **f**(*t*) := *N i*=1 **a***ibi*(*t*) we can see that the problem is a convex problem, and solving this vector valued problem is discussed in Yee (2015).

#### **12.2 Smoothing for CODA Time Series**

In this section we consider functional observations with image in *S D*, i.e. functions **u** : [*t*1, *tN* ] → *S D*. As before, we assume that only a set of discrete samples (*ti*, **x***i*) is given, with *i* = 1,..., *N* and **x***i* ∈ *S D*, where **x***i* is a sample of **u**(*ti*). To construct a smooth estimate **u**ˆ of **u**, we firstly define derivatives and smoothing splines in a compositional context. For a function **u** : [*t*1, *tN* ] → *S D*, its derivative at a time point *t* is defined as

$$\mathbf{u}'(t) := \lim\_{h \to 0} \frac{1}{h} \odot \mathbf{u}(t+h) \ominus \mathbf{u}(t). \tag{12.7}$$

Accordingly, one can define higher order derivatives inductively, e.g. **u**(*t*) := (**u** ) (*t*). For a reference on compositional calculus we refer to Pawlowsky-Glahn and Buccianti (2011). In accordance with the previous section, define **u**ˆ as

$$\hat{\mathbf{u}} := \arg\min\_{\mathbf{u}} \sum\_{i=1}^{N} \left\| \mathbf{x}\_{i} \ominus \mathbf{u}(t\_{i}) \right\|\_{A}^{2} + \lambda \int\_{t\_{1}}^{t\_{n}} \left\| \mathbf{u}''(t) \right\|\_{A}^{2} dt,\tag{12.8}$$

where λ > 0 is again a fixed smoothing parameter controlling the smoothness.

Using the continuity of ilr**V** and (12.2), it follows that

$$\begin{aligned} \operatorname{ilr}\_{\mathbf{V}}(\mathbf{u}')(t) &= \operatorname{ilr}\_{\mathbf{V}}\left(\lim\_{h \to 0} \left\{\frac{1}{h} \odot \mathbf{u}(t+h) \ominus \mathbf{u}(t) \right\} \right) \\ &= \lim\_{h \to 0} \operatorname{ilr}\_{\mathbf{V}}\left(\frac{1}{h} \odot \mathbf{u}(t+h) \ominus \mathbf{u}(t) \right) \\ &= \lim\_{h \to 0} \frac{\operatorname{ilr}\mathbf{v}(\mathbf{u}(t+h)) - \operatorname{ilr}\mathbf{v}(\mathbf{u}(t))}{h} \\ &= \operatorname{ilr}\mathbf{v}(\mathbf{u})'(t) \end{aligned}$$

holds. With the same arguments, the equation ilr**V**(**u**)(*t*) = ilr**V**(**u**)(*t*) follows.

Therefore, defining **f** := ilr**V**(**u**), Problem (12.8) can be reformulated using the latter, as well as the properties (12.2) and (12.4):

$$\arg\min\_{\mathbf{u}} \sum\_{i=1}^{N} \left\lVert \mathbf{x}\_{i} \ominus \mathbf{u}(t\_{i}) \right\rVert\_{A}^{2} + \lambda \int\_{t\_{1}}^{t\_{n}} \left\lVert \mathbf{u}^{\prime\prime}(t) \right\rVert\_{A}^{2} dt \tag{12.9}$$

$$\begin{aligned} \xi & \Longleftrightarrow \arg\min\_{\mathbf{u}} \sum\_{i=1}^{n} \left\| \text{ilr}\_{\mathbf{V}}(\mathbf{x}\_{i}) - \text{ilr}\_{\mathbf{V}}((\mathbf{u}(t\_{i})) ) \right\|\_{A}^{2} + \lambda \int\_{t\_{l}}^{t\_{n}} \left\| \text{ilr}\_{\mathbf{V}}((\mathbf{u}^{\prime\prime}(t)) ) \right\|\_{A}^{2} dt \end{aligned} \tag{12.10}$$

$$\iff \arg\min\_{\mathbf{f}} \sum\_{i=1}^{N} \left\lVert \text{ilr}\_{\mathbf{V}}(\mathbf{x}\_{i}) - \mathbf{f}(t\_{i}) \right\rVert\_{E}^{2} + \lambda \int\_{t\_{\mathrm{l}}}^{t\_{\mathrm{a}}} \left\lVert \mathbf{f}''(t) \right\rVert\_{E}^{2} dt. \tag{12.11}$$

The latter is a vector valued smoothing problem in R*D*−1 for the data (*ti*, ilr**V**(**x***i*)), see Problem (12.6), and it can be solved accordingly.

Given a solution ˆ**f** to (12.11), a solution to (12.8) is then **u**ˆ = ilr−1 **V** (ˆ**f**) per definition of **f**. In the case that different solutions to (12.11) exist, e.g. ˆ**f**1 and ˆ**f**2, we know from the equivalence chain before and from the fact that ilr**V** is isometric, that also ilr−1 **V** (ˆ**f**1) and ilr−1 **V** (ˆ**f**2) are different solutions to Problem (12.8). Equally, having two different solution of (12.8) leads to different solutions of (12.11). This means that if (12.11) is uniquely solvable for a chosen **V**, we get that **u**ˆ is also uniquely determined. Therefore, the choice of **V** is irrelevant. With the exception of some very degenerate settings, Problem (12.11) is uniquely solvable in most applications.

#### **12.3 Outlier Detection in Compositional FDA**

In the univariate case we can think of outliers as observations being very far away from the main mass of the data set, thus far away from the data centre with respect to the scale (Maronna et al. 2006).

The outlyingness of a multivariate observation **x** ∼ *P***X**, where *P***X** denotes the distribution of a *p*-dimensional random vector **X** and **x** a realisation, can be built on the univariate case by means of projection onto a line defined by **r** ∈ R*p*, with **r** = 1, thus **r X**. As discussed in Donoho et al. (1992), the outlyingness of an observation **x** of the projection **r x** can be measured by

$$\frac{|\mathbf{r}'\mathbf{x} - \text{median}(\mathbf{r}'\mathbf{X})|}{\text{mad}(\mathbf{r}'\mathbf{X})},\tag{12.12}$$

where "mad" denotes the median absolute deviation, i.e. the median of |**X** − median(**X**)|. Taking the supremum of (12.12) over all **r** with **r** = 1 yields a measure of outlyingness for any **x** independent of the direction **r**. Adjusting (12.12) for skewness—see Hubert and Vandervieren (2008) for adjusted boxplots of skewed distributions in the univariate case—the adjusted outlyingness (AO) is defined as

$$AO(\mathbf{x}, P\_{\mathbf{X}}) := \begin{cases} \sup\_{\|\mathbf{r}\|=1} \left( \frac{\mathbf{r}^{\mathbf{'}\mathbf{x}} - \operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}})}{w\_{2}(\mathbf{r}^{\mathbf{'}\mathbf{X}}) - \operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}})} \right) & \text{if } \mathbf{r}^{\mathbf{'}\mathbf{x}} > \operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}})\\\sup\_{\|\mathbf{r}\|=1} \left( \frac{\operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}}) - \mathbf{r}^{\mathbf{'}\mathbf{x}}}{\operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}}) - w\_{1}(\mathbf{r}^{\mathbf{'}\mathbf{X}})} \right) & \text{if } \mathbf{r}^{\mathbf{'}\mathbf{x}} \le \operatorname{median}(\mathbf{r}^{\mathbf{'}\mathbf{X}}), \end{cases}$$

where w1 and w2 are functions that allow to adjust for the skewness of the univariate distributions, see Hubert et al. (2015b) for an exact definition of these two functions.

To obtain a measure of outlyingness in the FDA case, e.g. for the data (**f** : [*t*1, *tN* ] → R*p*) ∼ *P***F**, Hubert et al. (2015b) propose to use the functional adjusted outlyingness of a FDA point **f**:

$$FAO\left(\mathbf{f}, P\_{\mathbf{F}}\right) := \int\_{t\_1}^{t\_N} AO\left(f(t), P\_{F(t)}\right)dt,$$

where *Pf* (*t*) denotes the marginal distribution of **F** for fixed *t*.

In a compositional functional data context, where the compositions are functions of the form **u** : [*t*1, *tN* ] → *S D*, with distribution *P***U**, we propose to define the compositional functional adjusted outlyingness as

$$CFAO\left(\mathbf{u}, P\_{\mathbf{U}}\right) := \int\_{t\_{\mathrm{l}}}^{t\_{\mathrm{N}}} AO\left(\mathrm{ilr}\_{\mathrm{V}}(u(t)), P\_{\mathrm{ilr}\chi\left(\mathbf{U}(t)\right)}\right) dt.\tag{12.13}$$

For Definition (12.13) to be a valid measure of outlyingness it needs to be checked that it is well defined, i.e., this measure needs to be independent of the choice of the basis matrix **V**. As **V** ilr**V**(**x**) = clr(**x**) holds by definition for a matrix with orthonormal columns **V**, we have for a different matrix **V** with orthonormal columns ilr**V**(**x**) = **V** clr(**x**) = **VV** ilr**V**(**x**), see Filzmoser et al. (2018). As the matrix **VV** ∈ R(*D*−1)×(*D*−1) is of full rank *D* − 1, we get, for any fixed *t*

$$AO\left(\text{ilr}\_{\mathbf{V}}(\boldsymbol{\mu}(t)), P\_{\text{ilr}\mathbf{V}(U(t))}\right) = AO\left(\mathbf{V}^{\dagger}\mathbf{\tilde{V}}\,\text{ilr}\_{\mathbf{V}}(\boldsymbol{\mu}(t)), P\_{\mathbf{V}}\mathbf{\tilde{V}}\,\text{ilr}\_{\mathbf{V}}(U(t))\right) \tag{12.14}$$

$$=AO((
\mathbf{V}
\text{'V})(
\text{ilr}
\text{\"y}(
\mu(t))), P\_{(
\text{V}
\text{'V})(
\text{ilr}
\text{\"}
\text{\"}
(U(t)))})
\qquad (12.15)$$

$$=AO\left(\text{ilr}\overline{\chi}\left(\mu(t)\right),P\_{\text{ilr}\overline{\chi}\left(U(t)\right)}\right) \tag{12.16}$$

where the last equality follows from the affine invariance property of AO, see Hubert and Van der Veeken (2008); affine invariance means that *AO*(**x**, *P***X**) = *AO*(**Ax** + **b**, *P***AX**+**b**) holds for any regular matrix **A** ∈ R*p*×*p* and **b** ∈ R*p* for **x** ∈ R*p* with **x** ∼ *P***X**. As CFAO is defined as an integral over (12.16) it follows that the latter is equally invariant and thus well defined.

To visually find outliers in the FDA setting, Hubert et al. (2015a) introduced a functional outlier map (FOM). Assume that the evaluation of *K* multivariate functional data points **f**1,...,**f***K* is given at time points *t*1,..., *tn*, and denote *PK* the sample distribution of the functional data points, and *Pti* the sample distribution of the evaluations at time point *ti* . The FOM is defined as a two dimensional graph, plotting *FAO*(**f***k* , *PK* ) on the horizontal axis against

$$\frac{\sigma\_{i=1,\ldots,N}((AO(\mathbf{f}\_k(t\_i), P\_{t\_l}))\_i)}{(1 + FAO(\mathbf{f}\_k, P\_K))}\tag{12.17}$$

on the vertical axis, for *k* = 1,..., *K*, where σ denotes the standard deviation. The motivation behind this map is that when a data point **f***k* is a shift outlier, its according point in the FOM plot will be higher on the horizontal axis. If a data point **f***k* displays an outlying high variability in time, this will result in a high value on the vertical axis in the FOM plot. The denominator in (12.17) is necessary to correct for the effect that when a data point is shifted further, this is reflected in the standard deviation accordingly, see Hubert et al. (2015a).

Given the evaluation of the compositional functional data **u**1,..., **u***K* , *k* = 1,..., *K*, at time points *t*1,..., *tN* , we suggest equivalently to plot *CFAO*(**u***k* , *PK* ) on the horizontal axis, against

$$\frac{\sigma\_{i=1,\ldots,N}\left(\left(AO\left(\text{ilr}\_{\mathbf{V}}(\mathbf{u}\_{k}(t\_{i})),P\_{t\_{i}}\right)\right)\_{i}\right)}{\left(1+CFAO\left(\mathbf{u}\_{k},P\_{K}\right)\right)}\tag{12.18}$$

on the vertical axis. Again, the latter is independent of the choice of **V**, because *AO* as well as *CFAO* are affine invariant, see the reasoning for (12.16) and its conclusion.

#### **12.4 Application to COVID-19 Data**

In this section we use data from https://covid.ourworldindata.org, which are publicly available. This page contains for most countries of the world daily information related to the COVID-19 pandemic. Here we focus on European countries only, and on the following information:

**Fig. 12.1** COVID-19 data from Austria in the period April 1 until December 31, 2020. The plots show daily data for the 5 variables used for the analysis

We select the time period from April 1 until December 31, 2020, because from April onwards the information was consistently collected in the data base. However, for some of the European countries the information on some of the variables was not available, so that finally only 35 European countries could be used. Still, for some countries there were missing values (or shorter time periods with missings), which have been imputed by a weighted moving average imputation method, implemented as function na\_ma() in the R package imputeTS (Moritz and Bartz-Beielstein 2017).

As an example, Fig. 12.1 shows the data for Austria, and the data structure is similar in many of the other countries. Still, there might be countries with deviations in the multivariate data structure, and the task is to identify such countries. The focus here is on relative information in terms of log-ratios between the different variables.

Figure 12.1 reveals that the total number of cases starts to grow quickly in October 2020, and the same is true for the total number of deaths (per million). The number of tests grows steadily over the time period. The positive rate decreases at the beginning of this selected time period, but it increases drastically in October, followed by a decline in November/December. The reproduction rate fluctuates more, and has higher values than one in the summer and fall.

Multivariate functional outlier detection is here first applied to the data expressed in relative information, i.e. as ilr coordinates. In a second stage we also compare with an analysis based on absolute information, as reported in Fig. 12.1 for Austria. Naturally, the different treatment of the data will very likely lead to different results. As an example for relative versus absolute information, we may consider just the number of cases and the number of deaths (per million). For most countries, an increase of cases also implies an increase of deaths, probably with a different time delay. If one looks at relative information in terms of a log-ratio, however, differences between the countries might get more clearly pronounced. We will come back to this issue later.

**Fig. 12.2** Ilr coordinates of the data from Austria, together with the lines after smoothing. The smoothed lines (for every country) are the input for compositional functional outlier detection

For every country, the data are first ilr-transformed, resulting in time series of the ilr coordinates. Since the specific choice of the ilr coordinates is not relevant here, we use so-called pivot coordinates, where the first coordinate expresses all relative information of the first part to the remaining parts in the composition, see Filzmoser et al. (2018). Figure 12.2 shows the resulting ilr coordinates for the Austrian data; since there are 5 variables available, see Fig. 12.1, we end up with 4 ilr coordinates. Figure 12.2 also shows the lines after smoothing the data in ilr coordinates, thus after solving Problem (12.11). The information of these lines form the compositional functional data as they are used for multivariate outlier detection. Since we used pivot coordinates, only the first coordinate (denoted here by *z*1) has a clear interpretation in terms of all relative information of the total cases to the remaining variables. This coordinate is in fact proportional to the first clr coefficient (Filzmoser et al. 2018). We will show and discuss the corresponding clr coefficients later in Fig. 12.5.

Once the smooth functions are estimated for every country, compositional functional outlier detection can be performed. Figure 12.3 shows the compositional functional outlier map (CFOM). Every point in the plot corresponds to a country, and the line indicates the outlier cutoff. It can be seen that one (red) point (Iceland) slightly exceeds the cutoff, and another point (Belarus) is just below the cutoff. The sorted compositional functional adjusted outlyingness is again shown in Fig. 12.4 (left), with the corresponding country names added. The values for Iceland and Belarus clearly stick out, and the next biggest value originates from the data from Luxembourg. These countries are not particularly outlying in their variability in time, since their values in Fig. 12.4 (right) are not unusual.

Figure 12.5 is an attempt to identify the reason for outlyingness. The plots show the smoothed functional data in clr coefficients, which are simply obtained by a transformation from the functions in ilr coordinates, see Eq. (12.5). The function for Iceland is shown in red, and that for Belarus in blue. For example, the clr coefficients for the total cases (left plot) mainly show a strongly increasing trend at the beginning, and again at the end of the considered time period. This means that the cases have grown rapidly, relative to the remaining variables (on average). The function for Belarus (blue) shows a quite different behaviour, with very high values especially around May. This means that the total cases are very much dominating over the values of the other variables. The reason for this is not because of high values of

**Fig. 12.3** Compositional functional outlier map: the points represent the countries, and the line is the outlier cutoff. Iceland exceeds the cutoff value, Belarus is just below the cutoff, see also Fig. 12.4

**Fig. 12.4** Sorted compositional functional adjusted outlyingness (left), and sorted values from the vertical axis in Fig. 12.3 (right)

cases, but because of exceptionally low (reported) values of the remaining variables. Also the values for Iceland (red curves) are seen as atypical. For example, the clr coefficients of the total cases started to be the lowest in April, but then increased to be the highest in August. In a ratio, it can either be the change in the numerator or in the denominator, or in both, to get this behaviour, but in any case it turns out to be quite different compared to the other countries.

As a comparison, the following analysis is based on absolute information. Thus, the smoothed curves are directly estimated from the raw input data without any trans-

**Fig. 12.5** Functional data represented in clr coefficients. Every function represents the time series of one country; Iceland is shown in red, Belarus in blue

**Fig. 12.6** Functional outlier map (FOM) as a result of using the untransformed absolute data information

formation, see Eq. (12.6). Then multivariate functional outlier detection is applied, which results in the functional outlier map presented in Fig. 12.6. Here, one point clearly exceeds the outlier cutoff value, and this point is Luxembourg.

Details are presented in Fig. 12.7, where the left plot are the sorted values from the horizontal axis, and the right plot the sorted values of the vertical axis from the FOM of Fig. 12.6. Indeed, Luxembourg appears with an exceptionally high value of FAO, and neither Iceland nor Belarus are atypical in any of these plots.

Finally, Fig. 12.8 shows the raw functional data. The outlier Luxembourg is shown by green curves, Iceland in red, and Belarus in blue. Luxembourg shows a very clear difference in the total tests, which might be the reason for the multivariate outlyingness. The countries Iceland and Belarus, which were clearly different in the

**Fig. 12.7** Horizontal axis (left) and vertical axis (right) from Fig. 12.6 for functional outlier detection based on the untransformed absolute information

**Fig. 12.8** Smoothed curves for the untransformed (absolute) data, with Luxembourg in green, Iceland in red, and Belarus in blue

compositional analysis, follow the main data structure well and do no longer appear as atypical. This shows that both types of analysis indeed focus on different data aspects, and it will be based on the task and research question to determine which of the analysis is more appropriate.

#### **12.5 Summary and Conclusions**

Outlier detection has been a relevant task in data analysis already since the beginning of data collection, and it continues being important also for more complex data structures. The identified outliers may point at atypical events, and depending on the context even at possible cases of fraud; see, e.g., van Capelleveen et al. (2016) or Nian et al. (2016). Outlier detection methods are also useful for pandemic-related data, as they may guide policy makers to draw appropriate conclusions.

Here we have used publicly available time series data related to COVID-19, as they are reported from different countries. The multivariate information, here in terms of the number of cases, deaths, tests, the positive rate, and the reproduction rate, has been treated as compositional data, where relative rather than absolute values are processed in the analysis. Absolute values would refer to the data as they are reported, while relative information refers to the log-ratios between the values of the different variables. An outlier detection method which makes use of relative information thus will focus more on the differences of the developments over time between the variables, and not necessarily on extreme values in single variables. In fact, if there is a peak in one variable in a certain time period, and the peak also appears in another variable in the same period, the log-ratio would not show up as unusual. A temporal shift of the peaks, however, creates big log-ratios, and if the position or magnitude is different for one country compared to the others, this country will appear as a potential outlier.

The time trends of the COVID-19 data have been treated here as functional data. Functional data which are processed with tools from compositional data analysis commonly have a constant sum constraint, such as probability density functions or particle-size curves, see van den Boogaart et al. (2014) or Menafoglio et al. (2014). Here we considered the single variables of the multivariate data information as parts of a composition, and since the information is derived continuously over a domain (here time), such data are regarded as multivariate compositional functional data. As functional data are supposed to be smooth functions, the concepts from compositional data analysis already need to be taken into account when generating the compositional functional data. Thus, the original data information, which usually needs to be smoothed in order to represent functions, has to be presented in the appropriate geometry. Since we deal with multivariate information, smoothing also needs to be done in a multivariate context. Here we have used isometric log-ratio coordinates to move the data from the simplex to the standard Euclidean geometry, and we have shown that the specific choice of these coordinates is not relevant for obtaining the smooth functions.

Once the multivariate compositional functional data are available and expressed in the appropriate geometry, standard tools for multivariate functional outlier detection can be used. The application of the methodology to the COVID-19 data revealed that the outlyingness values for the two countries Iceland and Belarus were clearly higher compared to the other investigated countries. Diagnostics in clr coefficients, again referring to relative information, has shown that some of the functions for these countries indeed deviated clearly, at least in certain time periods. Because clr coefficients refer to log-ratios of a specific variable to the geometric mean, deviations can be caused either by atypical values of this variable, or by atypical values of the geometric mean, representing an "average behaviour" of all analysed variables. The analyst would then have to compare this information to that from the other countries, or even go back to the original data source for such a comparison. There could be many reasons for outlyingness: data reporting is done differently (probably only for some of the variables), the policy of the restrictions in the context of the pandemic is very different, the behaviour of the people to deal with the pandemic is very different, etc.

We have also compared such an analysis with multivariate functional outlier detection using the absolute information, where outliers are, for example, countries with extreme values of a function in a certain time period. This analysis led to different outliers, and it finally will depend on the underlying task and research question which type of analysis is most appropriate.

There are many further methodological challenges, which are revealed when considering real data applications as, for instance, the full COVID-19 data set provided from the source mentioned in the paper: zero values, missings, poor data quality, some countries do not provide information for some of the characteristics, etc. These issues are relevant already for estimating the multivariate smooth functions, and subsequently also for the purpose of outlier detection. Our future research will be devoted to such tasks.

**Acknowledgements** This research was supported by the Austrian Science Fund (FWF) under the grant number P 32819 Einzelprojekte.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 13 The Legal Challenges of Insuring Against a Pandemic**

**Rachel Hillier**

**Abstract** COVID-19 has raised, and continues to raise, questions about the traditional approach to insurance cover. For instance, business interruption insurance covering "pandemics" under all risks insurance policies are likely to be a thing of the past. With tensions between businesses and the insurance industry on the rise, what can be done to offer businesses some protection at a premium they can afford, without emptying insurers' reserves? In this chapter we talk about legal challenges related to traditional insurance against the risk of losses caused by a pandemic, and whether parametric insurance is the solution.

#### **13.1 Introduction**

The insurance industry has always been adept at developing insurance products to meet the changing needs of society, from the development of death benefit and personal injury insurance during the Industrial Revolution to motor vehicle insurance in the late 1890s,1 to aircraft insurance in the 1920s, to drone insurance in the 2010s. The ability to adapt insurance to meet new perils arising from changing human activity is a common thread in the history of insurance.

The COVID-19 pandemic as a new peril is no different. Living with the risk of loss associated with COVID-19 has become a reality and the insurance industry is working to provide financial protection to those affected. However, some of our most established and oldest types of insurance policies have been found wanting during the pandemic, particularly in relation to small businesses hit hard by financial losses during repeated lockdowns and trading restrictions imposed across the world in an attempt to curb the spread of the virus. This chapter looks at the weaknesses and strengths of the legal structure of traditional insurance policies, and how parametric

R. Hillier (B)

© The Author(s) 2022 M. C. Boado-Penas et al. (eds.), *Pandemics: Insurance and Social Protection*, Springer Actuarial, https://doi.org/10.1007/978-3-030-78334-1\_13

1The first motor policies in the UK were written around the time of the original London-to-Brighton car run, which took place on 14th November 1896.

Partner at Capital Law, Cardiff & London, UK e-mail<EMAIL_ADDRESS>
insurance, with the aid of new technologies could assist in providing new insurance to meet the needs of a COVID-19 world.

As a lawyer trained and qualified in England and Wales, I have concentrated in this chapter on challenges to the legal system in England and Wales. However, many other jurisdictions have the same or a similar insurance contracts and therefore similar challenges.

## **13.2 Summary of the Traditional Approach to Insurance**

#### *13.2.1 The Origins of Insurance*

Fire insurance was the start of commercial business insurance in England and can be traced directly to the Great Fire of London in 1666. A company called the "Fire Office" was set up by Nicholas Barbon (a building entrepreneur who made his money rebuilding London after The Great Fire). From 1680, the Fire Office also had a "company of men" to put out the fires. By the 17th Century, marine and life insurance policies were commonly purchased. By this time, the English courts were regularly hearing cases involving insurance disputes. The legal construct of an insurance contract has in many cases not changed significantly since the 17th Century and is predominantly based on the indemnity principle.

#### *13.2.2 The Insurance Indemnity Principle*

A definition for a "contract of insurance" is conspicuously missing from the central piece of legislation that regulates the insurance industry in the United Kingdom, the Financial Services and Markets Act 2000 (commonly known as FSMA). This allows the industry to adapt and develop new models of insurance to meet new types of risk. In this section, we will explore why the indemnity principle has become so entrenched in insurance law and why, in many cases, it is not fit for purpose in a pandemic.

Modern insurance has been defined in broad terms by the English courts as "*an agreement to confer upon the insured a contractual right which, prima facie, comes into existence immediately when loss is suffered by the happening of an event insured against, to be put by the insurer into the same position in which the insured would have been had the event not occurred, but in no better position*".2 This definition includes the principle of "indemnity"—that payment under an insurance contract puts the policyholder back in the position they would have been had the event not occurred, but in no better position.

2 Callaghan v. Dominian Insurance Co. [1997]2 Lloyds Rep. 541 per Sir Peter Webster.

The indemnity principle has served the insurance industry well for many years. It is a good tool to ensure that insurers pay only the value of what is lost, so that an insurance policy is not gambling, or merely a security entered into for the purpose of profit.

The fundamental purpose of insurance is to mitigate risk. Risk can be mitigated in many ways. For example, doing business through a limited company can limit the risk of its shareholders to the nominal value of shares owned. Another example is the presence of fire proof doors and smoke alarms in a building to reduce the risk of damage by fire. Mitigation of risk through insurance is achieved through the transfer of risk to another (to insurers and reinsurers). The insurance company ensures it has sufficient reserves from premiums collected from many insureds to protect the few who suffer a loss, so that those few have their losses reinstated, at the expense of all the insureds who paid the insurer their premiums. Thus, risk is transferred to insurers. If the insurer has not collected sufficient premiums to cover the insured losses, it is the insurer that must pay the difference. The insurer must therefore calculate not only the likelihood of a claim occurring, but the likely value of each claim. The indemnity principle provides a good method for insurers to learn from its previous claims as well as to collect information about insured property to build sophisticated models to work out how much each insured is likely to lose on average on an indemnity basis, whilst ensuring at the same time that it does not pay out more than it needs to.

Whilst often convenient to insurers, in the UK, there is no statutory requirement for the indemnity principle to be the basis on which claims are valued, as long as a policyholder has an "insurable interest" in what is being protected. That is, an insured must "*benefit by the safety or due arrival of insurable property, or may be prejudiced by its loss, or by damage thereto, or by the detention thereof, or may incur liability in respect thereof* ".3

Non-indemnity insurance (including parametric insurance) does exist (and has existed for some years), but such policies are small inlets compared to an ocean of indemnity policies. Section 27(3) of the Marine Insurance Act 1906 provides that an insurance policy may be "valued or unvalued" and that where it is valued, "*Subject to the provisions of this Act [...], and in the absence of fraud, the value fixed by the policy is, as between the insurer and assured, conclusive of the insurable value of the subject intended to be insured, whether the loss be total or partial.*" An example of such insurance is income protection insurance. The amount the insured will be able to claim each month if they are unable to work is fixed in advance, usually with reference to the insured's income at the time the insurance is taken out. There may be some indexed rise in that fixed amount built into the policy, but the agreed payment will be made whether the income actually lost is less or more than the pre-agreed fixed amount.

In the face of new threats to the modern world, such as cyber-crime, climate change and COVID-19, alongside the ability of modern computing power, machine learning and artificial intelligence to provide more accurate model future risks, the

3 Marine Insurance Act 1906, Section 5.

indemnity principle, if not obsolete, may start to lose favour as the primary measure of claims payments in the insurance industry.

#### **13.3 The Effect of COVID-19 on the Insurance Industry**

On 10th January 2020, the World Health Organization reported that Chinese authorities had determined that an outbreak of a pneumonia type disease, first reported in December in the Wuhan province, was caused by a novel coronavirus, see WHO (2020). The virus spread rapidly into and across Europe. On 23rd March 2020, the UK prime minister, Boris Johnson, announced that people in the United Kingdom must "stay at home" and that non-essential businesses must close.

The closure of manufacturing plants, bars, restaurants, retail establishments, entertainment venues and other places of business resulted in significant business interruption losses and an influx of claims under commercial business insurance policies. But it was not all bad news for insurers. People used their cars less and accidents on the roads reduced significantly, reducing claims under vehicle insurance. Meanwhile, deaths due to COVID-19 spiralled (in June 2020, more than 1,000 deaths a day were recorded due to Coronavirus for 22 consecutive days). The majority of deaths were older people, and many were above the age range for life insurance cover. Yet, the population as a whole was faced with its own mortality, and uptake of life insurance for those who were within normal age ranges for life insurance soared.

# *13.3.1 The Effect of COVID-19 on Business Interruption Insurance Policyholders*

When Coronavirus hit, those businesses that had purchased specialist pandemic insurance, for example, the All England Lawn Tennis Club (host of the world famous Wimbledon Tennis Championship) who had learned from the SARS epidemic and taken out specialist pandemic insurance in 2013, did receive payments under insurance policies. However, as they were indemnity based policies, calculating loss was a long and complicated process. Richard Lewis, Chief Executive of the All England Lawn Tennis Club said "It's a wide-ranging policy but part of the reason it takes so long to work through is that everything is looked at, so there's no blanket payout.", Gangcuangco (2020). A Daily Mail report quoted Lewis as explaining "It's looked at line by line quite literally, every cost, expenditure, bit of income, revenue, whether it has to be repaid, all that sort of thing".

Many small and medium sized businesses (SMEs) thought they were covered under their all risks commercial business insurance policies. However, most insurers believed that these policies were not designed to cover a pandemic and there was no cover under them. This dichotomy of understanding between insured and insurer and the resulting court cases that ensued across the world have often been in the press during 2020/21. The insurance industry has not covered itself in glory by refusing to pay claims under these policies and taking claims through to the bitter end of the court process in many countries.

In the UK, SMEs complained to the Financial Conduct Authority (FCA) that insurers were refusing the vast majority of claims under all risk business insurance policies. This led to the FCA seeking permission from the UK courts to submit test cases relating to common policy wordings, following widespread concern over the lack of clarity and certainty for businesses. The FCA hoped to obtain some certainty for hard pressed SMEs. The Supreme Court heard the case. It considered a representative sample of 21 types of policy wordings from eight different insurers as test cases. The FCA applied to the court for a decision as to whether the business interruption sections of each policy covered losses due to business interruption flowing from COVID-19 closures. The Court's decisions were broadly in favour of business owners. However, despite that decision, insurers continue to decline claims as policy wordings are often slightly different to those wordings considered by the court. Similar test cases have been considered by courts in other countries across the globe—France,4 Germany,5 Ireland,6 Australia.7

Even where insurers have accepted a policy covers economic loss due to the disease, many policy terms have exclusions and conditions that mean claims were legitimately rejected. For example, many policies only cover losses flowing from closure due to the presence of disease on the premises. During the first wave of the pandemic "test and trace" in the UK was not in place, and in the second wave was found to be wanting. Insureds therefore often have difficulty proving that COVID-19 was actually present on the premises and claims were rejected. Even where liability has been accepted, in many cases payments have not been made. Assessment of loss under the indemnity principle is leading to long delays in payment, because not only does the policy wording have to cover the loss event, and causation have to be proved, but loss must also be proved. Seven weeks after the Supreme Court judgement, the UK regulator reported that Hiscox (whose policy wording was considered by the court) had paid just 151 of its customers affected and was still deciding more than 4,500.

Insurers are vilified in the press, but as Hiscox said in response to criticism, "These are complicated claims that require comprehensive financial information and discussions with customers in order to settle them fairly.", see BBC News (2021). Quantifying economic loss on an indemnity basis is a time-consuming, administra-

4 On 22 May 2020, for example, Paris's commercial court ordered Axa SA unit Axa France IARD S.A. to pay e45,000 to restaurant group Maison Rostang SAS over the closure of one of its restaurants.

5 On 1 October 2020 the Munich regional court ordered insurer Versicherungskammer Bayern to pay e1.01 million to the operator of the Augustinerkeller beer garden.

6 In October 2020, the Irish High Court heard business interruption cases between insurer FBD Holdings PLC and four pubs.

7 On 18 November 2020, the NSW Court of Appeal handed down its much-anticipated decision in HDI Global Specialty SE v Wonkana No. 3 Pty Ltd [2020] NSWCA 296 ("HDI Global").

tively burdensome and costly task, particularly when a once in a life time constantly evolving world-wide pandemic, waxes and wanes, and the same businesses go in and out of lockdown in response pandemic waves (more on the difficulties of quantifying economic loss below). Expecting SMEs to provide detailed evidence of loss in an ever-changing environment of Government led opening and forced closure, whilst they tackle additional and evolving requirements for COVID safe environments and ongoing staff absence due to COVID-19 illness or being required to self-isolate, is creating even further stress on business owners. Many businesses are becoming insolvent and failing before they can prove their losses, even when they have a valid claim.

Ongoing delays and disputes concerning claims for business interruption losses resulting from COVID-19 are happening across the world. For example, in the United States, by August 2020 over 1,000 COVID-19-related insurance coverage lawsuits had reportedly been filed with early outcomes suggesting different judicial interpretations of key issues and limited potential for any consolidation of proceedings, see Covington (2020).

Meanwhile, ongoing loss to SMEs and increased premiums at renewal (as insurers adjust premiums to recoup claims paid and ensure adequate reserves) means many businesses are now finding themselves unable to afford increased premiums without agreeing to large deductibles. Those that can afford cover are often unable to procure insurance against future COVID-19 related losses due to policy terms offered.

#### *13.3.2 The Effect of COVID-19 on Insurers*

For insurers (and reinsurers), COVID-related business interruption claims have meant many potential losses and uncertainty as to whether policies are valid, affecting levels of reserves required. As of March 2021, publicly available data puts an estimate of total losses due to the pandemic worldwide at \$34.523 billion, reported in Reinsurance News (2021). However, the use of the indemnity principle as a measure to pay claims means that this figure is an estimate, and the true figures will take years to be known as evidence of economic loss claims is painstakingly gathered and claims are slowly paid. Greater uncertainty as to both liability and quantum means greater uncertainty in traditional claims modelling, resulting in insurers having to hold more cash to top up the reduction in the value of assets held, to meet reserves and regulatory capital requirements.

In my view, the legal costs, and the costs of payments of claims is nothing compared to the reputational damage the insurance industry has suffered as a result of high-profile court cases across the world. Many SMEs have paid substantial premiums for many years, never making a claim, and now feel let down by their insurers in their time of need. This has led to a worldwide loss of faith in the insurance industry by the small business community. Insurance is built on trust, and that trust has been seriously diminished.

We are now starting to see that many insurers are adding wide ranging exclusions to their all risks business insurance so that SMEs have no cover against further waves of COVID-19. It remains to be seen what the appetite of businesses for renewal of these policies will be, particularly where premiums are increased from the previous year as insurers seek to recover loss.

# **13.4 Life Insurance Versus Business Interruption Insurance**

2020 brought an interesting comparison between the fortunes of non-investment life insurance providers and business interruption insurers.

Whilst insurers of business interruption policies have become embroiled in expensive litigation, claims payments have been slow, renewal prices have increased and many insurers have withdrawn from insuring the risk, in comparison, life insurers have been largely unaffected by the pandemic, despite the unprecedented loss of life in 2020. There is no evidence of insurers withdrawing from the life insurance market. Despite high numbers of claims, insurers have not sought to exclude death due to COVID-19, premiums have remained fairly static, many claims have been paid swiftly and life insurance remains a relatively cheap insurance.

Why is that? We all know of the devastating number of deaths due to COVID during 2020. The demographics of those deaths in the UK, according to the Office of National Statistics, is as follows, see Office for National Statistics (2021):

The majority of UK life insurance policies are purchased when individuals take a mortgage and are designed to expire upon mortgage repayment. Although the age at which people repay their mortgages is increasing and of course there are life insurance products available for those in their 80s, the majority of people have repaid their mortgage and their life insurance policies have ended by the time they reach 80 years. This means the demographics of the COVID-19 death rates have been kind to the life insurance industry. Also, whilst there is no doubt that the number of claims under life insurance has increased, at the same time, life insurers have seen a significant increase in the sale of life insurance, which has meant healthy cash reserves to set off against claims.

There is also a stark difference in the mechanics of life insurance products compared to all risks commercial business insurance. Life insurance is a relatively simple product, not based on the indemnity principle. Policyholders decide at the outset of the policy what the payment should be upon the event that is insured—their death. The trigger for payment is simple and easily proved by production of a death certificate. Whilst claims may be investigated where fraud is expected and death due to pre-existing conditions are often excluded, payments are usually made quickly on production of a death certificate. There is nowhere near the administrative burden relating to claims payments, which also keeps premiums lower. There is no ongoing loss—there is either a death, or not a death.

# **13.5 Why Existing Indemnity Based Pandemic Insurance Products Are Not Working**

Advances in epidemic risk analytics, including monitoring and modelling tools, have helped insurers better understand their risks and improve response strategies for indemnity based claims. Technology has enabled better risk measurement, monitoring, mitigation and claims management. There are innovative Insurtechs offering tech-based solutions in several areas of indemnity based insurance (UK examples are, Distribind which provides an automated bordereaux system, and Concirrus which uses onboard sensors to provide real time analytics of marine shipping risk). But technology cannot entirely mitigate the time it takes to predict and evidence loss for a new indemnity based risk.

Specialist pandemic insurance was available to businesses prior to COVID-19. In 2018, Marsh McClennan launched a specialist pandemic insurance which provided indemnity based cover, using triggers such as mortality or infections in a defined area.8 The policy could be tailored to the policyholder to provide coverage for specific expenses, geographies, types of disease, or portions of a calendar year, but until COVID-19 it reportedly had very little uptake, see Collins (2020). I suspect that the slow uptake was due to the perceived gap in the cost of the insurance compared to the risk of suffering losses due to a pandemic. Hindsight is a wonderful thing, and it must be remembered that the last global pandemic was the influenza pandemic of 1918. Whilst Ebola and SARS were modern epidemics, they were seen in the Western World as foreign diseases of little relevance or risk to communities in the West.

Even for specialist pandemic insurance using the latest technology, the methodology used to determine future loss was not well developed. Insurers had some knowledge of epidemics such as SARs and Ebola, but both were different to each other and each was different to COVID-19. The unpredictability of epidemiology meant that actuaries considering risk models for pandemic insurance were involved in more art than science. Furthermore, modelling of the economic input was based on sparse evidence and on economic simulations rather than empirical data. Unlike natural catastrophes and other crises, pandemics and epidemics typically do not cause

8 https://www.marsh.com/ca/en/press-centre/marsh-to-help-businesses-minimize-financial-lossfrom-pandemics.html.

immediate physical damage, and so they are difficult to model because every business will have bespoke economic loss, dependent on their circumstances at the time.

Then there is the challenge of anticipating behaviour of both governments (severity of lockdowns, enforcement of restrictions, availability of vaccines) and of individuals (super spreaders, mask wearing, social distancing, ability and willingness to work from home).

Most businesses did not hold specialist pandemic insurance, but did hold all risks commercial business insurance which covers any peril in relation to a particular category of risk, unless excluded in the policy. To give an example, an all risks business insurance policy may cover any loss arising from theft, but exclude theft by way of deception, so a break-in would be covered, but fraud would not. This approach to insurance has led to long and complicated insurance policies with many exclusions9 and there is often a mismatch of understanding between insured and insurer as to what these policies cover.

Insurers viewed commercial business insurance policies as protection against losses flowing from physical damage to insured premises, subject to restrictions and limitations and that only very few of such policies covered economic loss flowing from COVID-19. For policyholders, they read their commercial business policies as including cover for economic losses relating to closure of their business due to disease and had the expectation that, having paid insurance premiums for years, they would be covered for losses to their businesses flowing from COVID-19.

This mis-match of expectations stems from the nature of all risks insurance policies which started life as simple insurance cover against physical damage to property resulting from fire. Many additions of cover and exclusions over time mean these policies are often repetitive and inconsistent. Add to the mix some unfortunate poor drafting, and it is unsurprising that these types of insurance policies have become a hot bed for litigation in the last year.

Looking at some common policy wordings, you can see why commercial business insurance policyholders had the expectation that they were covered for economic loss flowing from COVID-19:

We shall indemnify you in respect of *...*

9 For example, the RSA business combined policy is 72 pages long.

Reading the extracts above, you could be forgiven for siding entirely with the policyholders making claims for losses during the COVID-19 pandemic. But all-risks policies will also have exclusions, for example,

One of the biggest areas of dispute in COVID-19-related business interruption claims is whether COVID-19 has caused damage to business properties from which economic loss flows. This is because most indemnity based business insurance policies have, at their core, the principle that the insurance covers loss flowing from physical damage. However, over time additional cover against economic loss has been added, such as economic loss due to closure by a local authority and in amongst the list, often loss arising from interruption of trade due to closure of the premises on the occurrence of disease, either on or near the premises. Policyholders argue that COVID-19 has damaged their business and their premises and so they are covered when they have to close as a result.

Where claims were made on the basis that COVID-19 had physically damaged the premises, the insurers' arguments against this were varied and included:

Where the words "physical damage" are used, insurers felt they had a strong case and this has been born out in recent case law.10 However, some policies refer only to

10 TKC London Limited v Allianz Insurance plc [2020] EWHC 2710 (Comm).

"damage", and so the argument then is, whether COVID-19 has caused "damage" to the business.

Where policies clearly and separately include cover for loss flowing from business interruption due to closure because of disease, insurers have argued as to the meaning of "disease", "notifiable disease", and "human disease". Many wordings refer to "notifiable disease", but it is often not defined. Medical practitioners must report "notifiable diseases" to the government under statutory duties in the Public Health (Control of Disease) Act 1984 and the Health Protection (Notification) Regulations 2010, and on 5 March 2020, COVID-19 was added to the list of notifiable diseases. Where notifiable disease is not defined, is that what was meant? Some policies are even more vague—referring only to "human disease" or just "diseases", particularly with reference to local authorities shutting businesses down.

Cover often includes losses due to "closure by an authority" because of disease. Insurers have argued that an "authority" relates to a local authority, and it was the UK Government, Welsh Government and Scottish Government that initially "advised" and then required businesses to close, so there is no cover.

For good measure, where policies do not specifically exclude loss due to a pandemic, insurers argue that commercial business insurance simply is not designed to cover pandemics and that policyholders must have realised that when they purchased the insurance, because of the price of the premium. Whilst the concept of applying a construction of a policy that makes commercial sense for both parties is established law, in my view raising such arguments both in court and in the press, does nothing but harm the reputation of the insurance industry.

Where primary liability is proved by the insured, under the indemnity principle, policyholders then have to prove causation between the insured event that has occurred and the economic loss suffered. This principle has led to further disputes between the insurer and the insured, particularly in relation to economic damage suffered by SMEs during lockdown due to the presence of COVID-19 on or near the insured premises. Insurers have argued that in many cases, the economic loss of businesses has been caused by a national lockdown and an economic downturn, not by the presence of COVID-19 on the premises. Insurers have required evidence from their insureds of COVID-19 physically being present on the premises, which, particularly during the first wave of COVID-19 is impossible for most as testing was not widely available and there was no means of tracing where infected people had been. Even where COVID-19 on the premises could be proved, policyholders had to show that this caused the closure of the premises and that such a closure would not have happened irrespective of the COVID-19 instance on their premises. In many cases this is an impossible hurdle to climb.

A perfect storm has brewed between insurers and their policyholders. It is not that the insurers are the corporate bad guys, and nor do the policyholders have unrealistic expectations of cheap property based insurance. The problem is the structure of the indemnity policy, which does not meet the fundamental needs of risk transfer in a pandemic. It is little wonder that insurers are choosing to change policy wordings on renewal to remove all cover for losses related to disease, COVID-19 or otherwise.