Tracking the dynamics and allocating tests for COVID-19 in real-time: An acceleration index with an application to French age groups and départements

An acceleration index is proposed as a novel indicator to track the dynamics of COVID-19 in real-time. Using data on cases and tests in France for the period between the first and second lock-downs—May 13 to October 25, 2020—our acceleration index shows that the pandemic resurgence can be dated to begin around July 7. It uncovers that the pandemic acceleration was stronger than national average for the [59–68] and especially the 69 and older age groups since early September, the latter being associated with the strongest acceleration index, as of October 25. In contrast, acceleration among the [19–28] age group was the lowest and is about half that of the [69–78]. In addition, we propose an algorithm to allocate tests among French “départements” (roughly counties), based on both the acceleration index and the feedback effect of testing. Our acceleration-based allocation differs from the actual distribution over French territories, which is population-based. We argue that both our acceleration index and our allocation algorithm are useful tools to guide public health policies as France might possibly enter a third lock-down period with indeterminate duration.

Denote P i t = t τ =1 p i τ and D i t = t τ =1 d i τ the cumulative numbers of positive and diagnosed persons up to date t. Finally, defineP i t = P i t /P i T andD i t = D i t /D i T for t = 1, . . . , T , which are the fractions of, respectively, positive and diagnosed persons at date t relative to that at end date. In more technical term, dividing the historical times series by the most recent entry amounts in our setup with non-negative numbers to perform min-max normalization (see See Han,Kalber,and Pei [4], section 3.5.2). Our object of interest is the relationship betweenP i andD i over time in the context of a pandemic, when testing is the only way to detect confirmed cases, which is depicted in the scatter-plot of Figure 2.
A first-order Taylor expansion atD i T = 1 gives: Equation (3) shows that the derivative (f i ) (1) is essentially an elasticity, given the normalization of cumulative numbers by end date values: it measures how many positives follow an increase in the number of tests at end date, as a fraction of end date values. For example, values such that (f i ) (1) > 1 mean that the pandemic is accelerating, since a given fraction of the total of tests performed up to date T is associated with a larger fraction of positives who are detected with those tests, in percentage of the cumulative number of positives at T . On the contrary, (f i ) (1) < 1 implies that the pandemic is decelerating. This elasticity is what we have labeled the acceleration index and labeled ε T in Section 2.1.
All of the above implies that, given the estimate of the first-order derivative, ε T ≡ (f i ) (1), equation (3) can be rewritten in terms of the numbers of positive and tested persons, that is: In other words, equation (4) can be used to decompose the effect of tests on positives in levels, that is, how many additional positives are detected given additional tests, between T and T + dt: Equation 5 is identical to equation 1 in Section 2.1, where ε T is estimated as the ratio of variations of cases and tests between T and T − 1. Similarly, the above decomposition in levels holds for any date t < T , as follows: From equation (6), the effect of tests on positives in percentage terms from the perspective of date t is therefore written as: The elasticity of the number of positives with respect to the number of tests is now, because it is evaluated at date t as opposed to end date T , the product of the derivative at the relevant point, times the ratio of average positive rates -that of date T over that of date t.

A.2 Decomposition and Acceleration index: Exponential Case
This section explores what the decomposition stated in Section A.1 reveals when time is assumed to be continuous and when the number of cases grows exponentially over time, as usually assumed in epidemiological models, of SIR type and related for example. Although typically absent in the latter strand of literature, we have to introduce tests and we assume that they also grow exponentially.
More formally, using the notation in the previous section, suppose that the number of cases per unit of time is denoted by p(t) = αe βt while the number of tests per unit of time is d(t) = γe νt , where the growth rates β and ν are assumed to be positive for the sake of illustration. Cumulated cases and tests are then noted P (t) = t 0 p(τ ) dτ and D(t) = t 0 d(τ ) dτ , respectively. It is easy to derive, by straight integration, the expressions: It follows that our acceleration index is given, as function of time, by: From equation (9), then, one infers that two cases occur. When β = ν, that is, when both cases and tests grow at the exact same rate, then our acceleration index equals 1 at all dates. When the two growth rates differ, however, ε(t) converges, when t goes to infinity, to the ratio of growth rates β/ν, independently of the scale parameters α and γ. As an illustrative example, suppose that β > ν, so that positives grow faster than tests. Then the pattern of our acceleration index ε(t) over time will have two regimes: it first grows almost linearly and eventually reaches the upper bound β/ν > 1.
Obviously, in that case both the daily positivity rate p(t)/d(t) and the average positivity P (t)/D(t) grow over time, and the former quantity exceeds the latter all the time so that acceleration prevails.
This closely resembles the pattern following early August to early October in Figure 3, as underlined in the main text.

A.3 Statistics for Age Groups
In Table 1 we report a few statistics for all age groups, as of October 25, 2020. In the second and third columns we report the numbers of cumulated cases and tests, respectively. The fourth column depicts that average positivity rate, defined as the ratio of cumulated cases and cumulated tests, while the actual test shares appear in the fifth column. Finally, the last column shows the share of cases by age group, which is defined as the ratio of cumulated cases.