Quadratic growth during the 2019 novel coronavirus epidemic

The number of infections and the number of fatalities in the 2019 novel coronavirus epidemics follows a remarkably regular trend. Since the end of January, the ratio of fatalities per infection is about 2% and remarkably stable. The increase appears to be roughly exponential, but with an e-folding time that gradually increases from just two days at the end of January to about ten days by February 9. Here we show that, since January 20, the number of fatalities and infections increases quadratically and not exponentially, as widely believed. At present, no departure from this behavior can be seen, allowing tentative predictions to be made for the next 1-2 months. Contrary to the usual exponential growth, a quadratic growth is the result of control interventions. It can be explained by spreading on the periphery of a bulk structure, which can be geometrical or sociological nature.


Introduction
Daily news reports suggest a steady, nearly exponential increase in the number of fatalities and infections in the 2019 novel coronavirus (2019-nCoV) outbreak. 1, 2 This wealth of quantitative data must be contrasted against the qualitative judgments of what may appear like a sudden increase in the number of cases from one day to the next. In the popular press, there are also speculations about a large number of unreported cases, casting doubt on the usefulness of the reported numbers. Although this is justified in view of the length of the incubation period of about 6 days in this case, 3 it affects all data uniformly. Instead of such subjective descriptions, it could be more valuable and informative to quote, for example, the current e-folding time and the perhaps the time scale of its change. Ideally, however, it would be desirable to quantify the parameters of the underlying differential equation describing the growth of fatalities and infections.
Significant effort is currently also going into quantitative modeling of the 2019-nCoV epidemic. Particularly noteworthy in this connection is the time delay model, 4 the prediction model, 5 and a model involving the basic reproduction number. 6 We feel, however, that at the current time, more detailed diagnostics is needed to monitor the change in the epidemics. This is the purpose of the present paper.

Results
The daily news reports provide an easily accessible source of information that appears to reveal a remarkably regular trend; see the 2019-nCoV situation reports of the World Health Organization 7 and the DEVEX 8 and worldometer 9 data bases. In Table 1, we list the number of fatalities, infections, and the ratio of fatalities per infection.
A graphic representation of the increase in the number of fatalities and infections is given in Figure 1. It is tempting to fit the data with an exponential growth, 10 n(t) = n 0 exp(t/τ ), but it is clear that the instanta-   neous e-folding time τ would need to increase gradually for achieving a reasonable fit both at early and late times.
In fact, a possible fit to the data is provided by a model in which the e-folding time is allowed to change linearly time, i.e., n(t) = n 0 exp[t/τ (t)], where τ (t) = ǫ(t − t 0 ) models a linear increase and ǫ is a constant factor. In Figure 2, we plot τ (t) = (d ln n/dt) −1 along with the fit. One can see that τ increases by one day every two days, i.e., ǫ = 1/2. This finding will turn out to be significant.
Let us now discuss how the growth with a variable efolding time could be described in terms of a population dynamics model. At the level of an ordinary differential equation, a strictly exponential growth corresponds to the equation dn/dt = n/τ . If we now assume τ (t) to increase linearly in time, as stated above, we have with the solution where τ is an integration constant. Given that ǫ = 1/2, the growth in Equation (2) is quadratic and we can state the final equations in the following explicit form: and, because 0.7 × √ 0.022 ≈ 0.1, we have

Conclusions
The present work has demonstrated that for the 2019-nCoV epidemic, the available data are accurate enough to distinguish between an early exponential growth, as was found for the 2009 A/H1N1 influenza pandemic in Mexico City, 10 and the quadratic growth found here. It was expected that the growth would not continue to be exponential, and that it would gradually level off in response to changes in the population behavior and interventions, 11 as found in the 2014-15 Ebola epidemic in West Africa. 12 However, that the growth turns out to be quadratic to high accuracy is rather surprising and has not previously been predicted by any of the recently developed models of the 2019-nCoV epidemic. 4-6 It is also a remarkable that the fit isolates January 20 as a crucial time in the development of the outbreak. At that time, the actual death toll was just three and the number of confirmed infections just a little over 200; see Table 1. The 2% fraction of fatalities per infection has not yet been established.
It is rare that an epidemic provides us with such clean data as in the present case of the 2019-nCoV outbreak. At the moment, the quadratic growth of the epidemic does not show any sign of a decline, and so Equation (3) predicts a continued increase and a death toll of about 10,000 by April 1. This number is still significantly less than 0.1% of the population of Wuhan, but it may be hoped that control interventions 11 prevent this from happening.

Introduction
In spite of the drastic confinement efforts, the 2019 novel coronavirus (2019-nCoV) epidemic continues to claim lives at a high rate. 1-3 As is well known in population dynamics, because of such control efforts, the number of fatalities N follows a subexponential growth. 10-12 Ideas to explain the resulting growth of N in terms of a linear increase in the characteristic time scale are not fully satisfactory; see the main part of the paper. They predict an algebraic increase with time t, i.e., N ∝ t γ , but they do not rigorously constrain the value of the exponent γ, which must instead be established empirically (see the main part of the paper).

The model
Here we propose a model where the continued confinement efforts prevent the spreading of the bulk of the infected population, but they cannot prevent spreading on its periphery; see Figure 4 for a sketch. The rate dN/dt, with which N increases with time, is therefore equal to the number of infected people on the periphery divided by a characteristic spreading time τ . We therefore arrive at the following simple differential equations: where n ≈ 2 √ πN is the number of people in the periphery. Here, the prefactor depends on the geometry and we would have n = 4 √ N for a rectangular geometry. We may therefore set n = α √ N , where α ≈ 3.5 for a circular geometry. Inserting this into Equation (5) yields with the solution Earlier empirical work suggested a growth of the form N (t) = (t/T ) 2 , with T = 0.7 days (or about 17 hours) and t being the time in days after January 20, 2020. This implies that the spreading time is τ = αT /2 = 1.2 days for a circular geometry.
The idea of a geometrically confined bulk with a surrounding periphery may need to be generalized to sociological or network structures that can follow similar patterns. 13 In the present work, we do not make any attempts to analyze this aspect further, but refer instead to recent work in Ref., 14 who analyzed the spatial patterns of the 2019-nCoV.

Monitoring the spreading
To monitor the spreading of the disease, it is useful to plot √ N versus time, as is done in Figure 5. The data points have been assembled from the DEVEX website 8 and the latest updates have been taken from worldometer. 9 The data is tabulated in Table 1, except for the latest number of fatalities of 1384 on February 14, 2020. The extrapolation of the line (t/0.7 days) 2 suggests that the N = 10, 000 mark will be crossed by the beginning of April.
We reiterate that the present model does already take control intervention into account. This strengthens the validity of our exploration and emphasizes the need for even more drastic changes in the type of control interventions.

Further implications
One might wonder why this quadratic growth is not generally discussed in the literature. There is a large variety of theoretical models; see Ref. 10 for a review. Several such models have already been adapted to the 2019-nCoV epidemic. 4-6 More importantly, the idea of control interventions has been discussed in detail, 11 but the present epidemic provides us with an unprecedented data record with large numbers of infections occurring on an extremely short timescale. This timescale is short enough so that the quadratic growth law is not yet strongly affected by any changes in the control intervention strategy.
It is important to notice that during the 2019-nCoV epidemic, there was so far never an episode of exponential growth. By the time the quadratic growth law commenced on January 20th, the city of Wuhan was already under quarantine. Given the unaltered quadratic growth since then, there is no evidence to suggest that earlier human interventions could have changed the course of history. Conversely, unless there is a more drastic change in the type of interventions, we must expect the current growth to continue until other (e.g., seasonal) factors begin to play a role within the coming months.