How lethal is the novel coronavirus, and how many undetected cases there are? The importance of being tested.

Ugo Bastolla

The ratio between the number of death and the number of confirmed cases is clearly an underestimate of the lethality, because it assumes that all the presently infected people will eventually recover; but it is also an overestimate, because it assumes that all the extant infections are detected. To mitigate the first problem, it is possible to extrapolate the lethality to the infinite time when all the infected cases are resolved. The asymptotic lethality rate d is the fraction of present confirmed cases (N ) that are already deceased (D) plus the presently infected persons I = N − R − D that will eventually die, d = (D + d 1 I)/N ⇒ D/N = d − d 1 (N − R − D)/N (here R is the cumulative number of recovered persons). If we assume that d 1 is constant, we can estimate the parameters d and d 1 through linear regression of the apparent death rate D/N versus the infected fraction 1 − (R + D)/N . The assumption of constant d 1 is not realistic in the initial stage of the epidemics but it becomes more and more accurate as time passes, as the data from China shows. A technical point: I regularize the linear regression with the method of rescaled ridge regression [5] . This regularization produces accurate fits while limiting the values of the fitted parameters. Without regularization, the extrapolated death rate d would be rather large, while the assumption of constant d 1 is expected to lead to overestimate d, in particular in the initial stages of the epidemics. The fitted value of d 1 is similar to, but always smaller than, d, which appears reasonable: In fact, the persons that die first belong to the classes with higher death rate, such as the elder and the sick. Figure 1 shows the curves of D/N as a function of the infected fraction. These curves are reasonably linear for large time for most countries. The apparent death rate D/N 2 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. strongly depends on the number of performed tests, as I shall show below, and since this number is not constant, as for instance data from Italy show. Therefore, I fit the lethality rate at the time t=12 March at which data on the tests are available for most countries [6] (solid lines in Fig. 1 ). From these fits, I extrapolate the parameter d at the date of 12 March, provided that two conditions hold: at t the ratio between cumulative recoveries and cumulative deaths R/D is at least 0.9 and D(t) ≥ 10. When these conditions are not met the asymptotic death rate d is clearly overestimated. If the above conditions are not met, I perform a regularized fit over the whole time range (dashed lines in Fig. 1 ), and again I accept the result only if R/D > 0.9 and D ≥ 10 at the last available time. For only seven countries I could find data on the number of tests and at the same time I could extrapolate the death rate fulfilling the above conditions: Japan, South Korea, Malaysia, Italy, Spain, France and Switzerland. The extrapolated d is plotted in Fig.2 as a function of the number of confirmed cases N divided by the number of tests T for the seven countries for which d can be reliably estimated. You can see that all countries fall on the same straight line, d = d 0 + a(N/T ). This means that the huge apparent differences between the low mortality rates of countries like South Korea or Germany and the high mortality rate of countries like Italy can be fully expl