A memorandum on the observed number of infected cases

If we simply think that people randomly contact each other, and if the proportion of infected people is still small, then a) the increase in the number of infected people $I(t)$ is proportional to $I(t)$ itself. b) On the other hand, a portion of $I(t)$ is removed from $I(t)$ by healing spontaneously (or by dying without being tested). In most simple case, this can also be expressed as being proportional to $I(t)$, as seen in the classical SIR model. c) In addition, by testing for infections, people found to be infected are isolated from the population, for example, by being hospitalized. In an extreme case where a certain number of randomized tests per unit time performs, the rate of finding infected people is also proportional to $I(t)$. In another case where a certain portion of $I(t)$ becomes severe and only such patients are selectively tested (as long as the testing capacity is not saturated), confirmed cases will again be proportional to $I(t)$. Therefore, it is reasonable to think that this case is also proportional to $I(t)$. Eventually, the temporal change of $I(t)$ is $$\frac{dI}{dt} = \lambda I(t) = (a-b-c) I(t) \qquad (a, b, c > 0),$$ where $\lambda = a - b - c$, and $a$ is the rate of infection per time, $b$ is the rate of recovery, and $c$ is the rate of isolation by testing. Therefore, $I(t)$ is simply an exponential function, $$I(t) = I_0 e^{\lambda t},$$ where $I_0$ is the initial number of infected people at $t = 0$. $\lambda$ is the reciprocal of the “time constant” of the increase (i.e., the function is $e$-folded for every $1/\lambda$ of time), and when $\lambda > 0$, the function increases with time, and when $\lambda < 0$, it decreases. In the following discussion, only $a - b$ is meaningful and there is no particular need to distinguish between $a$ and $b$ in the formulas, but we use different symbols assigned to each for ease of understanding.

The number we can observe as the cumulative number of infected people is not $I(t)$ itself, but those who are removed and isolated from $I(t)$ by testing at $c I(t)$ per time. Let this be the “observed number of infected people” $H(t)$. Therefore, $$\frac{dH}{dt} = c I(t)$$ and when we put $H(0) = 0$ at $t = 0$, then $$H(t) = \frac{c I_0}{\lambda} (e^{\lambda t} - 1) = \frac{A}{\lambda}(e^{\lambda t} - 1) \qquad (\lambda \neq 0).$$ where $A = c I_0$. When $\lambda > 0$, this asymptotically approaches an exponential function at $t \rightarrow \infty$, which has the time constant $1/\lambda$ that is same as $I(t)$, and the ratio to the unobserved (hidden) infected people $I(t)$ is $I(t)/H(t) \rightarrow \lambda/c$.

Physicist Alessandro Strumia and the Financial Times journalist John Burn-Murdoch presented semi-logarithmic charts of the cumulative number of observed case by each country, aligned over time when the number reached a certain reference count[1][2]. In these plots, we can see a remarkable difference in the movements of Japan and European countries. The following is a similar plot based on data up to April 3, 2020 (data source: CSSE, Johns Hopkins University). Note that it is not the number of observed cases, but the number divided by the population of each country.

European countries such as Italy, Germany, the United Kingdom, and France have shown a similar trend, with a steep tilt (short time constant) around the reference value and then gradually slowed down in the tilt, while Japan has not seen such an initial rise and has generally remained at a constant tilt. As a result, there is great difference in the numbers of observed cases between Europe and Japan.

Let us consider to represent $H(t)$ of the model above in this plot. When $\lambda > 0$, let $t^* (> 0)$ be the time when the reference value $H^∗ (> 0)$ is reached. That is, $$H^∗ = \frac{A}{\lambda} (e^{\lambda t^∗} - 1)$$ Then, the observed number of infected persons $\tilde{H}(t)$ which is shifted so that it becomes a reference value $H^∗$ at time $t = 0$ is $$\tilde{H}(t) = \frac{A}{\lambda} (e^{\lambda (t + t^∗)} - 1) = H^∗ e^{\lambda t} + H(t)$$ Drawing $\tilde{H}(t)$ on the semi-logarithmic plot for the cases $A = 1$ and $A = 10$, assuming $H^∗ = 1$ and $\lambda = 1$, we get the following:

When we represent $\tilde{H}(t)$ in the semi-logarithmic plot as above, the slope of the plot that is proportional to the inverse of the time constant (and doubling time) is $$\frac{d}{dt}\log\tilde{H}(t) = \frac{1}{1-e^{-\lambda(t + t^*)}}\;\lambda = \frac{\lambda H^* + A}{\lambda H^* + A (1 - e^{-\lambda t})}\;\lambda \qquad (\log \mbox{is the natural logarithm}),$$ and especially at $t = 0$, we get $$\left.\frac{d}{dt}\log\tilde{H}(t)\right|_{t=0} = \frac{A\;}{H^*} + \lambda,$$ that is, the initial slope has a term proportional to $A$ in addition to $\lambda$.

The plot showed above seems to illustrate well the difference between the European countries and Japan based on actual data. That is, if there was no significant difference in the initial number of infected cases $I_0$, then the difference in the two plots was primarily due to difference in the rate $c$ of testing. The early rapid increase in Europe shows the effect of “revealing” by extensive testing, and the uniform tilt in Japan is a natural consequence of the small number of testing. In Europe, extensive testing and social distancing policies were taken, while in Japan, calls for behavioral change were made mainly by locating the attributes of clusters, so the data in the latter half of the time are influenced by them. On the other hand, the slope of the plots in the first half of the data and at some distance from the time constant $1/\lambda$ would reflect the constant $\lambda$ assumed in this model. Based on this idea, it can be seen that the speed of spread in Japan was slightly slower than in Europe[3]. However, the difference is not so much.

Again, we can largely explain the differences in the plots by difference in the amount of testing between the two, and we cannot say that there was a remarkable difference between Europe and Japan in the actual number of hidden infected persons $I(t)$ on the basis of the initial increase in Europe. And to the extent that they appear in the data, both are currently far from being successful in preventing the spread of infection.

In this memorandum, the model is kept as simple as possible for ease of analysis. A more refined model would be needed in order to analyze it in line with the actual data.

(This article is a translation of the post on April 4, 2020)

• [1] Alessandro Strumia, A tweet on 2020-03-05.
• [2] FT Visual & Data Journalism team, Coronavirus tracked: the latest figures as the pandemic spreads, Financial Times.
• [3] Yoh Tanimoto, A tweet on 2020-04-03.