CoViD‐19 epidemic follows the “kinetics” of enzymes with cooperative substrate binding

Abstract

The Hill equation, which models the cooperative ligand‐receptor binding equilibrium, turns out to be useful in modeling the progression of infectious disease outbreaks such as CoViD‐19. The equation fits well the data for total and daily case numbers, allows tentative predictions for the half‐point and end point of the epidemic, and presents a mathematical characterization of how social interventions “flatten the curve” of the disease progression.

1. INTRODUCTION

Those who followed the CoViD‐19 epidemic (and who did not?) watched with trepidation as the number of cases rose steadily. A common question was “when will this end and life return to normal?” The answer was invariably “who knows?” Well, it turns out, biochemists know. A glance at the plot of total number of cases vs. day showed a sigmoidal rise. This curve is familiar to biochemists as the saturation plot for a protein that binds ligand cooperatively, that is, with a Hill coefficient ≥ 2.

There are a number of sophisticated mathematical models of infectious disease progression. ¹ , ² , ³ , ⁴ They can be stochastic or deterministic, statistical or mechanistic. The most complex models can account for variables like reproduction number (R0 = infectivity), steady state, vaccination, social distancing, hospital capacity, etc. But wouldn't it be nice if there was a simple, familiar mathematical model that we all had access to and could use to model the severity and timeline of a pandemic? Enter, the Hill equation.

The Hill equation describes how ligand binding to a protein receptor saturates with the concentration of free ligand, [L]:

$$f_{\mathrm{b}}=\frac{\text{occupied receptor}}{\text{total receptor}}=\frac{\left[P{L}_n\right]}{{\left[P\right]}_{\mathrm{tot}}}=\frac{{\left[L\right]}^n}{K_{\mathrm{d}}+{\left[L\right]}^n}=\frac{1}{1+{\left(\frac{K_{\mathrm{d}}}{\left[L\right]}\right)}^n}$$ (1)

where f _b is the fraction of total receptor that has bound n ligands, and K _d is the PL _n dissociation equilibrium constant; the exponent n is known as the Hill coefficient. Because K _d is the free ligand concentration that gives f _b = 0.5, also known as [L]₅₀, Equation (1) can be written as.

$$\left[P{L}_n\right]=\frac{{\left[P\right]}_{\mathrm{tot}}}{1+{\left(\frac{{\left[L\right]}_{50}}{\left[L\right]}\right)}^n}$$ (2)

For an enzyme, the enzyme‐substrate complex (E _n = P _n ) goes on to catalyze the reaction, so the Hill equation can be transformed to describe the dependence of initial reaction rate (v ₀) on initial substrate concentration, [S]₀:

$$v_0=\frac{V_{\max }}{1+{\left(\frac{K_m}{{\left[S\right]}_0}\right)}^n}=\frac{V_{\max }}{1+{\left(\frac{{\left[S\right]}_{50}}{{\left[S\right]}_0}\right)}^n}$$ (3)

where V _max is the asymptotic v ₀ at saturating [S]₀. A Hill coefficient ≥ 2 signifies the cooperative binding of two or more substrate molecules at the same time. Equation (3) reduces to the well‐known Michaelis–Menten equation when n = 1 (i.e., non‐cooperative binding of a single substrate molecule); in this case, the Michaelis–Menten plot, v ₀ vs. [S]₀, saturates hyperbolically at the asymptote V _max. For n ≥ 2, cooperative v ₀ vs. [S]₀ plots saturate sigmoidally.

To model the sigmoidal rise of the total number of disease cases (e.g., CoViD‐19) with time, one would use this version of the Hill equation:

$$\text{cases}\left(t\right)=\frac{{\text{cases}}_{\text{final}}}{1+{\left(\frac{t_{50}}{t}\right)}^n}$$ (4)

Here t represents days, t ₅₀ is the day that you reach half of the final number of cases (cases_final), and n controls the steepness of the sigmoidal rise. Together, t ₅₀ and n provide information on how long the outbreak will last.

For example, Figure 1a shows sigmoidal plots for n = 3, 5, and 8, given cases_final = 100 and t ₅₀ = 40 days. Note that a lower n flattens the curve, so that the rate of increase of cases is shallower. Although this does not decrease the final number of cases, it is advantageous in that it would decrease the number of hospital admittances on a particular day. The curve is also flattened by decreasing cases_final (Figure 1b) and by increasing t ₅₀ (Figure 1c).

FIGURE 1.

Sigmoidal saturation plots for various values of (a) n = 3, 5, 8 (cases_final = 100 and t ₅₀ = 40 days); (b) cases_final = 40, 60, 100 (n = 5, t ₅₀ = 40 days); and (c) t ₅₀ = 20, 40, 60 (n = 5, cases_final = 100)

Many print media stories that discussed flattening the CoViD‐19 epidemic curve illustrated the effect not with total cases vs. time curve, but with the bell‐shaped daily cases vs. time curve. Daily cases as a function of time is the first derivative of the total cases function:

$$d\left(\text{case}\mathrm{s(t)}\right)/\mathit{dt}=\#\text{daily cases}\left(t\right)=\frac{n·\mathrm{cas}{\mathrm{es}}_{\text{final}}·{\left(t_{50}\right)}^n}{\left(t^{\left(n+1\right)}\right)·{\left[1+{\left(\frac{t_{50}}{t}\right)}^n\right]}^2}$$ (5)

In Figure 2 we can see how the three fit parameters affect the shape of the first derivative plot. Lowering cases_final (Figure 2a) decreases the amplitude, that is, the number of daily cases on each day, but leaves the relative width and time‐axis position of the bell‐curve unchanged. Lowering n (Figure 2b) increases the relative width of the curve, and shifts its peak time to the left (to shorter times). This has the effect of increasing the number of daily new cases early and late in the epidemic (before day 25–30, and after day 55–60), while dramatically decreasing this number at the peak of the epidemic. Finally, increasing t ₅₀ (Figure 2c) also increases the relative width of the curve, while shifting the peak time to the right (later times). This spreads out the number of new daily cases, decreasing it early in the epidemic, while increasing it later on.

FIGURE 2.

First derivative (daily cases) plots for various values of (a) cases_final = 40, 60, 100 (n = 4, t ₅₀ = 40 days); (b) n = 3, 5, 8 (cases_final = 100 and t ₅₀ = 40 days); and (c) t ₅₀ = 20, 40, 60 (n = 4, cases_final = 100). The vertical dotted line in (a) and (b) marks t ₅₀

From Figure 2c we can see that the curve is asymmetric—it rises faster than it falls. Furthermore, this asymmetry declines as n increases (Figure 2b, n = 8). Another surprising observation regarding the first derivative curve is that its peak is always ≤ t ₅₀ (vertical dotted line in Figure 2a,b). The peak approaches t ₅₀ as n increases, rising from about 0.6 × t ₅₀ for n = 2, to 0.975 × t ₅₀ for n = 8.

To explore some real epidemic data sets, Figure 3 presents a plot of the number of CoViD‐19 cases in China and South Korea. Clearly, the sigmoidal fits to Equation (4) are good, although not perfect: For example, the early undershoot and late overshoot of the China curve are due to the fact that Chinese authorities changed their method of diagnosing the disease after day 24, suddenly increasing the number of cases.

FIGURE 3.

Total number of CoViD‐19 cases in China (black points, January 19–March 31, 2020) and South Korea (blue points, February 17–March 31, 2020). Data are fit to Equation (4) using nonlinear regression (Kaleidagraph program); fitted parameters are listed in Table 1

Before we go any further, it is important to state up front that fitting infection data to the Hill equation is purely a statistical process. There is no theoretical model explaining why this equation fits the data, thus there is no way to connect the Hill equation parameters to viral or human biology, or to social policy implementation. On the other hand, one might hypothesize that faster viral reproduction inside the host would raise both n and cases_final; the same would be expected for anything that raises infectivity, for example, more infectious viral particles, easier viral spread from asymptomatic individuals, higher frequency of close human contact, etc.

Hill equation fits are quite good for almost all data sets (R ² = 0.9940–0.9997). Table 1 lists Equation (4) fitted parameters for data sets from 35 countries (fitted plots for each data set can be found in the Figures S2–S38). The Hill coefficients (n) vary about two‐fold (from 4 to 8), and half‐times (t ₅₀) vary about five‐fold (from 20 to 100 days); on the other hand, cases_final vary over 3,000‐fold, from 384 for Taiwan to 1.4 million for the United States.

TABLE 1.

Equation (4) fit parameters for CoViD‐19 total cases vs. day #

	# casesfinal	t 50 (day)	n	R 2	% completion a
China	74,780 ± 120	29.76 ± 0.08	6.03 ± 0.09	0.9975	102 b
S. Korea c	10,594 ± 26	20.58 ± 0.07	4.85 ± 0.08	0.9995	101
Taiwan d	384 ± 4	25.90 ± 0.13	5.06 ± 0.13	0.9986	100
Hong Kong d	1,054 ± 9	34.11 ± 0.09	7.41 ± 0.13	0.9993	98
Austria	15,380 ± 50	32.47 ± 0.08	5.91 ± 0.07	0.9995	99
Australia	6,709 ± 18	34.91 ± 0.06	7.81 ± 0.09	0.9996	100
Thailand	3,000 ± 15	30.80 ± 0.10	4.97 ± 0.07	0.9993	98
Luxembourg	3,847 ± 25	25.15 ± 0.13	4.13 ± 0.08	0.9989	97
Switzerland	30,700 ± 150	33.51 ± 0.11	4.57 ± 0.05	0.9995	95
Germany	171,400 ± 900	42.44 ± 0.12	5.48 ± 0.06	0.9996	94
Germany e	174,100 ± 800	42.77 ± 0.11	5.36 ± 0.06	0.9996	96
Israel	16,760 ± 200	46.69 ± 0.26	6.13 ± 0.14	0.9984	94
Czechia	8,070 ± 50	35.72 ± 0.15	4.57 ± 0.06	0.9995	92
Norway	8,295 ± 70	34.72 ± 0.23	3.82 ± 0.07	0.9990	92
Spain	(2.51 ± 0.04) × 105	43.8 ± 0.4	4.91 ± 0.12	0.9980	92
France	191,200 ± 2,100	46.90 ± 0.25	5.84 ± 0.11	0.9989	90
Italy	(2.26 ± 0.03) × 105	48.9 ± 0.3	4.44 ± 0.08	0.9989	89
Malaysia	6,690 ± 60	41.94 ± 0.23	4.30 ± 0.06	0.9994	87
Portugal	28,200 ± 400	40.8 ± 0.3	4.49 ± 0.09	0.9990	85
NY state	366,000 ± 5,000	40.0 ± 0.4	4.18 ± 0.07	0.9991	82
NY state e	378,000 ± 4,000	40.84 ± 0.3	4.06 ± 0.06	0.9992	80
Iran	115,000 ± 2,500	50.5 ± 0.7	3.92 ± 0.10	0.9978	81
SF Bay Area	6,010 ± 60	42.92 ± 0.24	4.08 ± 0.04	0.9997	81
Turkey	149,900 ± 1,100	39.15 ± 0.15	4.51 ± 0.04	0.9998	80
USA	(1.29 ± 0.03) × 106	56.8 ± 0.5	5.84 ± 0.13	0.9986	79
USA e	(1.48 ± 0.04) × 106	59.9 ± 0.6	5.35 ± 0.13	0.9983	79
UK	210,000 ± 2,500	59.05 ± 0.26	6.16 ± 0.07	0.9997	77
UK e	236,000 ± 4,000	61.6 ± 0.4	5.72 ± 0.09	0.9993	79
Belgium	60,800 ± 1,000	47.6 ± 0.4	4.48 ± 0.07	0.9993	77
Netherlands	51,450 ± 800	48.2 ± 0.4	4.04 ± 0.06	0.9994	74
Canada	72,200 ± 2,700	64.1 ± 0.9	5.66 ± 0.14	0.9984	67
Ireland	29,900 ± 1,400	52.21 ± 1.1	4.84 ± 0.16	0.9973	66
Japan	21,350 ± 1,400	74.5 ± 1.5	6.85 ± 0.29	0.994	64
Calif. state	88,200 ± 7,500	54.8 ± 2.7	3.32 ± 0.13	0.9970	52
Calif. state e	114,100 ± 9,000	62.9 ± 2.9	3.08 ± 0.10	0.9978	49
Russia	211,000 ± 8,000	60.9 ± 0.6	7.11 ± 0.09	0.9997	44
Russia e	(3.48 ± 0.24) × 105	69.1 ± 1.3	6.26 ± 0.12	0.9992	42
Sweden	44,600 ± 2,500	69.7 ± 2.0	3.38 ± 0.06	0.9993	42
India	70,050 ± 3,300	64.8 ± 1.0	5.51 ± 0.08	0.9996	42
India e	117,000 ± 8,000	75.6 ± 1.8	4.97 ± 0.09	0.9994	36
Peru	(9.0 ± 1.7) × 104	64 ± 4	5.14 ± 0.23	0.9966	32
Peru e	(6.3 ± 5.8) × 105	109 ± 26	4.44 ± 0.16	0.9979	7
Brazil	(4.4 ± 1.5) × 105	89 ± 10	4.05 ± 0.12	0.9984	15
Brazil e	≈2 × 106	≈ 130	4.02 ± 0.12	0.9984	≈6
Mexico	(1.9 ± 1.4) × 105	100 ± 22	3.97 ± 0.13	0.9983	8
Mexico e	(1.2 ± 0.3) × 105	84 ± 8	4.23 ± 0.13	0.9983	19

Note: Fitted curves for each country can be found in the Figures S2–S38. Unless otherwise specified, all fits are performed by nonlinear regression, using the Kaleidagraph software program, and the last reported data point is from April 27, 2020.

^aLast reported total number of cases ÷ cases_final × 100%.

^bThe final reported value exceeds cases_final (>100% completion) due to case influx from abroad.

^cDays 25–55 are omitted from the fitting process, as discussed below.

^dHong Kong, Taiwan data were fit with non‐zero y‐intercept (120 and 35 cases, respectively).

^eThe final reported value is from May 3, 2020.

Above, we speculated that anything that enhanced infection rate might raise both n and cases_final. In fact, the fitted data from 35 countries show no strong relationship between any of the three fit parameters (Figure S1). For example, the United States had a cases_final value almost six times higher than the next‐highest country, but its t ₅₀ and n _Hill are in the middle of the range of values. In general, countries with the lowest and highest values of t ₅₀ span the entire range of n _Hill; the same is true of the lowest and highest values of cases_final. There may be a weak correlation between t ₅₀ and cases_final (Figure S1): Countries with the lowest t ₅₀ values (< 36 days) all had low cases_final (< 30,000); conversely, countries with the highest cases_final (> 170,000) all had moderate to high t ₅₀ values (≥ 40 days). It seems logical that strong social intervention, including wide‐spread testing, contact tracing, and quarantining, would lower both t ₅₀ and cases_final.

Only three data sets were not well‐fit by Equation (4), and their deviations allow some interesting speculation on the effectiveness of social intervention. The Taiwan and Hong Kong (Figure 4) data sets had to be fit with a non‐zero intercept. This could be because their interventions, after the first scores of cases were identified, were so effective that these initial cases did not lead to a sigmoidal rise; only after the first 120 cases appeared in Hong Kong did the disease spread as an epidemic, presumably via community transmission.

FIGURE 4.

Hong Kong CoViD‐19 cases fit to Equation (4) with a zero y‐intercept (red curve), and with a non‐zero intercept (black curve) of 122 ± 4 cases

The South Korea data set showed a sigmoidal rise in cases until day 25, after which the cases numbers rose more slowly (Figure 5). One could speculate that South Korea, which was the second nation to experience an outbreak of CoViD‐19, managed to mount an intensive social intervention around day 25, including for example, extensive testing, contact tracing, quarantining, etc. This would be another example of “flattening the curve”. Note that although the curve was indeed flattened from day 25–55, the number of cases_final was unchanged by this intervention: 10,594 ± 26 for the blue dashed curve (i.e., the normal sigmoidal rise without the intervention) vs. 10,550 ± 110 for the red solid curve. In this case, flattening the curve did not decrease the eventual number of cases, it just slowed the rate at which those cases appeared every day, allowing hospitals to accommodate the influx of patients.

FIGURE 5.

South Korea CoViD‐19 cases fit to Equation (4) with all points (red solid curve), and with days 25–55 omitted (blue dashed curve)

As mentioned above, in the midst of a pandemic, one question on everyone's mind is “when will it be over?” The nature of an asymptote is that you never actually reach it. However, one can ask how long it would take to get close to the end, for example, within 90 or 95% of the maximum value. Manipulating Equation (4), we can show that if f is the fraction of the total you are interested in (e.g., 0.9 or 0.95), then the time to reach that value (t _f) can be calculated:

$$t_f=\frac{t_{50}}{{\left(\frac{1}{f}-1\right)}^{1/n}}$$ (6)

We can see from Equation (6) that, unsurprisingly, t _f increases linearly with t ₅₀ , but it also decreases with n (larger n gives a steeper rise, which gives a shorter t _f). For example, if n = 3 and t₅₀ = 20 days, then t ₉₀ = 42 days, but for n = 4, t ₉₀ = 35 days. Again, decreasing n flattens the curve, which spreads out the duration of the epidemic (Figure 2b). Using Equation (6) and fit parameters from Table 1, t ₉₀ for the United States and New York state come out to 92 and 72 days, respectively; that would be May 24 (2020) for the United States as a whole, and May 6 for New York state.

A better way to judge if you are close enough to the end of an epidemic to ease social restrictions is to identify a target maximum number of new daily cases. The World Health Organization has stated that an outbreak is deemed “contained” when < 10% of tests for the disease are positive; to be even safer, we can aim for < 1%. Harvard University's Global Health Institute has opined that the “bare minimum” of testing capacity in the United States should be 500,000 per day (0.15% of the population), so 1% would be 5,000 positive tests per day. Taking the fit parameters from Table 1 and using the first derivative equation (Equation 5), we can predict that the outbreak could be deemed contained in the U.S. by day 99 (May 31). For New York, the target would be < 300 new daily cases (0.015% × 20 million), which should occur by day 106 (June 9).

Of course, it is easy to fit an asymptotic curve when your data approach the asymptote, as they do in Figure 3. But what if you are in the middle of the epidemic, and you want to predict the end? This is more difficult. And here the function and limitations of data‐fitting must be considered. By fitting an entire data set to a mathematical equation, we can then use the fit parameters to predict the value of the dependent variable (y) for a value of the independent variable (x) that has not been measured, but lies within the range of measured values. Furthermore, the R ²‐value gives you a notion of the quality of the fit, and if we know the uncertainties in the fit parameters, we can calculate the uncertainty in the predicted y‐value.

However, that is not the situation in the middle of an epidemic. Here we are collecting data points one day at a time, and trying to use the fit equation to predict y‐values appertaining to x‐values that lie far beyond any that we have measured so far. In other words, we are using nonlinear regression to attempt to fit an incomplete data set. In this case, the R ² value and the fit parameter uncertainties do not mean the same thing that they do when working with a complete data set.

Consider the countries that were below 70% completion as of April 27, 2020, that is, the last ten countries in Table 1. One can see (Figure 6) that the relative uncertainties of the cases_final and t ₅₀ parameters are substantially higher than for the countries closer to the end of their epidemic. On the other hand, n _Hill relative uncertainties are only slightly higher for these bottom 10 countries, compared to the other countries (Figure 6). Table 2 affords a closer look at this effect.

FIGURE 6.

Relative uncertainties for cases_final (red), t ₅₀ (blue), and n _Hill (black), plotted against the percentage completion of the case load for all countries listed in Table 1

TABLE 2.

Average relative uncertainties (in %) for cases_final, t ₅₀, and n _Hill for all countries in Table 1

Completion	≥90%	70–89%	<70%
casesfinal	0.7%	1.5%	4% for Canada (67% complete) to 74% for Mexico (8% complete)
t 50	0.4%	0.8%	1.4% for Canada (67% complete) to 22% for Mexico (8% complete)
n Hill	1.7%	1.7%	1.3% for Russia (44% complete) to 4.5% for Peru (32% complete)

Note: Relative uncertainties are highest for the lowest completion % data sets.

It turns out that even the high uncertainties for the low‐completion data sets are woefully underestimated. And it goes almost without saying that an R ² value above 0.99 for fitting a data set that is less than 50% complete is meaningless. One way to examine this important issue is to take a complete data set, and remove points from the end, examining how this affects the fit. For example, Germany's data set was very well‐fit by Equation (4), with R ² = 0.9996, and its final reported number of total cases (from April 27) was 94% of the fitted cases_final value. Thus, its fit parameters are extremely well‐determined: cases_final = 171,400 ± 900; t ₅₀ = 42.44 ± 0.12 days; and n _Hill = 5.48 ± 0.06. Removing points from the end of the data set allows us to examine the fit parameters at 90% completion, 80%, and so on.

In Figure 7 we plot the fitted values for cases_final at completion percentages of 94% down to 14.5% (below 14% the fitting process failed). The first thing to notice is that the uncertainties in cases_final returned by nonlinear regression, plotted as vertical error bars, do not reach the value obtained for the full data set (dotted line). From this we learn that when using nonlinear regression to fit an incomplete data set, the reported uncertainties are greatly underestimated. The second observation is that below 30% completion, the cases_final value is far too low, by 2‐ to three‐fold. If your set of reported total cases ended within this range, that is, well below t ₅₀, then one cannot expect the fitted parameters returned by nonlinear regression to be accurate (compared to values obtained from the full data set). Similar errors were noted for fitted values of t ₅₀ and n _Hill below 30% completion (data not shown).

FIGURE 7.

Fitted cases_final values for the Germany total cases data set, from which points were removed from the end of to lower the completion % from 94% to 13%. Error bars are the uncertainties reported for nonlinear regression fits to Equation (4)

Another way to look at this problem is to examine t ₉₀ predictions at various points of completion %. Figure 8 plots t ₉₀ (calculated from the Equation (4) fitted parameters, using Equation (6)) for the Germany data set ending on day 31 (14.5% completion), day 32 (17% completion), etc., up to the full 68‐day data set (94% completion). Note that t ₉₀ rises dramatically from day 31 to day 39 (39% completion), after which it continues to rise more slowly all the way to the end of the data set on day 69 (94% completion). This receding “goal line” is another example of the dangers of trying to fit an incomplete data set.

FIGURE 8.

Calculated t ₉₀ values the Germany total cases data set, from which points were removed from the end of to lower the completion % from 94% (full set of points, 68 days) to 13% (30 days)

Consider the Germany data set ending on day 34, which is 22% complete. From fitting this incomplete data set, one would expect t ₉₀ to be day 47 (Figure 8), and the final number of total cases to end up around 75,000 (Figure 7). But moving forward 13 days to day 47, t ₉₀ is still 12 days away (talk about running and getting nowhere!), and the predicted final number of cases has more than doubled, to 155,000. Here is another example of this problem: An earlier draft of this paper included data up to March 31 (2020). At that point (day 39), the U.S. case load was 213,000, and fitting suggested cases_final = 440,000 ± 14,000 and t ₉₀ = day 51. As I write this on May 1 (day 69), the current U.S. case load is 1.1 million, and t ₉₀ = day 83!

The bottom line is that one has to take these fitting results with a grain of salt, especially when conditions such as testing frequency and social intervention change during the course of the epidemic. Having said that, the predictions do become fairly reliable above 60% completion. Of course there is an endless loop problem here, because one does not know the actual cases_final until the data set is complete, so your calculation of % completion will have an uncertain denominator. However, Figure 7 allows us to calculate that this error should be less than ≈10%.

A word is in order concerning the use of diagnosed cases, as opposed to hospitalizations or deaths. I have chosen to use reported cases because they are a current indicator of the epidemic, whereas deaths and hospitalizations are trailing indicators. However, it is definitely true that increasing testing during the course of the epidemic skews the fits. This explains, at least partially, the modest rise in cases_final seen in Figure 7 after 40% completion. The skewing effect of increased testing can also be seen in the slight upward tilt of the California total cases curve after day 48 (Figure S31), and in the increase in cases_final and t ₅₀, and decrease in n _Hill and completion % from April 27 to May 3 for almost all of the countries that were re‐fit on May 3 (Table 1). The good news is that hospitalization and death data sets follow the same trend as total cases (only delayed by 1 or 3 weeks, respectively), and can also be fit to the Hill equation (total cases, Equation 4) and its first derivative (daily cases, Equation 5).

Let us close with an example of the usefulness of this fitting procedure. On April 8 (2020), the governor of New York announced that, based on the models his experts were using, he expected the peak of the COViD‐19 outbreak would be reached in mid‐June. Assuming that by “peak” he meant the maximum of the new daily cases (first derivative) curve, simple plotting and fitting would have shown him that New York state had in fact already passed its peak of new infections earlier that week, on April 4!

Some experts predict that the world will have to deal with a resurgence of CoViD‐19 in the fall of 2020, and perhaps even in years afterward. Even if not, we are only one interspecies virus hop and genetic recombination away from the next epidemic. In the meantime, biochemistry students can be regaled with the unexpected usefulness of the Hill equation in predicting the progression of epidemics.

Supporting information

Figure S1: Neither t₅₀ (a) nor cases _final (b) vary in a regular way with n, nor does cases_final depend on t₅₀ (c). These parameters seem to vary independently of each other, although there may be a weak correlation between cases _final and t ₅₀ (c): countries with low t ₅₀ (< 36 days, inside the red rectangle) all have low cases _final (< 3000); all countries with the highest cases _final (> 150,000, inside the black rectangle) all have moderate to high t ₅₀ (≥ 40 days). The correlation is weak, because other countries with t ₅₀ > 40 days have low cases _final.

Figure S2: Total CoViD‐19 cases for China, from 19 Jan.–27 April 2020.

Figure S3: Total CoViD‐19 cases for S. Korea, from 17 Feb.–27 April 2020. Red open circles and solid line include all data; this is not a particularly good fit. From day 25 on it looks like the virus spread was dampened, perhaps by social intervention, e.g., extensive testing, contract tracing, quarantining, etc. The small blue solid circles and dotted line are the fit with days 25‐55 removed. Note the much‐improved fit.

Figure S4: Total CoViD‐19 cases for Taiwan, from 9 March–27 April 2020. Note the non‐zero y‐intercept of 35 cases (34.6 ± 2.7) on day zero. This signifies that Hong Kong had a total of about 120 cases present on the island before the epidemic began spreading via normal community transmission.

Figure S5: Total CoViD‐19 cases for Hong Kong, from 19 March–27 April 2020. Non‐zero y‐intercept of 122 cases.

Figure S6: Total CoViD‐19 cases for Austria, from 26 Feb.–27 April 2020.

Figure S7: Total CoViD‐19 cases for Australia, from 29 Feb.–27 April 2020.

Figure S8: Total CoViD‐19 cases for Thailand, from 9 March–27 April 2020.

Figure S9: Total CoViD‐19 cases for Luxembourg, from 9 March–27 April 2020.

Figure S10: Total CoViD‐19 cases for Switzerland, from 3 March–27 April 2020.

Figure S11: Total CoViD‐19 cases for Germany, from 26 Feb.–27 April 2020

Figure S12: Total CoViD‐19 cases for Israel, from 20 Feb.–27 April 2020.

Figure S13: Total CoViD‐19 cases for Chechia, from 1 March–27 April 2020

Figure S14: Total CoViD‐19 cases for Norway, from 26 Feb.–27 April 2020.

Figure S15: Total CoViD‐19 cases for Spain, from 26 Feb.–27 April 2020

Figure S16: Total CoViD‐19 cases for France, from 25 Feb.–27 April 2020.

Figure S17: Total CoViD‐19 cases for Italy, from 20 Feb.–27 April 2020.

Figure S18: Total CoViD‐19 cases for Malaysia, from 29 Feb.–27 April 2020.

Figure S19: Total CoViD‐19 cases for Portugal, from 2 March–27 April 2020.

Figure S20: Total CoViD‐19 cases for New York state, from 16 March–27 April 2020.

Figure S21: Total CoViD‐19 cases for Iran, from 20 Feb.–27 April 2020.

Figure S22: Total CoViD‐19 cases for San Francisco Bay Area, from 29 Feb.–27 April 2020.

Figure S23: Total CoViD‐19 cases for Turkey, from 9 March–1 May 2020.

Figure S24: Total CoViD‐19 cases for the United States, from 23 Feb.–27 April 2020.

Figure S25: Total CoViD‐19 cases for the United Kingdom, from 10 Feb.–27 April 2020.

Figure S26: Total CoViD‐19 cases for Belgium, from 29 Feb.–27 April 2020.

Figure S27: Total CoViD‐19 cases for the Netherlands, from 27 Feb.–27 April 2020.

Figure S28: Total CoViD‐19 cases for Canada, from 22 Feb.–27 April 2020.

Figure S29: Total CoViD‐19 cases Canada, from 28 Feb.–27 April 2020

Figure S30: Total CoViD‐19 cases for Japan, from 15 Feb.–27 April 2020.

Figure S31: Total CoViD‐19 cases for California state, from 4 March–27 April 2020.

Figure S32: Total CoViD‐19 cases for Russia, from 4 March–27 April 2020.

Figure S33: Total CoViD‐19 cases for Sweden, from 25 Feb.–27 April 2020.

Figure S34: Total CoViD‐19 cases for India, from 1 March–27 April 2020.

Figure S35: Total CoViD‐19 cases for Peru, from 6 March–27 April 2020.

Figure S36: Total CoViD‐19 cases for Brazil, from 3 March–27 April 2020.

Figure S37: Total CoViD‐19 cases for Mexico, from 10 March–27 April 2020.

Figure S38: Daily new CoViD‐19 cases for NY and California; last datum 8 April 2020.

Silverstein TP. CoViD‐19 epidemic follows the “kinetics” of enzymes with cooperative substrate binding. Biochem Mol Biol Educ. 2020;48:452–459. 10.1002/bmb.21397