Covid-19 Predictions Using a Gauss Model, Based on Data from April 2 ^†

Abstract

We study a Gauss model (GM), a map from time to the bell-shaped Gaussian function to model the deaths per day and country, as a simple, analytically tractable model to make predictions on the coronavirus epidemic. Justified by the sigmoidal nature of a pandemic, i.e., initial exponential spread to eventual saturation, and an agent-based model, we apply the GM to existing data, as of 2 April 2020, from 25 countries during first corona pandemic wave and study the model’s predictions. We find that logarithmic daily fatalities caused by the coronavirus disease 2019 (Covid-19) are well described by a quadratic function in time. By fitting the data to second order polynomials from a statistical ${\chi }^2$-fit with 95% confidence, we are able to obtain the characteristic parameters of the GM, i.e., a width, peak height, and time of peak, for each country separately, with which we extrapolate to future times to make predictions. We provide evidence that this supposedly oversimplifying model might still have predictive power and use it to forecast the further course of the fatalities caused by Covid-19 per country, including peak number of deaths per day, date of peak, and duration within most deaths occur. While our main goal is to present the general idea of the simple modeling process using GMs, we also describe possible estimates for the number of required respiratory machines and the duration left until the number of infected will be significantly reduced.

1. Introduction

Nowadays, numerous models to predict the spreading of infectious diseases like Covid-19 are available, for example the actively discussed susceptible-infected-removed (SIR) model [1,2,3,4]. Many of these models are either toy models using families of functions [5,6,7,8], machine learning [9,10,11], neural network [12,13], transmission [14,15], growth [16], agent-based [17], or Bayesian regression models [18,19] or they are so complex [4,20], by taking into account a wide range of factors, that simple predictions are not possible [21,22]. In times of the coronavirus epidemic, predictions such as the maximum number of fatalities per day or the date of the peak number of newly seriously sick persons per day (SSPs) are valuable data for governments around the world, especially those facing the beginning of an exponential increase of casualties, and we hope to serve the people in charge with the here presented approach. In particular, fast predictions on the course of the coronavirus disease are crucial for policy makers to optimize their management of the disease wave. To feed into the current debate on infectious disease models, we would like to propose a Gauss model (GM) as a simple, but effective description of fatalities caused by Covid-19 over time, similar to recent studies for the US [23] and for Germany [24]. In contrast to this previous work, we choose to use the logarithm of the reported daily death rates [25], instead of cumulative infections, as monitored input data and we also do not rely on doubling times.

The Gauss model maps time to the bell-shaped Gaussian function to fit existing data of deaths per day and country, and to use this fit to extrapolate the deaths per day to future times. Though the GM may appear too simple to be predictive, we can justify its use by several arguments: (1) the GM appears to be a special case of the SIR model, as recently suggested by a numerical study [26], (2) the GM is compatible with an agent-based epidemological model developed and presented in Appendix A, (3) the GM captures the data available today well, including the entire first epidemic wave in China, (4) epidemics are initially exponential and eventually saturating processes in cumulative quantities and thus give rise to bell-shaped daily quantities. A general ansatz for a sigmoidal time evolution involving a polynomial in the exponent of an exponential shows that coefficients of higher than GM order are not required to capture the existing data—all of which we will explain in detail.

The here explored GM does not present a model capable of similar mechanistic and causal richness compared with some of the existing infectious disease models. Our only addition is to note and use for predictions the macroscopic Gaussian nature of the time evolution of cumulative fatalities that is universal among all countries. The true performance of models and simulations in this pandemic might become clear only months or years from now [27]. Model predictions generally depend on the available data, which changes every day and makes systematic model comparisons challenging. For this reason, we set up a website (https://www.complexfluids.ethz.ch/corona) as part of our Supplementary Information that provides up-to-date data as well as predictions and compares these new predictions to the ones presented in this article based on data from April 2. The Supplementary Information also features GM fits for additional countries not considered in this work.

2. Results

2.1. Gauss Model (GM)

We model the time-dependent daily change of infections and daily change of deaths with their own, a priori independent, time-dependent Gaussian functions denoted by $i\left(t\right)$ and $d\left(t\right)$. Each Gaussianian is a bell-shaped curve, the black line in Figure 1A, characterized by three independent parameters: a width, a maximum height and a time at which the Gaussian curve attains this maximum height. For any value of these parameters, the general form of the Gaussian function—the bell-shaped curve in Figure 1A—is fixed, but the concrete fit to given data can be optimized, as illustrated in Figure 1B for varying parameters.

It must be emphasized that we model the daily change of deaths, in contrast to the cumulative number of deaths, more frequently available in public, because the change of deaths allows for a fit that emphasizes data around the peak time more than marginal data at the beginning or end of the pandemic wave, i.e., the time of interest for predicted quantities. We will explain this point in the discussion. The cumulative deaths are the sum of all previous daily deaths up to today, while the number of daily deaths in turn is the difference of two consecutive days in cumulative deaths. In Figure 1A, the red plot illustrates the cumulative number of deaths as a function of time for the respective daily number of deaths in the same panel.

2.2. Logarithmic Daily Fatalities Are Quadratic

Next, we fit a polynomial of second order to the logarithm of d of 14 countries as a function of time using a ${\chi }^2$-fit. The resulting quadratic fit is plotted in Figure 2. For the remaining 11 countries, similar fits could only be performed for the daily number of infections. It is worthwhile mentioning that we explored the possibility to obtain better fits using higher than second order polynomials, whose coefficient in front of the highest, necessarily even, power must be negative to ensure a sigmoidal cumulative number of fatalities. It turned out that all these higher order coefficients were negligibly small and varied considerably across countries, even taking on positive or negative signs for different countries. We concluded that these higher order coefficients fit more noise than signal. Therefore, we used a second order polynomial (the GM) to fit logarithmic daily deaths. Higher order coefficients might become of relevance for times well after the peak of the first wave of the pandemic, i.e., during times (i) for which predictions about required equipment are of much less relevance, and (ii) that may already be faced with the onset of a second wave.

We prefer to base any quantitative conclusions only on the number of deaths, and not on the number of infections per day. Deaths are better documented than monitored infections in nearly all countries. A death caused by Covid-19 is easier to count than an infection, which might cause no to moderate symptoms and hence might remain uncounted. Statistically, a constant fraction of infected die from Covid-19 at a later time after being registered as infected [27,28]. Thus, infection and death curves are equivalent descriptions of time evolution of Covid-19, and the coefficients characterizing their shape can be expected to be closely related. To demonstrate that infections and deaths follow the GM, we analyzed and show results for both measures.

2.3. The Fitted Parameters

Using the fitted polynomial coefficients we compute the three parameters of the GM, i.e., maximum height, time of maximum height, and curve width, for each country. For mathematical details, please refer to the Appendix A. To demonstrate the universal Gaussian nature of the daily fatalities over time d, we display them in Figure 3A,B, normalized so that all curves have unit width, maximum, and time of maximum. The same plots for the cumulative fatalities D are shown in Figure 3C,D. Daily infections i, daily fatalities d, cumulative infections I and cumulative fatalities D, all fit neatly onto the unit GM curve or its cumulative function, plotted in gray in the back for reference. China, which is the only country to provide data from its first pandemic wave for times greater than $0.6$ (in normalized units), fits to the GM well over the entire significant course of infections and fatalities. This sparks the hope that the used GM will have predictive power for the remaining countries also after the maximum. The fits already provide sufficient evidence that the part prior to the maximum is captured well by the GM. The resulting GM parameters are listed and plotted in

Table 1. Fitted GM parameters $w_d$, $t_{\mathrm{d},\max }$, and $d_{\max }$ for those countries, for which sufficient data about fatalities is available by the time of writing. Error bars reported for 95% confidence intervals. The table lists the three fitted GM parameters (width, time of peak, and peak value), followed by estimated cumulative number of fatalities $D_{\mathrm{total}}$, $D_{\mathrm{total}}$ per million people (MP), and the predicted dates $T_{\eta }$ from Equation (A10) at which the number of daily infected people had reduced to the level of $\eta$ of its maximum value. Number of inhabitants according to OECD [29]. For a plot of the fitted parameters see Figure 4. ${}^{*)}$ For China we considered only the data during the first pandemic wave, i.e., until a minimum in daily fatalities was reached on March 12.

${\mathit{\alpha }}_3$ Code	Country	${\mathit{w}}_{\mathit{d}}$	${\mathit{t}}_{\mathit{d},\mathbf{max}}$	${\mathit{d}}_{\mathbf{max}}$	${\mathit{D}}_{\mathbf{total}}$	${\mathit{D}}_{\mathbf{total}}$/MP	${\mathit{T}}_{1\%}$
AUT	Austria	13.2 ± 1.5	April 7 ± 21	21.6 ± 4.7	500 ± 170	60 ± 20	April 25 ± 44
BEL	Belgium	13.3 ± 0.6	April 14 ± 18	430 ± 26	10,200 ± 1100	890 ± 100	May 3 ± 19
CHE	Switzerland	11.6 ± 0.5	April 5 ± 17	63 ± 12	1300 ± 300	156 ± 37	April 20 ± 1
CHN	China ${}^{*)}$	15.7 ± 0.3	February 17 ± 3	95 ± 3	2600 ± 100	1.9 ± 0.1	March 11 ± 4
DEU	Germany	13.1 ± 0.4	April 12 ± 10	340 ± 6	7900 ± 400	95 ± 5	April 30 ± 11
ESP	Spain	10.3 ± 0.2	April 1 ± 6	960 ± 70	17,500 ± 1600	380 ±35	April 13 ± 7
FRA	France	15.2 ± 0.3	April 11 ± 8	980 ± 320	26,000 ± 9000	390±140	May 4 ± 9
GRC	Greece	7.0 ± 0.2	March 27 ± 9	3.8 ± 1.3	47 ± 17	4.4 ± 1.6	April 1 ± 7
IDN	Indonesia	11.9 ± 0.8	March 19 ± 22	5.3 ± 4.7	111 ± 91	0.4 ± 0.4	—
IRN	Iran	15.3 ± 0.1	March 25 ± 2	150 ± 14	4100 ± 400	51 ± 5	April 17 ± 2
ITA	Italy	12.4 ± 0.1	March 27 ± 1.8	832 ± 60	18,300 ± 1400	300 ± 23	April 12 ± 2
NLD	Netherlands	9.8 ± 0.1	April 2 ± 4	144 ± 23	2500 ± 400	147 ± 26	April 13 ± 5
PRT	Portugal	6.0 ± 0.1	March 29 ± 4	24 ± 4	260 ± 40	25 ± 4	April 2 ± 4
SWE	Sweden	12.6 ± 1.2	April 15 ± 35	162 ± 12	3600 ± 600	370 ± 60	May 1 ± 35

and Figure 4. For most countries the GM width is within 10 and 15 days, roughly half of all countries have passed their peak of daily fatalities already and the peak is roughly below 20 fatalities per day and per million people.

2.4. Additional Predictions

Using the GM, one can obtain predictions for the further course of the Covid-19 pandemic analytically from the three descriptors. We here present two possible applications: cumulative fatalities as a function of time and the maximum required number of respiratory equipment as well as its time point.

First, the time evolution of the number of cumulative fatalities D, plotted in Figure 5, can be obtained by summing daily number of deaths d, predicted by our model. In this figure, we rescaled all curves back to normal times so that the future course of cumulative deaths can be easily read-off. One can compute analytically that the date after which the new deaths per day have decreased to 1% of their maximum lies roughly 2 widths after the maximum of daily deaths, and the values, denoted by $T_{\eta }$, for each country can be found in Table 1. Already from visual inspection Figure 5 suggests for Italy and Spain to plateau first, while France will have to face an increasing number of fatalities considerably longer. It also anticipates the cumulative number of fatalities per million people over the entire course of the Covid-19 disease to be highest for France, Spain, and Italy.

Next, we estimate the number of required respiratory machines per date for the Covid-19 epidemic. We start by assuming the number of respiratory machines per day to be equal to the cumulative number of active seriously sick persons on the given date, where active means not yet recovered by that date. Each new seriously sick person per day (SSP) requires a respiratory machine for some days or even weeks before passing away or recovering from Covid-19. According to other works [28], people who have died from Covid-19 occupied respiratory equipment for an average of 7 days prior to their death, but respiratory equipment may be in use for up to about one month for cases that later recovered. Thus, we may roughly estimate the number of active SSPs per date as the sum of people that became seriously sick within the past 10 days. Please note, however, that we try to only conceptually link the GM to useful quantities, we leave a thorough search of exact numbers to the reader.

As a final step, we need to relate SSPs and deaths. Assume that each SSP dies with a constant probability $\gamma$ after some days, i.e., $\gamma$ times SSPS gives the daily fatalities some days ahead. Taking again numbers from Ref. [28], we could use that each deceased patient had used a respiratory machine for an average of 7 days prior to death and thus estimate the daily number of SSPs at a given date by the number of daily fatalities 7 days in the future divided by probability $\gamma$.

The result of the above estimate reveals that the required number of respiratory equipment itself is a Gaussian curve, roughly centered around the same date as the daily fatality curve, and its peak value is proportional to a multiplicative factor that depends on the width of the Gaussianian and ranges between $0.5$ and $0.9$ for the widths we found, the total number of fatalities $D_{\mathrm{total}}$, and the ratio of passing away as a SSP $\gamma$.

3. Discussion

The preceding section demonstrated how a GM can be used to obtain statistical predictions for a pandemic such as the corona pandemic. The presentation was intentionally conceptual, to convey only the principle idea of such a model. It is clear that such predictions might work better for some countries than others. This is also why the reader should not be distracted too much by the large error in the fitted or derived parameters of Table 1.

The question remains, though, how the GM could be justified. In the following we aim to present a number of arguments in favor of the GM that make use of sigmoidal nature of the saturating processes such as pandemics and include a numerical argument on the stability of models.

First, let us list some other ’microscopic’ models that can lead to a Gaussian dynamics of fatalities or infections. One of the more prominent ones recently appeared in the Washington Post [30]. Stevens investigated what happens when simulitis spreads in a town, if everyone in the town starts at a random position, moving at a random angle, infecting others upon collision, and recovering after a certain time. The simulated number of infected people rises rapidly as the disease spreads and tapers off as people recover—a bell-shaped curve. We recreated the simulations and found evidence for the applicability of the GM under many circumstances. These results are not reported here, but support our central assumption. From another recent work using a holistic agent-based model [31], where the agents adapt their behavior through artificial intelligence as part of the solution, there seems also evidence from the numerical results presented, that the number of newly infected may be well captured by a Gaussianian. We went on and developed a new agent-based stochastic model described in detail in Appendix B, that can be seen as an efficient implementation of the situation explored in the Washington post. Our implementation does not require studying trajectories and collision events. Under most conditions this model results in a perfectly Gaussian-shaped daily number of fatalities.

The key of these behaviors lies in the sigmoidal nature of saturating processes. The derivative of sigmoidal functions have a bell-shaped form, similar to a Gaussian function, but may be asymmetric in general. We here model the daily fatalities d, formally the derivative of the cumulative fatalities D. Since we expect cumulative cases D to be sigmoidal, this fixes the derivative, the daily fatalities d, to a bell-shaped form. Even though all pandemics thus give rise to bell-shaped d by this argument, the curve’s parameters might differ, influenced mostly by policy, health system and culture. The predictive power of our model rests on the assumption that these influences are encoded already into the early data of casualties, combined with the assumption that a certain principal shape or a certain principal model of all pandemics is fixed.

Why do we choose a symmetric bell-shaped form, the Gaussian function? We recognize that other models, such as Poisson or gamma functions that fade out slower after their maximum, might be alternatively used. However, a symmetric function results from an agent-based model (Appendix B), it is the simplest model among all bell-shaped functions and works well enough to convey the idea of such models. Second, the times of greatest interest to policy makers are until the bell-curve’s peak since once passed the health system should be able to cope. Sigmoidal functions are popular to model saturation processes such as cellular growth [32,33] or enzyme (Michaelis–Menten) kinetics. Numerous sigmoidal functions exist and they have been used in many variations to predict infectious diseases [34] and have even been linked directly to models such as the SIR model [35]. Not surprisingly, sigmoidal models are frequently used in times of coronavirus [36,37,38], most often using a (generalized) logistic function (Richard’s curve). One model particularly similar to our GM [23] also uses the Gaussian integral as a sigmoidal function with similar parameters. However, the study only applies to the US, for which we noticed a bimodal Gaussian nature of d and thus decided not to model it here. Moreover, the precise methodology remains elusive as of today when we submitted this manuscript, but we assume the author fitted the sigmoidal Gaussian integral function, not its Gaussian derivative. We believe this to be a crucial difference to our procedure, to be discussed in the next paragraph.

A major issue with sigmoidal models is that they are often prone to overfitting [39] and also we in our preliminary experiments found such sigmoidal fits to be sensitive to initial conditions and to often require a large number of parameters. Previously, people have tried to experiment with regularizations [35] to account for such instabilities. Instead, we here choose to fit the logarithm of daily change of cases d, not the cumulative cases D. The logarithm of d weights more evenly values close to the function’s maximum and disregards other values. We believe this leads to a more reliable model of d around its maximum, the turning point of D, and the time of interest since most relevant predictions such as peak of the pandemic, time point, and width of peak are focused around this maximum.

The above arguments explained the necessity for sigmoidal models, but we also see mechanistic problems in other types of models, such as exponentials. Many exponential models rely on doubling times [24], which require intense preprocessing of data, such as smoothing, and are model dependent. Please refer to the methods in Appendix A for further discussion of doubling times in the context of the here presented GM. Other exponential models report considerable deviations from an exponential nature, e.g., a power law behavior as the curve flattens [40]. Sigmoidal functions automatically account for exponential growth and subsequent flattening.

We must note that we aim to be compared with descriptive models, i.e., models that work only in the statistical limit of large data. In contrast, mechanistic models for infectious diseases [27] are able to study the effect of parameters such as policies, health system, or culture on the outcome of a pandemic, and thus provide more detailed predictions and recommendations.

4. Conclusions

The here presented GM allows for simple predictions of future course of the Covid-19 disease and we have provided first evidence that a GM is able to capture the time evolution of the daily fatalities and infections per country. Fitted models describe past data well, including data from the first pandemic wave in China. As long as conditions such as the degree of social distancing remain unchanged, the GM is able to capture model data created by an agent-based approach, as we have demonstrated.

The model is so simple that it can be reproduced and applied without detailed knowledge of epidemiology, statistics, or programming languages. There are many countries not yet drastically affected by Covid-19, which will likely change for many in the coming weeks, and the GM could for example be used to apply it to such countries as soon as sufficient data is available. Using the recipe presented here, interested readers are in the position to obtain estimates for the shape of the Gaussian curve for their country, state, community, and use this model to compute more quantities of interest, such as our sketch of how to estimate the maximum number of required respiratory machines and the date of this maximum demand. Knowing the time of maximum rush days of SSPs, the maximum number of SSPs and width could help the government and medical agencies in these countries to optimize the management of the disease wave by appropriate drastic actions for limited time. Moreover, fortunately, as our study here demonstrates, the time of peak of the disease wave differs among countries. Knowledge of these peak times and their durations allows other countries to help those who undergo the peak of the wave at a significantly later time, with breathing apparati and trained medical personnel for a brief predictable time.

On one hand we are afraid our predictions will become reality, on the other they are more optimistic than all (few) predictions we came across so far. Confronting these predictions and the method with reality will help to either establish or rule out the presented approach. The applicability of the GM to describe the pandemic and its peak times for all countries, we will be able to judge in the far future. We are monitoring this development as part of our Supplementary Information.

5. Note Added in Proof

By the time this manuscript was under review several of the online available GM predictions could be used and also verified. The total number of fatalities in Germany after the first pandemic wave, accompanied by border openings, was 7897 on April 15 [25], while the GM predicted $D_{\mathrm{total}}=7900\pm 400$. As of today, three weeks later, this number increased to 8598 due to a daily number of deaths that presently remains at a relatively low and almost constant level of about $d\approx 20$. Similarly, as of today, June 3, the cumulated number of fatalities are 670 (Austria), 9522 (Belgium), 29024 (France), 179 (Greece), 1698 (Indonesia), 33601 (Italy), 8012 (Iran), 27128 (Spain), 1921 (Switzerland), 5996 (Netherlands), 1447 (Portugal), 4542 (Sweden), to be compared with Table 1. The only two countries for which we couldn’t make predictions on April 2, the United Kingdom and the United States (

Table A1. GM parameters $w_i$, $t_{i,\max }$, and $i_{\max }$ for those countries, for which sufficient data about infections, but insufficient data about fatalities is available to us, as of April 2. The coefficient $i_{\max }$ we could not extract from the existing data with an error less than 100%. Parameters $w_i$, $t_{i,\max }$, $i_{\max }$ were used to create Figure 3.

${\mathit{\alpha }}_3$ Code	Country	${\mathit{w}}_{\mathit{i}}$	${\mathit{t}}_{\mathit{i},\mathbf{max}}$	${\mathit{i}}_{\mathbf{max}}$	${\mathit{I}}_{\mathbf{total}}$	${\mathit{I}}_{\mathbf{total}}/$MP
BRA	Brazil	11.1 ± 0.2	April 3 ± 8	800 ± 350	16,000 ± 7000	78 ± 35
CHL	Chile	11.7 ± 0.2	April 2 ± 7	321 ± 33	6600 ± 800	370 ± 45
GBR	Great Britain	20.0 ± 0.7	April 27 ± 14	—	—	—
JPN	Japan	38.0 ± 2.3	April 27 ± 22	195 ± 90	13,000 ± 7000	104 ± 54
SAU	Saudi Arabia	15.4 ± 0.6	April 2 ± 13	140 ± 23	3900 ± 700	120 ± 24
SRB	Serbia	10.3 ± 0.3	April 1 ± 8	113 ± 50	2100 ± 1000	290 ± 130
PAK	Pakistan	10.3 ± 0.3	March 28 ± 11	170 ± 80	3200 ± 1600	16 ± 8
PER	Peru	16.8 ± 1.3	April 9 ± 28	148 ± 64	4400 ± 2300	140 ± 70
POL	Poland	15.0 ± 0.4	April 7 ± 10	340 ± 50	9100 ± 1500	240 ± 40
ROU	Romania	18.9 ± 1.3	April 19 ± 26	690 ± 90	23,200 ± 4600	1200 ± 250
USA	United States	14.8 ± 0.2	April 14 ± 5	—	—	—

), as their coefficient $c_2$ of the GM had no sign within errors, turned out to suffer most from the pandemics, with 39811 and 107175 fatalities so far. A more detailed comparison is provided by our Supplementary website. The epidemological foundation of the GM will be a subject of our subsequent work.