Coronavirus growth rate and its possibly spurious resemblance to vapor pressure model

Question

I collected the latest data on the coronavirus from Johns Hopkins University as shown and fitted different curves to this data to model the relationship between the number of confirmed patients $P$ who are/were infected as function of time $T$ taking 20-Jan-20 as day 1.

As opposed to applying the standard SIR and SEIR I tried out simple curve fitting to see the general trend. The curve fitting software I used has well known models form different branches of science as well as we could build our own custom models which I did as shown in the image below. The score against the name of a model gives how well a model fits this data. The higher the score, the better the fit and the maximum possible score is 1000. Although we have data only for 18 days (as of 7-Feb 2 AM GMT) one model always kept appearing at top as the best fit and this was the vapor pressure model. After checking for various boundary conditions, I rejected many of models but I could not find any immediately reason to reject the vapor pressure model. Similarly when I modeled the number of reported deaths against time, the vapor pressure model once again gave the best fit which for which I could not find any obvious reasons to reject.

So I did some research on the vapor pressure model.

Basic concept of vapor pressure Because the molecules of a liquid $L$ are in constant motion and possess kinetic energies, at any moment some fraction of them has enough energy to escape from the surface of the liquid to enter the gas phase. This process, called evaporation, generates a vapor pressure $P_L$ above the liquid. Molecules in the gas phase can collide with the liquid surface and reenter the liquid via condensation. Eventually, a steady state is reached in which the number of molecules evaporating and condensing per unit time is the same, and the system is in a state of dynamic equilibrium. Under these conditions, a liquid exhibits a characteristic equilibrium vapor pressure that depends only on the temperature $T_L$. Volatile liquids are liquids with high vapor pressures, which tend to evaporate readily from an open container; nonvolatile liquids have low vapor pressures. When the vapor pressure equals the external pressure, bubbles of vapor form within the liquid, and it boils. We can express the nonlinear relationship between vapor pressure and temperature as an almost linear relationship using the Antoine equation.

$$ P_L = exp\Big(a + \frac{b}{T_L} + c\log T_L\Big) $$

Next I did some research what is known about how the coronavirus spreads and if it is related to liquids. Here is what I found.

How coronavirus spreads: When an infected person coughs or sneezes, they shed droplets of saliva, mucus, or other bodily fluids. If any of those droplets fall on you—or if you touch them and then, say, touch your face—you can become infected as well. Hospital for communicable diseases define exposure as being within six feet of an infected person for 10 minutes or longer. Time and distance matters.

The coronavirus spreads when it escapes from an infected person through microscopic droplets of liquid carrying the virus through air. I wonder this has anything to do with why the vapor pressure model keeps coming on the top as the best fit even though there is no apparent pressure or temperature as in the vapor pressure equation and I cannot see how they could be elated. May be all this just a mere coincidence.

As a bad scientist but a concerned human, I thought I must report this observation in case there is anything worth in it.

Question: Given this limited data what can we infer about the corona virus growth rate and how can we reject the vapor pressure model as a mere spurious correlation.

Note 1: I am well aware of spurious correlation. But with only 3 weeks of data, we many not be able to detect a different trend this early. Hence I am reporting the best fit regardless.

Answer 1

"If all you have is a hammer, everything looks like a nail." The dataset you have is small, possibly underrepresented, and of unknown quality, since it is argued that many cases could have not been diagnosed. You observe an exponential growth, a common phenomena in many natural and artificial processes. The curve fits well, but I'd bet that other similar curves would also fit well.

Notice that the the Antoine equation mentioned by you, is a very flexible one, since it can account for constant ($\alpha$), exponential ($b/T_L$), and linear ($c\log T_L$) growth curves. This makes it easy to fit to many datasets.

Moreover, with this kind of data, it may be harder to model it at early stages. Notice that you could fit a linear growth model to the earliest period. Later, quadratic may fit just fine. Later, exponential would fit better, where the exact rate may be hard to catch, since by definition "the more it grows, then the more it grows", and it may easy speed up quite rapidly. It may be easy to fit some curve to such data, but the best test of it, would be a test of time, i.e. validating it on future data.

Answer 2

The growth of infected cases $y$ is more or less exponential but the growth rate $c$ is not constant.

$$ \frac{\partial y}{\partial t} \approx c y$$

For instance, note in the graph how the change in cases from day to day depends on the number of cases in a particular day and the increase in cases is larger when the current cases are large. But, instead of a linear relation as with simple exponential growth, you get some curve that decreases in slope as $y$ becomes larger (or equivalent when time is further, the cause is not clear here).

There are many type of equations that model exponential growth where the growth rate $c$ is not constant. Many of these models look a lot the same when you are observing the growth for only a short period of time. Because then the variation in growth is not large and is easily approximated by one or the other. In our case, a simple polynomial fit is actually doing the best (in terms of less sum of square residuals).

But this fit should only be considered as an empirical relationship. There is not a strong underlying meaning, and if there is any meaning* then it is not tested by such fits and entirely hypothetical.

With these 18 data points, we only know that the relative growth has made some peak above 100% per day and after that decreased.

Is it because of some mechanism how the disease spreads, or is it because how the cases are reported (is the data clean)? When multiple interpretations are possible for the same curve (and the slight variations in residuals make not much difference in deciding which one makes a better explanation than the other) then we need more (and different) measurements in order to test the different interpretations.

Question: Given this limited data what can we infer about the corona virus growth rate and how can we reject the vapor pressure model as a mere spurious correlation.

There is indeed limited data to say which model is correct. However, for the case of the VPM model we can say that it is spurious and incorrect (we need not more data for this). We can say this using: logic of the mechanism (it doesn't make sense), expert knowledge, previous experience, the fact that curves look a lot the same on a small range (increasing the coincidence that curves look the same).

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*You could say that a relation like $ \frac{\partial y}{\partial t} \approx c y^n$ somewhat makes sense as some sort of growth based on a power of $y$ (a toy model would be the growth of a circle where the increase in the area of the circle relates to the circumference of the circle).

Comparison with larger data range

When we use data with a larger range (e.g. this data from wikipedia, which has at the time of writing points of 27 days and days 5-23 correspond to your data) then we can see how your VPM curve could coincidentally seem to fit(/explain) the data.

The VPM model is in the small range (from 5 to 23 days) approximately similar to a linear/polynomial model:

compare:

$$\left[ log(y) \right ]^\prime = \frac{y^\prime }{y} \approx a + bt$$

with

$$\left[ log(y) \right ]^\prime = \frac{y^\prime }{y} \approx a/t^2 + b/t$$

the latter can be approximately linear in a small range (note that the VPM actually already fails for the small values for your 18 data points, which you can see well on the log-scale, but these small values count less strongly in the sum of squared residuals; what the VPM seem to do well is fit the little jump with the 100% increase; in hindsight, we can say that this should be considered just fitting noise)

Predictions/extrapolation

Note 1: I am well aware of spurious correlation. But with only 3 weeks of data, we many not be able to detect a different trend this early. Hence I am reporting the best fit regardless.

It is not very useful to fit curves in this way. It doesn't tell much information when there is not a good underlying theory.

In the image below two models are extrapolated and they give a final number of 45.3k or 47.8k cases and at that point (after only 2 or 3 days) the growth rate is zero (according to those models).

This extrapolation is not very useful. We don't know whether the model is truly like a curve that happens to fit well (and we could devise many other curves that would fit equally reasonable).

There may be more parameters involved that we do not take into account. The fits with those polynomials are not expressing what happens outside the range. It is not difficult to imagine that the growth will be for a long time nonzero, this is a scenario that is not modeled by a 'random' fit with a polynomial or some other method like a curve fitting tool that just checks a limited set (67) of various models (that may have nothing to do with the situation).

This optimistic extrapolation with polynomial models looks even more dramatic when we look at the absolute growth of cases. Currently, this is in the ten thousands per day. The trend in the last days does not show that this is gonna decrease so quickly and it seems like we are gonna hit above 50 000 cases.