Exponential Math and the Corona Virus

Ignoring Lava's ridiculous anti-science conspiracy theories ... this is a thread about math. So let's look at the math.

The function I am using pretty closely describes the behavior in the past couple of weeks. It is

f(d) = 780 * 1.25992015 ^ d

This is the exponential function that starts at 780 and doubles every 3 days. To use, just enter 'd' as the number of days since 3/25. That is how I calculate the "predicted" values in the OP.

I put the word "predicted" in quotes, because I am not really making a prediction. At some point I know the numbers will start to diverge from this trend (hopefully soon).

However I am defining what exponential growth means. If we continue seeing exponential growth, then the numbers I calculated will be close to the actual numbers.

The reason this is important is.

1) There is a desire to minimize the numbers and to talk about the curve flattening or reaching a peak. As long as the number continue the trend of exponential growth this is only wishful thinking.

2) As long as exponential growth continues, the impacts that people are hoping for aren't happening yet. These claims that we are reaching a peak, or that social distancing is working are worse then speculation, they contradict the actual data.

3) This math is scary. If the current trend of exponential growth doesn't change, we will reach a million American deaths in April. No one knows for how long exponential growth will continue. But we do know that exponential growth has been the trend for the past couple of week.

Don't minimize this. I accept that there is uncertainty. Sure, things could get better; there could be a surprise cure or the weather could magically fix the disease or some unknown magical demographic effect could save us.

There is no guarantee they will get better.

2 Replies

@maxdancona,

maxdancona wrote:

Ignoring Lava's ridiculous anti-science conspiracy theories ... this is a thread about math. So let's look at the math.

If you re-read my post https://able2know.org/topic/546733-1#post-6978791 you will understand why the curve slows down as awareness of the virus grows.

I use plain English, but if you really understood math, you would be able to see how the variables I describe affect the growth curve.

When you are using math to predict social phenomena, it doesn't do any good to pretend there is accuracy in numbers. There are too many variables and unique situations at the empirical level to have measurements and quantitative analyses describe anything resembling actual reality.

The best you can do is try to identify factors that will change your assumptions, e.g. that the virus spreads at one rate while people are unaware of it, and that the rate changes once people become aware and start social-distancing and quarantine.

Obviously humans aren't perfect, so you're not going to achieve either perfect avoidance of contamination or maintain contamination conditions that occurred before people knew the virus was spreading.

So you know that contamination will slow but not how much. Like I said, go back and read post https://able2know.org/topic/546733-1#post-6978791

0 Replies

Best guess from COVID-19 Expert Forecast Survey #6 is:

245,000 deaths in the US, with the peak coming in April or May.

There's still quite a lot of variability in the estimates though.

https://works.bepress.com/mcandrew/

2 Replies

@oralloy,

oralloy wrote:

Best guess from COVID-19 Expert Forecast Survey #6 is:

245,000 deaths in the US, with the peak coming in April or May.

There's still quite a lot of variability in the estimates though.

https://works.bepress.com/mcandrew/

I am looking now. It looks like Survey #5 (the predictions they made 2 weeks ago) were way off... on the low side. If I am reading correction, they predicted an 85% chance it would be under 20,000 new cases by March 23. The actual number was over 39,000.

1 Reply

@maxdancona,

It looks like they are doing a survey once a week. Although possibly if things get bad they will conduct more frequent surveys.

I expect that the estimates will get better and more accurate over time. But regardless, it should be a good resource for keeping track of what the current best estimates are.

0 Replies

@maxdancona,

I agree with your description and analysis of the spread of this infection. However, please recognize that this exponential growth applies to the cumulative total number of infections, and not to the number of active (i.e. infectious) cases present at any time. Eventually nearly everyone will become infected before we reach a point of herd immunity, though that may take a long time. We have already slowed the exponential growth of the infections by the social distancing recently applied ( the resulting growth is still exponential in character., but with a smaller exponent - and longer doubling time).

It appears that most of the governments actions are focused on slowing the rate of infections so that we can avoid over taxing our medical treatment facilities while we progress along the fairly long road to herd immunity

The reported data is subject to a host of indirectly related variables, mostly involving the effectiveness of our ability to detect and report new infections. Our recently expanding testing program will likely create a rise in the fraction of cases detected and reported, some of which appears to already show in reported data for the last few days. In addition, recoveries to this infection (which lasts up to three weeks from infection to resolution) will continue growing and limiting the growth of active cases.

Our collective herd immunity to the disease is likely a lot higher than the reported data suggests precisely by the likely large and growing number of mild, unreported recoveries (and immunities) out there. We've seen reports of an easily collected and processes blood test for the detection of virus antibodies. This will identify all active and recovered cases, and will in a number of weeks give us all a better handle on our current situation.

As others have noted above the introduction new and better treatment modalities (several possibilities out there, though none yet proven) may limit fatalities and the overall effects of the disease.

I've seen conflicting reports about the susceptibility of the virus to mutations. No or few mutations means long term effectiveness of vaccinations (when they become available), but it also implies continued high transmission rates (some viruses mutate their way out of existence or potential to harm us).

Ac an interesting aside, China's reported data suggests it has geographically confined the disease to the original Wuhan area, and has seen only a trickle of new cases over the past ten days with a continuing decline in the number of active cases. Is this true? I'm suspicious. With Active cases down to a very low level they're reporting a death rate (= deaths/closed cases) of 4%- much higher than other countries nearing this stage of containment. Does this suggest the number of cases might be much higher. China still runs the risk of a nucleus of infections reaching other densely populated urban areas and new explosions of the virus as a result.

2 Replies

@georgeob1,

Quote:

We have already slowed the exponential growth of the infections by the social distancing recently applied ( the resulting growth is still exponential in character., but with a smaller exponent - and longer doubling time).

This is simply untrue (at this point). Right now we are seeing the same exponential growth that we saw two weeks ago. We expect and hope that the social distancing will "flatten the curve" in the near future. But right now we don't see this.

The exponential function is the same now as it was two and a half weeks ago. The actual data is not showing any "longer doubling time".

The data is the data, you can't make up something that isn't there simply because you want it to happen. Part of the point of the thread is to show what exponential growth looks like (I used a doubling time of 3 days which may have been a bit slow).

Tomorrow will be the first calculated point... we will see how close the actual number is to the rate predicted by constant exponential growth.

1 Reply

@maxdancona,

maxdancona wrote:

Quote:
We have already slowed the exponential growth of the infections by the social distancing recently applied ( the resulting growth is still exponential in character., but with a smaller exponent - and longer doubling time).

This is simply untrue (at this point). Right now we are seeing the same exponential growth that we saw two weeks ago. We expect and hope that the social distancing will "flatten the curve" in the near future. But right now we don't see this.

The exponential function is the same now as it was two and a half weeks ago. The actual data is not showing any "longer doubling time".

The testing rate is also going up, and the people choosing to get tested are possibly more likely to have been exposed and that is the reason they are choosing to get tested.

0 Replies

@georgeob1,

Since this is a math thread... I am putting aside the fact that this is about human lives, but that is why understanding the scale of this is so important

Quote:

However, please recognize that this exponential growth applies to the cumulative total number of infections, and not to the number of active (i.e. infectious) cases present at any time

There is an interesting feature of exponential functions. In calculus terms we say that the derivative of an exponential function is itself. To simplify you can think of the "derivative" is the "rate of change".

This means that if the number of cases is an exponential function, then the rate of increase of the number of cases is an exponential function with the same growth rate as the cumulative number of cases.

You can look at the data here https://www.worldometers.info/coronavirus/country/us/. You will see that the "number of cases" and the "daily new cases" follow the same curve with the same growth rate and the same "doubling time". This is not surprising. As long as the growth rate is truly exponential this must be the case.

1 Reply

@maxdancona,

Thanks for for the unrequested tutorial on elementary Calculus. Yes, it is well known that if f(x) = e exp x, then , also for the derivative, f'(x) = e exp x .

However my central my point was that the cumulative case load which is indeed described by that exponential function, behaves very differently from the even more significant function describing the trajectory of the active case load i. e. the number of people able to transmit the disease to others. That value reaches a peak when new infections first equals new recoveries. It follows a bell curve with a maximum that coincides with the moment the total case curve reaches an inflection point at which the growth rate in case load first starts to decrease ( a sign change in the in the second derivative of the caseload logistics function marking the end of the exponential growth phase - which inevitably occurs. . Any sustained decline in the growth rate of active cases is an indicator that the inflection point for total infections and the maximum for active cases is near.

Right now we're in an early stage of the epidemic, just as you noted, and the rate of recoveries as reported is still small, but now starting to grow very fast. However we and, all the other countries involved, have large, but as yet unmeasured cadres of infected people who experienced very mild symptoms, who have recovered, and whose infection and recovery have gone unreported. These are a cadre of people with immunity for the disease and unable to transmit it to others

There are quick and easily administered blood tests near deployment that will identify people with antibodies for the disease in their blood: these are folks who either have the disease or who have recovered from it. The recovered subset of this cadre is unable to transmit the disease to others and, in addition their plasma can be used to very effectively aid in the treatment of other newly infected people.

1 Reply

@georgeob1,

OK. I have some quibbles with your description of math (particularly your use of the term "bell curve" which generally means a Gaussian distribution), but what you are saying is that at some point the rate of new deaths will go down. I agree with that.

It is a fact that right now we have constant exponential growth. We don't know when this "inevitable" inflection point will happen. The rest of what you are saying is speculation that we are not (yet) seeing in the data.

Looking at the data and the current trend, do you agree that we will likely have more than 10,000 US deaths by April 6th (actually more than 12,000 if the current exponential growth continues)?

Do you also see the possibility that there could be more than 1 million US deaths by the end of April, and that is with some divergence from exponential growth?

I feel you are minimizing the mathematical reality. So what do you think is a realistic prediction for the number of US deaths in April?

1 Reply

Code:





Number of US deaths from Covid-19

               predicted          actual

3/25          -----                780

3/28          1,560                1706                      

3/31          3,120 

4/03          6,240 

4/06         12,480 

4/09         24,960 

4/12         49,920 

4/15         99,840 



4/21        399,360 

4/27      1,597,440

In the OP on March 25, I calculated how many US deaths there would be assuming constant exponential growth. I used an exponential growth rate of 0.25992 (which corresponds to doubling every 3 days).

The actual data for the first prediction was 1706. This is higher than the prediction, mostly because yesterday was an especially bad day.

As of today, there is no evidence that the rate of exponential growth is decreasing.

I believe that it is very likely we will have over 10,000 deaths by April 6th. As people have pointed out, at some point the data will show a divergence from exponential growth as we reach a peak. However, I believe it is very possible for the number of US deaths to reach one million by the end of April (even with the curve flattening).

(the data for the actual number of US deaths is from https://www.worldometers.info/coronavirus/country/us/

1 Reply

@maxdancona,

maxdancona wrote:

Code:Number of US deaths from Covid-19 predicted actual 3/25 ----- 780 3/28 1,560 1706 3/31 3,120 4/03 6,240 4/06 12,480 4/09 24,960 4/12 49,920 4/15 99,840 4/21 399,360 4/27 1,597,440

In the OP on March 25, I calculated how many US deaths there would be assuming constant exponential growth. I used an exponential growth rate of 0.25992 (which corresponds to doubling every 3 days).

The actual data for the first prediction was 1706. This is higher than the prediction, mostly because yesterday was an especially bad day.

As of today, there is no evidence that the rate of exponential growth is decreasing.

I believe that it is very likely we will have over 10,000 deaths by April 6th. As people have pointed out, at some point the data will show a divergence from exponential growth as we reach a peak. However, I believe it is very possible for the number of US deaths to reach one million by the end of April (even with the curve flattening).

(the data for the actual number of US deaths is from https://www.worldometers.info/coronavirus/country/us/

You should make a point of noting that the testing rate is going up and some people who would otherwise die of other infections as a complication to their health being compromised overall by other factors are now being included in the COVID19 statistics without noting that their deaths are not directly caused by COVID19.

Statistics distort reality by failing to attenuate readers to the fact that the thing being counted/measured may or may not be the determinant factor in each unit of data reported.

To give a very blatant example, if we made a statistical analysis of how many people breathed nitrogen as part of a terminal condition that led to their death, the statistics would be very high, but since everyone breathes nitrogen all the time as the largest component of atmospheric air, it is obvious that breathing nitrogen is a negligible factor in the dying process. Nevertheless, you can count the number of dying people breathing it and find an extremely strong, significant correlation between breathing nitrogen and death.

So statistical research has the ability to mislead people into assuming conclusions from the data that aren't warranted.

1 Reply

@livinglava,

LivingLava is being ridiculous, but there is some interesting math.

1. A property of exponential growth is that subtracting an error rate doesn't change the rate of growth. You can try it yourself; subtract ten percent from each of these numbers... it doesn't change the trend and we still end up with over a million US deaths.

The function I used was

f(x) = 780 * (1.25992)^x

With a 10% rate of error (which seems quite a high error rate when you are talking about false positive for people being dead).

f(x) = 702 * (1.25992)^x

Note: that an error rate doesn't change the rate of growth. All it does is change the starting point. Math is great because you can check your answers to see if they are true. If you are so inclined, try the new function in your Spreadsheet and see if this function get the original answers - 10%.

2. The number of people who have died is the most certain number we have. People who die of Covid-19 will likely be in the hospital where there will be a post-mortem to confirm the cause of death. And... these are extra deaths in addition to the normal amount of deaths, so you can compare the number of deaths this month to the normal number of deaths.

1 Reply

@maxdancona,

maxdancona wrote:

LivingLava is being ridiculous, but there is some interesting math.

1. A property of exponential growth is that subtracting an error rate doesn't change the rate of growth. You can try it yourself; subtract ten percent from each of these numbers... it doesn't change the trend and we still end up with over a million US deaths.

The function I used was

f(x) = 780 * (1.25992)^x

With a 10% rate of error (which seems quite a high error rate when you are talking about false positive for people being dead).

f(x) = 702 * (1.25992)^x

Note: that an error rate doesn't change the rate of growth. All it does is change the starting point. Math is great because you can check your answers to see if they are true. If you are so inclined, try the new function in your Spreadsheet and see if this function get the original answers - 10%.

2. The number of people who have died is the most certain number we have. People who die of Covid-19 will likely be in the hospital where there will be a post-mortem to confirm the cause of death. And... these are extra deaths in addition to the normal amount of deaths, so you can compare the number of deaths this month to the normal number of deaths.

What do you say to the people who are pointing out that total death rates are not higher than usual due to COVID19?

Doesn't that imply that the people dying would have probably died of something else if the COVID19 virus hadn't been present to finish them off?

Maybe death can come in many different cloaks, and COVID19 is just one such cloak. How do you prove that right or wrong using only math?

1 Reply

@livinglava,

I don't know whether to move this over to the conspiracy thread or not. You are rejecting reality. (I would like to see a link to someone claiming what your latest claim).

1) Hospitals are being flooded with patients. Hospitals in New York are getting patient loads they have never seen before, and they are reporting more deaths then they have ever had to deal with before. This is reality being experienced and reported by real people. The math backs up this reality.

2) I would like to see a link for your crazy new claim that we aren't seeing a change in "total death rate". I just googled, and I don't see anyone claiming this. It is possible that people on a conspiracy website are either misinterpreting or misstating the data. I suppose if some idiot compares the 1,706 deaths so far from corona virus to the 2.8 million deaths in a normal year they might make a wrong conclusion...

That isn't a problem with the mathematics. It is a problem with the idiot.

1 Reply

@maxdancona,

maxdancona wrote:

I don't know whether to move this over to the conspiracy thread or not. You are rejecting reality. (I would like to see a link to someone claiming what your latest claim).

You have to know what 'reality' is in order to know whether someone is rejecting it or not. You don't know reality. All you know is how to play with categories and math.

Quote:

1) Hospitals are being flooded with patients. Hospitals in New York are getting patient loads they have never seen before, and they are reporting more deaths then they have ever had to deal with before. This is reality being experienced and reported by real people. The math backs up this reality.

Math backs up nothing. Math is an analytical tool. If you do math on BS, (BS+BS)/2 is still BS.

Quote:

2) I would like to see a link for your crazy new claim that we aren't seeing a change in "total death rate". I just googled, and I don't see anyone claiming this. It is possible that people on a conspiracy website are either misinterpreting or misstating the data. I suppose if some idiot compares the 1,706 deaths so far from corona virus to the 2.8 million deaths in a normal year they might make a wrong conclusion...

Which wrong conclusion do you mean, exactly?

Quote:

That isn't a problem with the mathematics. It is a problem with the idiot.

Apparently you think that because IQ scores are numerical, your math skills make you a psychologist. Good luck with that.

The bad news for you is that you can't add your IQ scores from different tests to get your total IQ. Your math might be right, though.

0 Replies

@maxdancona,

I agree with your assessment of the growth rate of deaths from this disease. Ours are indeed high and the trajectory hasn't yet shown the declining rates now evident in the data for Italy France and Spain. The Death rate ( = deaths/(deaths + recoveries) is itself distorted by the unknown number of "hidden" or unreported infections and recoveries. It will naturally start out very high and then decline as the active case load approaches a peak. Ours is high now at 38%, but rates in Italy, France, the UK , Netherlands, Norway and other countries are a good deal higher. All will decline as the epidemic progresses and recoveries catch up with infections. Much of this, of course depends on the number of days that have passed since the onset of the epidemics in the various countries. It is natural to compare simultaneous data among various countries, but more accurate to line up the data in terms of # days since the onset in each country. Unfortunately that['s hard to do.

The description of the normal trajectory of active cases in an epidemic as having the characteristics of a Bell curve is generally accurate, and consistent with the mathematical models used to describe epidemic. However the origin of them in this case is not from a Gaussian or normal statistical distribution. In science and engineering one frequently encounters the same differential equations describing analogous dynamics operating in very different physical applications. For example the surface wave resulting from a moving object at the interface between two mediums (think of a ship) obeys the same wave equation that governs the formation of a shock wave around an object moving through a single medium at supersonic speed. This has long been a technique used in the design of supersonic airfoils, using Schlieren imaging of a model of the airfoil immersed vertically in a shallow sheet of water flowing through a test rig. A cheap and accurate way to detect the details of the associated shock wave at various Mach numbers (thought takes some work to line up and scale the various physical parameters involved).

In the case at hand the Logistics curve and the associated differential equations accurately describe a number of physical phenomena of ehich epidemics is only one.

I'm not making forecasts of the deaths we may accumulate from this virus because there are simply too many variables that could affect the outcome.

3 Replies

@georgeob1,

georgeob1 wrote:

I'm not making forecasts of the deaths we may accumulate from this virus because there are simply too many variables that could affect the outcome.

Thank you for providing Maxadona with an example of how to include honesty regarding goodness-of-model-fit as part of mathematical analysis.

0 Replies

@georgeob1,

Quote:

he description of the normal trajectory of active cases in an epidemic as having the characteristics of a Bell curve is generally accurate, and consistent with the mathematical models used to describe epidemic.

I am almost certain that this is mathematically incorrect. This is math, so you could show me mathematically what you are talking about.

You can find the function for a "bell curve" and then show me how this matches even an idealized peak for the typical incidence rate in an epidemic. I don't think you can make this argument. But I am game.

Show me the math. Maybe we should make this another thread. But I am pretty sure you are wrong.

1 Reply

Exponential Math and the Corona Virus

Quick Links

My Account

able2know