Modeling COVID-19: When will the peak occur in the US?

CDC Statement on first possible community transmission case in the US

CDC Statement on first possible community transmission case of COVID-19 in the US

So, given that the CDC recently confirmed the first possible instance of community transmission of COVID-19 (Novel Coronavirus) in the US, I thought I’d guesstimate roughly when the peak of the epidemic would occur in the US with some (extremely) rough modeling.

I’ll be modeling it using the usual logistical model (which I think turns out to be the wrong model for a virus… but let’s just run with it for now) you all learned in your first Differential Equations course. I’ve recently been brushing up on my differential equations (have been getting rusty) in Khan Academy: https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/logistic-differential-equation/v/modeling-population-with-differential-equations

The rate of change of the population N (in this case, Coronavirus cases) with respect to time can be given as:

dN/dt = r*N*(1-N/k)

Where r is the exponential constant (related to doubling-time) and k is the “carrying capacity”, i.e. max number of cases (not actually a good definition for a virus… but again, let’s run with it).

This is solved as: 

N(t) = N0*k/((k-N0)*e^(-r*t) + N0)

Where N0 is the population at time = 0.

At the beginning, the number of cases rises exponentially. Early research says the doubling-time of COVID-19 was 7.4 days in the early days of the outbreak in China, according to: Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia

This is related to the exponential constant by:
r = ln(2)/(7.4 days) = 0.09366853791 (1/days)

According to Wikipedia, 28% of the US population became infected with the Spanish Flu (carrying capacity?): https://en.wikipedia.org/wiki/Spanish_flu

And US population is currently about 327 million people, so we’ll use 91.56 million as our “carrying capacity”:

k= 9.156*10^7 or 9.156e7 (in more compact notation)

And since community transmission just started, we can set N0 = 1. Therefore our equation becomes:
N(t) = 9.156e7/((9.156e7-1)*e^(-0.09366853791 (1/days)*t) + 1)

If we plot t in days: 

modeled infected population vs time

modeled infected population vs time

https://www.wolframalpha.com/input/?i=9.156e7%2F%28%289.156e7-1%29*e%5E%28-0.09366853791*t%29+%2B+1%29+for+1%3Ct%3C365

So sometime before 200 days from now, COVID-19 should have peaked in the US. Taking the derivative with respect to time, we see there will be a period of about two months when the number of new infections per day will be super high:

US infected per day vs time

US infected per day vs time (simple COVID-19 model) link

This compares fairly well with the peak of deaths for Spanish Flu in the US:

(Thanks again Wikipedia: https://en.wikipedia.org/wiki/Spanish_flu)

We can try overlaying these, and we see that the width of the peak of infections is fairly similar to the width of the peak of deaths from Spanish Flu in the US.

About two months of chaos, potentially. And we have about 5-6 months until this peak.

My model is pretty terrible. A virus doesn’t really have a carrying capacity in the same way… But it does seem to have pretty similar characteristics. I know almost nothing about virus modeling, this is COMPLETELY an amateur, toy model. A guess. There are professionals (like the CDC and the WHO) who do this for a living and you should listen to them, not me. Also, obligatory relevant XKCD webcomic: https://xkcd.com/793/


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Chris Stelter

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11 Responses to Modeling COVID-19: When will the peak occur in the US?

  1. gbaikie says:

    I don’t know about it, but flu season is in the winter, and US is leaving the winter.
    And looks like the model is peaking in the summer.

  2. Jardinero1 says:

    I read the mortality stats yesterday. The stats are based on diagnosed cases in China. For diagnosed cases the overall mortality rate is about two percent, a little higher for men and a little lower for women. It has killed no children under 19. The mortality rate rises with age of the infected and co-morbidity factors like smoking and poor health to start with. There are also believed to be many un-diagnosed cases and the likelihood is that the un-diagnosed cases survived which lowers the mortality rate further. The other thing I read is that the virus does not survive long in the open air when the temperature is above 80 degrees. I think the infection rate is likely to drop as summer approaches and also less of a problem in equatorial regions that lack air conditioning.

  3. Chris Stelter says:

    Summer might slow down the spread but won’t eliminate it. Come fall, it will spread even faster.

    It has already spread in warmer areas. It’s spreading all over Italy and Iran, and cases have been found in Brazil and Australia.

    And let us hope it mutates to a milder strain over time. The Spanish Flu did the opposite.

  4. Utterly_Clueless says:

    Interesting article. Can the carrying capacity be better estimated by looking at the Diamond Princess numbers? I see something like 18.68% on the cruise ship.

  5. gbaikie says:

    “It has already spread in warmer areas. It’s spreading all over Italy and Iran, and cases have been found in Brazil and Australia.”

    Wuhan is just above 30 degree North [30.5928° N}. And it’s spreading in Northern Italy.
    Qom Iran is a bit higher north than Wuhan {34.6416 N}. Either Italy or Qom could matter in terms what particular weather conditions they happen to be having.
    But in coming weeks Wuhan and Qom could become significantly warmer {depending weather].
    Let’s do weather of Qom, highs of 60s and lows getting close but not at freezing for next week.
    Wuhan: looks raining for next week, lowest night time low: 44 F and highest week forecast high: 61 F {today} cooling to mid 50s for daytime highs. But warmer nights than Qom.

  6. gbaikie says:

    I have question {related this predicted spike}.
    The US has average temperature of somewhere around 12 C- sometimes it’s said to be around 11 C {or less} but whether include Alaska sway the average by 1 K or more. And I talking about most of US population or where most US population or people are or the continental US {48 states of US}. Or average US citizen’s average temperature of their region is about 12 C {12 C = 53.6 F}.
    If US average temperature is about 12 C, what is average summer time or say May thru Sept average temperature.
    Such average temperature doesn’t mean much- if want it have more significant meaning, you weight higher population density {major cities regions} in terms of their average temperature. Or where most Americans are when the peak occurs and what is average temperature when growth rate infection is increasing to large numbers.

  7. peterh says:

    If this projection is accurate, a peak in late August. Allowing for a slowdown in warm weather, this could be peaking in October or November.

  8. Robert Clark says:

    Some good news on the virus:

    CHINA REPORTED ZERO NEW LOCAL CORONAVIRUS CASES YESTERDAY
    “IT’S LIKE THE WAR HAS ENDED.”
    BY JON CHRISTIAN / 2 HOURS AGO
    https://futurism.com/neoscope/china-zero-cases-coronavirus

    Can you use the length of time the virus lasted in China and South Korea for estimating it for the U.S.?

    Bob Clark

  9. Wayne Maksylewich CIH, CSP says:

    Mr. Stelter has done some nice work. The newspapers have been reporting infection rates of 35-80%; and these seem to be LCL/UCL numbers from some mathematical models. However, my experience in modeling exposures to hazardous substances always shows significant differences between measured exposures and model predictions. The nominal infection rates in Wuhan were 0.3% and 20% for the Diamond Princess. The vessel was an excellent experiment in infecting population and the 20% is similar to the quoted number for Spanish Flu (see above from Wikipedia as 28% for the USA).

  10. peterh says:

    The data coming out of china is Not To Be Trusted.

  11. gbaikie says:

    –The data coming out of china is Not To Be Trusted.–
    True.

    –Can you use the length of time the virus lasted in China and South Korea for estimating it for the U.S.?–
    I think so.
    And I would look number of active cases.
    One could say purpose of measures taken is to avoid medical system over load.
    To avoid what happened in Europe, particular in regards to France and Spain.
    I would say Italy was sort of like New York State. Both could not even handle a normal Flu season. And both didn’t even take minimal steps in the beginning.
    Or both didn’t seem aware that they were incapable of handling a normal Flu Season- if they were, and could see a possibility of global pandemic, then they should have been at red alert at the start of this.
    I gear everything in regards to worst condition in any State, or at moment it is New York State and it might be possible that New York State improves and some other State is the worst.
    Right now, New York State has 75,629 active cases, if cases were decreasing and total cases was less than 10,000. The question become what States are the main problem.
    New Jersey has 21,900 active cases. And is smaller State and is increasing faster in terms new case. So those two states by them self is a reason to be basically doing what we are doing “national lockdown of varying degrees”.
    In terms of other States, it seems that Michigan is quite worrisome at the moment and we will see if Michigan can turn it around in next few days.
    I would if New York State can get below 10,000 cases and New Jersey gets below 5000 cases and all states can get or remain below 5000, then in terms of threat of over loading hospitals seems to be quite low.
    And US could consider helping other nations in some “serious way”, which at this point in time, we can’t- without adding significant risk to US population.
    I happen to think Brazil could be in trouble fairly soon. And it’s possible we might have to remain in our “general lockdown” because Brazil or other countries are screwing up- but talking about a delay in terms of less than a week. And in terms a goal of lower all death in US and lowering risk of an unexpected outbreak, extending for less than a week [to save perhaps millions of lives in other countries and possibly saving as much as a million American lives] would be a good plan. And it makes economic sense.
    The other aspect of general lockdown of US is it’s giving time {and US “lost time” due to our “testing problem”}. And other aspect is lockdown is lower overall death rates from all causes though as we go towards summer, the overall death rate of US normally lowers- or lockdown will not lower this rate as much in a month from now.

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