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 Coronavirusâ€“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:
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:
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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