Doubling – flattening the curve – part 4

After a month of indecision, action has come. By restricting peoples’ movement and association, transmission of the virus can be delayed, or stopped. Makes sense in theory — the rapid growth of cases will peak and then subside. That’s the idea.

But what does that mean practically?

From a numerical regression, or any kind of curve fitting, it is much better to work with a straight line than an exponential curve. Taking the logarithm of the total number of world wide cases, excepting China, gives a nice straight line. This corresponds to a exponential doubling of cases every four days. I have talked about this number 4, in my previous series of doubling posts.

Well, the curve flattening corresponds to the doubling number getting progressively greater, from 4 days, to 5 days, to 6 days, etc. In plain language, the number of cases will double at progressively further intervals. If the curve levels out, the doubling number will approach infinity, i.e. it will take a long, long time before the number of cases doubles. I am looking at the John Hopkins charts

Lets look at some real numbers. Today is March 20, 2020. There are 8398 cases reported in New York State. If we use the 4 doubling number, we get the million case days, formula,

(%i2) covid_million(c):=((6*log(10)-log(c))*4)/0.69
                          (6 log(10) - log(c)) 4
(%o2) covid_million(c) := ----------------------
(%i4) covid_million(8398),numer;
(%o4) 27.70876345924717

Or 27.7 days until there are million cases in New York, or four weeks, or April 17.

If the curve really does flatten, say by one, then the doubling number would be 5, or

(%i5) covid_million(8398)*5/4,numer;
(%o5) 34.63595432405896

The number of days until a million has been lengthened by a whole week, to April 24.

This is a very simple model based on the first order assumption that the rate of increase of the number of cases depends proportionally on the number of cases,

(%i7) diff(f(x),x)=gamma*f(x);
(%o7)                       -- (f(x)) = f(x) gamma

For large population sizes, this is a good assumption to make. Looking through my Elementary Differential Equations textbook, I find that epidemics have been modeled using the logistic equation which becomes important as the number of infected people reaches a significant fraction of the overall population. This is the study of population dynamics. I am concerned here with altering the current trajectory through the public in common action.

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