In the last blog it was shown that knowing Imax, the peak value for the number of Infected individuals, helped reduce the number of parameters needed to estimate the values of the other parameters for a curve fit. In the case of a Covid-19 fit the peak value is not always immediately known. At the peak we know that dI/dt=(rS-a)I=0 which implies that rS-a=0 but we don't know what the values of S are and only have values of I to work with. However we can evaluate d(lnI)/dt=rS-a and look for a zero of this function.
In the above plot of Moscow I data we see that this function is about to cross the zero horizontal axis if it hasn't done so already. To get an estimated set of parameters for the fit the value of Imax was arbitrarily assumed to be 11000. Then a search of the remaining parameters was done to minimize the value of the rms error for the fit.
New data will allow us to improve on this fit and get a better estimate of the maximum value of I.
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