An algebraic approach to that lockdown paper

I just posted about this paper and I realised there is a simpler way to poke a hole in it.

The paper says

We define the dependent variable as the daily difference in the natural log of the number of confirmed cases, which approximates the daily growth rate of infections

and then defines a linear model for g. The details of the model are not important, let's just assume that nothing changes at all. What happens if we let the epidemic play out with a fixed g?

The important point is that they have used the cumulative confirmed case numbers. So let C_n be the cumulative daily totals of cases. Then
g = ln(C_n+1) - ln(C_n)
g = ln(C_n+1/C_n)
e^g = C_n+1/C_n
C_n+1 = e^gC_n

So their model for COVID19 with everything else held fixed, gives exonential growth directly in C_n. This is actually OK if the growth initial rate never changes but of course the whole point of this paper is to study changes in growth rate. C_n+1 is just C_n plus tomorrow's new cases. That depends only the reproductive rate of the virus and how many cases there were about a week ago. How many cases there were a few weeks or months ago has no place in the calculation. This is clearly not a valid model.

This is a much simpler way to see the flaw in the paper but unlike my earlier post, it doesn't give any insight into how this error skews the results in favour of earlier interventions.