Developing a simple mathematical model for the Covid data

Having seen my straight line log graph for the Covid data, I set out to develop a mathematical model. The famous R number is the number of people that one person infects. So if the R number is 3 and there are 5 people with Covid, then each will infect 3 and the number infected by them all is 3 x 5 =15. This means that at each stage the number infected = R x N, the number at the previous stage. To find the increase in number you do RN - N, the number at the end - number at the beginning of each stage. In the picture below I developed that into dN/dt, the rate of change of the number. I then integrated to write N in terms of t. There's a mistake in the last equation that I've just noticed. I've changed the N to a t in red.

Now the problem then becomes "what is t?" In the model above it is the stage in virus transmision from one person to the next. It isn't time in days. It's not a conventional concept of time. But all the data I have is measured in days - numbers of people on particular days. So I adapted the equation by inroducing a term T which is the number of days that it takes 1 person to infect R other people. Then t in my equation below is in days. I took natural logarithms and rearranged to get the y=mx+c form for a straight line graph.  

The gradient is (R-1)/T. Problem is that we don't know T. We do know that the gradient is -0.0282. We also know that nationally the R number is between 0.7 and 0.9. Putting R=0.7 into the equation gives T as 10.6 days. Putting R=0.8 gives T as 7.1 days. Putting R=0.9 gives T= 3.5 days. These values are consistent with given incubation times. I believe that it is methods like this that are used to calculate the R value and the different models used must have different values for T. I have found more data to use to test my model.