R Stuff Kind Of?

We want to consider what this R metric is and if anyone can calculate it from the data given. Let us begin with the SIR model.

S= susceptible individuals

I= Infected individuals

R= Recovered individuals

Let us assume that we want to see these as a function of time and that there are no births or deaths.

We know that:

where N is the total number of individuals. We further assume we have no change in N with time and further that all individuals are the same, no difference genetically, ages, etc.

Thus we can say:

Now let us look at each rate of change separately.

 For S we have:

 where r is the infection rate. This is just posited as a model. r is assumed constant and also we assume again that all people are equal.

For I we have:

Now a is the removal rate, namely what gets cured.

 Finally for R we have:

Now we also have:

The reproduction rate is defined as:

If R0>1 we have an epidemic and if R0 < 1 we have a stable infection. Thus we get the R0 term. It follows from an analysis of the stability of these nonlinear differential equations as shown in phase space.

The question is: how do we determine R0 from data collected? Namely we need to estimate a and r. How is that done?

 1. a is the rate at which the infected recover. Thus is we look at the curve of rate of recovery versus number of infected we can estimate a is we assume we use this highly simplistic model. BUT, and this is critical, we assume a is a constant and is equal for all people everywhere, a highly unlikely assumption.

 2. r is a bit more complicated. We can write the equation in I(t) as follows:

Thus we can plot the ratio of the rate of change of I to I, offset by a, as a function of S(t) and get the slope as an estimate of r.

Therefore we have a way to get a and r and thus R0. Kind of!

 There are fundamental flaws in this model.

 1. What is the susceptible base? Everyone, some, groups.

 2. a and r are most likely time and spatially variant. You cannot use the same values for everyone. This is the most serious element of the model's defects.

 3. What about deaths? This model does not include any.

4. The biggest problem is the data itself. This model may have a chance to work if and only if the data has integrity. There are two elements of integrity. First we have been able to measure all the people to ascertain the infection rate. We have not. Second, that the data is correct in a timely manner. It is not. We have repeatedly shown that. Thus any measures based upon the data are fundamentally flawed.

5. When I then try to calculate the results I get absolute junk. Why, because of the above plus the fundamental flaw in all the assumptions.

 The list goes on.

 Thus one wonders why we use this metric for anything. In New Jersey we have people using this as the determinant when it is grossly fallacious. It represents nothing. In New York we have transparent metrics such as new cases, new deaths, hospital beds etc. Everyone can see and understand them. In New Jersey we have a Governor who refuses to detail his Rt metric, a variant of R0 one could assume. Thus in New Jersey it appears we have a faulty variant of a fallacious metric. Great!

One truly wonders what brains ever put this together. They persistently state science and numbers and this is at best a guessing game based on total unknowns.

We have used the data from the State to plot R0 and its components calculated as we have described above.

Note we are at R0 of 1.0. It is not clear how the State calculates their Rt since many seem to refer to it but it without there being a standard.