Coronavirus: refining the model

The simple model described in the post from 4/2/20 is an idealized, but unrealistic representation of reality. In fact, we can only observe the data that we measure, in this case by testing people for coronavirus. There is no uniformity as to who gets tested or when, or whether some people are infected, and can spread the virus, yet do not show up in reported statistics. We can make some simple additions to our model to provide, again idealized, methods for incorporating these factors. To do this, we make an assumption that people will mot be tested for coronavirus until some number of days after they are infected. This accounts for incubation period, delay in getting tested once symptoms appear, as well as the lag in obtaining test results.

To do this, we make an educated guess as to a reasonable number of days, call this N, to account for these lags and delays, say for example six days. We then go to our spreadsheet and create a new column which we will begin N+2 rows from the top. In this cell we put the value from the same row of the column representing the new cases, C in our example. However, we also want to account for the fact that not everyone who is infected will be tested or have a positive result. This may be because the test result is a false negative, or because the person never develops concerning symptoms, because of logistic constraints, or social factors, etc. We account for this by multiplying the daily new cases by some factor. for example 0.75. In the cell below we add this value to our adjustment factor times the next daily new case value. Highlight the cell and drag the right hand corner down to the end of our simulation. This column now reflects a modeled population of test positive coronavirus patients. although it assumes no false positive results.

If we do this, we note that the initial ratio of observed daily new cases to total cases overstates the value of r, but as the epidemic proceeds, this value tends to zero while the value of r decays to some baseline number. We also note that, as might be expected the actual peak in the daily new cases occurs N days before the measured peak. this difference is shown in Figure 1 which compares the actual number of new infections (red) per day to the measured daily change in the number of infections; i.e. the numbers that are actually reported (green).

Figure 1.

This gives some insight into the effects of testing, and the various omissions and delays associated with tracking the spread of the virus using reported numbers.