Okay, after that long and boring Part 1, today gets even better because we have some math. The first post discussed the scheme and flow of the Minnesota model for the coronavirus epidemic. This post looks at how the model actually works as it is run against the population. So you have to have a starting point. The model is designed to operate over time, so you need a time”step” or unit of time in which your calculations are performed. The Minnesota model, like most, uses a day as the time unit and it starts on March 23, 2020, and ends its current model runs on March 22, 2021.
Now you have to decide what the world looks like on Day 1, when you start your model. You have your population, which is the State of Minnesota. You have loaded certain information about that population, primarily age and whether or not they have any one of several comorbidities. Those are fairly easy inputs to load. It is harder to identify how many people are infected on that Day 1? That is a big unknown. The modelers’ approach was to take the number of infections based on positive test results on that day, 169 Minnesotans, but assume that not every actual infection had been picked up. So how many actual infections were there? First they say that they multiplied this number by a factor (not described) “to reflect the probability that an infected individual would be detected at that time given testing policies and availability.” Sounds reasonable but they must not have liked the result they got, because then they switched to making the detection rate fit death counts in Minnesota. They worked backward by saying we think the infection fatality rate is X%, Minnesota had Y number of fatalities on April 5, so for the model to fit those fatalities, there must really have been Z number of infections existing on March 23rd. In doing this they used the very high infection fatality rate assumptions from other models, rates which are now clearly wrong. In any event, they decided that only 1% of all infections had been detected on March 23rd, so the number of Infected persons the model starts with is 16,900.
Now remember in the first post I mentioned that little delta input, the detection rate, that was used to account for asymptomatic cases in determining peoples’ destination from the Infected bucket to Hospitalized, ICU or Recovered. From March 23rd on the modelers set this input far higher than 1%, believing this better fit reality between March 23rd and early April, although 1% better fit reality on March 22nd. They set the detection rate for the base case at 75%. So somehow, magically on March 23rd and going forward, instead of only 1% of infections being detected by test results, we went to 75% being detected. That is pretty impressive. It also didn’t happen, and the increasing evidence is that in fact, as few as 5% or even 2% of cases have been identified by testing. As we pointed out yesterday, this is potentially the greatest flaw in the model–the assumption that we are only missing 25% of all actual cases.
So now we have our starting population of 16,900 Infected Minnesotans and they are out there in society on Day 1 mixing with the rest of the population according to the contact model. So our Susceptible group on Day 1 is the population of the state minus those 16,900 already Infected persons. Every day the model is run the Susceptible population is going down by the number of people who are Exposed on that day according to the contact model. So every day the Susceptible number is going down and the Exposed bucket is getting people added to it.
Also, starting on Day 2, you have some people moving from the Exposed bucket to the Infected bucket. It is not many initially, because while there is a range of times that a person might spend in Exposed, the average assumed was a 5 day incubation period. So on average people sit for 5 days in the Exposed bucket, but some go to Infected earlier, some go later, than 5 days. So every day, some people are going from Susceptible to Exposed and some people are going from Exposed to Infected. The sole regulator of how fast people move from Susceptible to Exposed is the contact model and the sole regulator of how fast people move from Exposed to Infected is the five day incubation period. And as we pointed out yesterday, everybody in Susceptible moves to Exposed at some point (again unless there is some limit in the model that is not apparent from the materials) and everybody in Exposed moves to Infected. The size of the Infected bucket started at 16,900 people and we will see in the next post the dynamics of how the size of that bucket changes.
Now keep in mind that at the end of any given day, the total population must equal Susceptible plus Exposed plus Infected plus Hospitalized plus ICU plus Recovered plus Dead. A person can’t be in two places at one time. Except in this model, as we pointed out yesterday, they can apparently be nowhere at all, in limbo, and that is what happens to 25% of the population. That is pretty impressive too.