Copyright © 2020 jsd

Some Simple Models of the Spread of Disease

1  Overview

These diagrams pertain to New York City. Figure 1 shows – roughly – what would have happened if there had been no stay-at-home order. It is plotted on a linear scale. The area under the curve represents the number of deaths. In particular, the magenta shaded area shows how bad this scenario is, compared to the nicer scenario shown in figure 5.

In this scenario, the disease runs its course. That is, it keeps spreading until essentially everyone has been infected, and has either recovered or died.

Actually it’s much worse than that, because the disease runs its course so rapidly that the healthcare system gets overwhelmed. This simple model does not (yet) take that into account.

The only thing limiting the spread is herd immunity, i.e. the fact that you can’t give the virus to somebody who has already had it.

Figure 2 uses a semi-log scale to present additional detail about how the calculation was done. Details coming soon.

nyc-cv-400-400-linear   nyc-cv-400-400-log
Figure 1:   Figure 2:

Figure 3 shows – roughly – what should happen if the stay-at-home order (imposed March 20th) is 50% effective at reducing the virus’s ability to spread. The crucial Rk0 parameter has been cut in half.

The thin purple line shows the previous scenario (no intervention) just for reference.

It must be emphasized that cutting the rate in half does not cut the number of deaths in half. The disease still runs its course. Almost everyone still gets infected.

nyc-cv-200-200-linear   nyc-cv-200-200-log
Figure 3:   Figure 4:

Figure 5 shows – roughly – what should happen if the stay-at-home order reduces the Rk0 parameter by a factor of 4 relative to the no-intervention scenario.

This has a much more dramatic effect. Not everybody gets infected.

nyc-cv-100-100-linear   nyc-cv-100-100-log
Figure 5:   Figure 6:

Figure 7 extends the previous scenario by showing – roughly – what should happen if the stay-at-home order is simply lifted on May 1st, as the #pussygrabber is calling for. The magenta-shaded region shows the number of deaths caused by this depraved decision.

The only way to lift the lockdown without causing a disastrous secondary outbreak is to have in place comprehensive testing, vigorous contact tracing, scrupulous isolation of persons who might be contagious, and numerous other measures. Alas, as far as I can tell, none of those measures are properly implemented now, and we are not even on a path to get them implemented any time soon.

nyc-cv-100-400-linear   nyc-cv-100-400-log
Figure 7:   Figure 8:

2  How the Models Work

Details coming soon.

The calculation starts from the published data on the number of deaths. We completely ignore the data on “confirmed cases” because it is grotesquely unreliable.

There is a differential equations that calculates the number of new cases tomorrow, based on the number of active cases today.

This involves a convolution that takes into account the fact that each infected person is contagious for a number of days. There is a short incubation phase before the contagious phase begins.

The parameter Rk1 applies before the stay-at-home order. The parameter Rk2 applies after.

3  Discussion

It is worth noting that the concept of flattening the curve is valuable, but it is not synonymous with stopping the spread of the virus.

At this point, the possible scenarios fall into three main categories. We can give them letter grades as follows, based on the number of deaths that result:

C: The spread of the virus gets stopped. Not everyone gets infected.

D: The infection runs its course. Virtually everyone gets infected.

F: The infection runs its course so quickly that healthcare systems are overwhelmed. This greatly increases the amount of suffering and death.

Within these main categories there are sub-categories, including C-minus, D-minus, and F-minus, based on how much needless economic damage is caused.

The opportunity for grade-A and grade-B outcomes was squandered many weeks ago.

Copyright © 2020 jsd