SIR Model Revisited

I've gone back and cleaned up my basic SIR Model as described in a previous post. I have added some features and put some more thought into it. I hope to describe it here and what might be some of the ramifications. Given the current state of political affairs here in the United States and being the election season it has helped me wade through some of the effluvia that is always present in politics. Especially concerning this matter.

First off let me re-visit the basic SIR Model. Assuming only that the disease is quite transmissible and that I don't put any limits on the susceptibility of the population I get this picture for the susceptible (S), the infectious (I), and the recovered (R).

Basic SIR Model With No Quarantine.

The model indicates about what you would expect. Nearly everyone gets the disease and then is considered no longer susceptible and the number of the infectious drops to zero. The recovered curve assumes that immunity is conferred on previously infected people, and it also includes deaths. It should probably be called no longer susceptible as it has other implications.

In this case I have set the beta and mu for transmission and recovery rates such that I now have R0=2 or the rate of spreading to previously uninfected members of the population. I then looked at what the model says if we have a quarantine such that R0 goes to just less than 1 which implies the disease will eventually stop being transmitted. The model looks like this. Once again, the results are not surprising. Eventually the number of infectious people drops to zero.

Basic SIR Model with Quarantine.

In real life scenarios however a quarantine is a difficult thing to maintain so I took a look at a broken quarantine scenario. In this case I assumed the initial quarantine reduced the R0 to below 1 and then as conditions changed the R0 jumped back up to the value of 2. It looks like the following.

Basic SIR Model With Broken Quarantine.

What it shows is that the susceptible and recovered curves start heading back to what they were in the no quarantine case but there is a secondary bulge in the infectious curve. When I looked at that portion of the model a little more closely I observed this.

Which is the second wave that has been discussed by others. The key point in this graph is the time scale. For the model it's arbitrary. In real life time is not arbitrary so a quarantine is very time dependent. The question becomes how long can you maintain an effective quarantine and what does it cost you if it's broken.

I would posit that the highest forms of cost during a disease outbreak is death, intensive medical treatment and lost work time for sick people. I'll just look at the numbers for deaths which tends to be a harder number in spite of the metrics that some are using for making that determination. Deaths come in the no longer susceptible portion of the model or the recovered curve.

To make some sort of sense about the potential for deaths I needed to know about the numbers of people in the recovered portion of the model. If I integrate the curves over time in some fashion I get the total numbers of people at risk and assuming that I get a constant percentage of deaths I can evaluate different scenarios. 

Integrals of Recovery Curves

For the no quarantine case (R0=2) I can see I will get about twice the number of deaths for a quarantine that achieves a R0 just less than 1. For a large population that is a number that could be in the millions. For the broken quarantine scenario where I reduce the R0 to less than 1 and then have it go back to 2 I get approximately 2/3's of the deaths that I would have had for the no quarantine scenario.

Let me state one important caveat about this series of posts. At no time did I try to match any numbers or predictions. To get a good understanding of the problem I felt a solid qualitative understanding was the first priority.

So, let me state some preliminary conclusions. Could we have done better in the United States? Almost certainly, a quarantine for a longer period of time would have reduced the numbers of infected. Some jurisdictions allowed contacts to go unabated in high-population density areas. Additionally some very irresponsible decisions were made concerning already infected people. Some allowed the movement of infected people into susceptible populations which is diametrically opposed to any idea of a quarantine. As usual a spreadsheet containing most of this is available for download on my Google Drive.

Which leads me to my next post on the matter. How much does a quarantine cost? I've been exploring some economic models on this matter and hope to have some thoughts on them in the near future.