Reproduction Number (R₀) and Contagion
by Patrick H. Ruane, April 13, 2020
The Reproduction Number (R₀), pronounced “R-naught” is a mathematical symbol that indicates how contagious an infectious disease is. R₀ tells you the average number of people who will catch a disease from one contagious person. It specifically applies to a population of people who were previously free of infection and haven’t been vaccinated.
If a disease has an R₀ of 10, a person who has the disease will transmit it to an average of 10 other people, as long as no one has been vaccinated against it or is already immune to it in their community.
If R₀ has a value of less than unity (R₀ < 1) then the disease will die out, an R₀ value of 1 means the disease will continue to exist but will not cause an epidemic. It is values of R₀ > 1 that are worrisome as this often leads to epidemic or, asin the case of SARS-CoV-2, a pandemic. The CDC has reported the R₀ for SARS-CoV2 to be 5.7 (95% CI 3.8 -8.9), this basically means that with 95% confidence, we know that between 4 and 9 people will be infected by each person that has SARS-CoV2.
Typical R₀ values for familiar diseases are approximately given as:
- Influenza 1–3
- Ebola 1–3
- Common Cold 2–3
- HIV 2–5
- Smallpox 3–6
- Mumps 4–7
- Polio 5–7
- Chickenpox 10–12
- Measles 12–18
- Stupidity 42
All this is well and good but we think the public care less about R₀ values and more about questions like — “When can we go back to normal?”. To this end we have attempted to build a mathematical model that is predictive.
Building a Covid-19 Mathematical Model
Building a cogent mathematical model for Covid-19 is not a trivial thing, and we admit up-front that we are amateurs and that the professionals will do a much better job. However, we also find that the professionals tend to over communicate to the point that the point itself is missed. Below we outline how we built the model, so feel free to skip to the results (David) if you are so inclined.
All models have their limitations and this model certainly has its flaws; however, we believe it has merit. We used the reported daily cases from Wikipedia (periodically this was checked against other sources and confirmed to be largely accurate). We recognize that factors such as incorrect government reporting, differing population densities, testing, social distancing, hygiene etc. have profound effects on the actual number of cases reported. Ultimately, we sought a single mathematical equation that roughly fits data sets for each geographical area (Country, State, County, City) and if successful we could use that equation to confidently predict when we/you can get back to normal.
We started with South Korea (sorry China, we didn’t fully trust your data). So, we plotted Figure 1 below:
Figure 1. South Korea
Looks like an S-Curve and luckily, we know the equation for S-Curves:
- Y = A * (1-e(-k*tⁿ))
- A = Number at the end of the S
- e = Euler’s number
- k = Rate of Growth
- n = 1, 2, 3 or 4
We guessed a value for A and solved the equation and plotted the data and got Figure 2, it certainly seems that in a contained system (in this case South Korea), SARS-CoV2 was growing like an S-Curve. We got a value of 3 for n and note that this is similar to how crystals grow in 3-dimensions within a system like a jar of water.
Figure 2. S-Curve Fit
Then we waited for the South Korea reported daily cases to drop to zero and of course they didn’t, the reason being that South Korea is not a simple contained system like a jar of water. Rather it is of course a complex society interacting with other complex societies. The data from South Korea after 52 days is shown in Figure 3. Try fitting that curve to a simple equation — we can’t and we don’t think you need to.
Figure 3. All Data
What we need for the Model to be predictive is a way to estimate the true value of A for each country, we assume other countries will have a prolonged lag phase of growth after the S-Curve analogous to South Korea. We identified an Inflection Point, where the S-Curve starts to flatten out, Figure 4.
Figure 4. Inflection Point
For South Korea the Inflection Point = 7,134. We then waited until we had a good handle on the true value of “A” — see Figure 3.
“A” is approximately 1.5 times as big as the value of the inflection point.
7,134 x 1.5 = 10,701
With values for A and n, we can now plot the South Korea model as if it were a simple system, recognizing that it is wrong with respect to what actually was reported, but also recognizing that it may very well be a good approximation, Figure 5.
Figure 5. South Korea Model
We could write pages and pages about the pros and cons of this model, but ultimately, we believe people fundamentally care as to whether it is predictive or not and quite frankly we don’t know yet, it might just be complete bollox. We are also not afraid to be wrong, and, besides this is much better than watching Netflix all day slowing down my internet while we are all in quarantine.
Our first prediction in adherence to Occam’s Razor is that we are probably wrong and will be adjusting the Model next week.
Our methodology is simple, we wait for the Inflection Point, multiply by 1.5 to give “A” and plot the S-Curve using a value of 3 for “n”, this gives us a value for “k” and that is really what we are interested in as it helps us understand how each geographical region did compared to others. If this works, this will generate a single objective value that speaks to directly as to whether you sucked as a nation/state/region or won a gold star.
Example 1 Italy
So, what are we saying?
We are predicting that in 30 days, May 13th 2020 that daily increases in the number of cases in Italy will be negligible and we guess that within another 30 days after that, things can go back to some semblance of normalcy.
So, what about other countries, states, counties, cities? The problem is, we cannot determine the true value of “A” until we see the S-Curve start to flatten and this has only taken place is a few regions.
Example 2 Australia
Australia certainly seems to have things under control and should be looking good by mid-May for a mid-June re-start. One caveat is that Australia is now entering its flu season, so the possibility of a significant uptick is definitely possible.
Growth Rate “k”
We have only identified the following geographical regions where an accurate value of “k” could be determined:
- China (CN)
- South Korea (KR)
- Italy (IT)
- Washington State (WA)
- Spain (ES)
- Germany (GE)
- France (FR)
We normalized the k values for ease of comparison and set KR at a value of 1, all other values are reported relative to KR. The higher the value the less well a region was in control of the situation. It is this plot that we look forward to as it will provide insight into how well your country did in fighting SARS-CoV-2 — don’t listen to talking heads on the TV, it’s mostly chewing gum for the mind, look at the data and decide what to do from there.
A snap shot of the data we have to-date is below in Figure 6.
Figure 6. Do you suck or not?
*A value of 14 does not mean you suck, it might, but we suspect we will see significantly higher values in the future.