SIR models and Covid-19 spread
3 Estimating the parameters
Let us first gather the information we have some idea of. First there are the various exponential growth rates, γ. These are ranging between 0.15 and 0.24. I would add, that the exponential fit seems better in the mid-March period when the government was still not apparently doing much to stem the tide. Let us take 0.2 as this factor. Further, the reproduction number (R0) range was estimated by experts at 2–3.5. I will initially use the 2.5 value, again as I did mid-March.
The reproduction number gives us that:
\[\alpha S = 2.5\beta\]
We can then stuff this into:
\[\gamma = + \alpha S - \beta = 1.5\beta = 0.2\]
This can be solved to give \(\beta = \frac{2}{15} = 0.1333\ldots\) The average time to “recovery” is then \(\left\langle d \right\rangle = \frac{15}{2} = 7.5\) days. This seems to me a reasonable value, remembering that in some sense this is the period when someone is infectious, and “recovery” in that sense is when they get visibly sick and their contact with others is hopefully very restricted (~5 days according to reports). Full recovery for serious cases seems to take a number of weeks.
Feeding back the β value one gets:
\[\alpha S = 2.5\beta = \frac{1}{3} = 0.3333\ldots\]
Mathematical details (click to reveal)
For general values of γ and R0, more abstract algebra gives:
\[\beta = \frac{\gamma}{R_{0} - 1}\]
\[\alpha S = \frac{\gamma R_{0}}{R_{0} - 1}\]