This is technical note about the assumption of linearity of infection risk with exposure time
A concern about this idea is that people get infected very quickly by the presence of an infectious person. Thus it makes no difference if 5 days of work are shortened to 2. This concern is relevant if mean transmission rates per social connection is large, on the order of q~1/day or above.
This concern does not apply to the 4 work/10 lockdown strategy, which doies not depend on shortening exposure time but rather allows most of those infected during work to cease becoming infectious during lockdown. It is a valid concern for the 2 work/5 lockdown strategy.
A brief calculation suggests that, given what we know now, average transmission rate q is not large enough for this concern to be relevant. To estimate q, note that an infectious person infects R people on average during the infectious period of duration D. If that person has an effective social group of size K, namely meets K people routinely, the average transmission rate per person is q=R/DK. Typical estimates for COVID are R=2.4 people, D=3 days and K~10 people or more, and thus, transmission rate per social contact is q~0.08/day or less. Thus, risk of infection is almost linear in exposure time for exposures of hours or a few days. For example, exposure for 2 days is about 1.9 as infectious as exposure for one day, since (1-exp(-2q))/(1-exp(-q))= 1.92. Spending 2 days instead of five days at work reduces the probability of infection by an infected co-worker by approximately 0.4. Note that q is different from the SIR model parameter beta which is total rate of transmission, typically around beta~1/day, because beta is not normalized by the size of the contact group K.
This low rate of transmission per person explains why cases do not typically infect most of their household members. The probability of a household member to be infected after D=3 days of contact during lockdown is 1-exp(-qD)~0.3 given q=0.08/day.
However, there may be scenarios where this argument does not work, and infections can be quick. We just don’t know enough right now. For example, certain people or interactions might have much higher q, and others much lower q, so that average q is not a good measure. In that case, a 4/10 strategy is more robust.
