Should I do laundry and get grocery delivery at the same day or separate days?

Posted by Yuling Yao on Mar 29, 2020.       Tag: decision theory  

So I have stayed in my room for more than three weeks, and by stay in the room I mean strictly staying in the room without even touching the knob of the outdoor once – at the risk of sounding creepy.

It seems I only have two exposures to outside risks: I need grocery delivery from whole foods market, and I need do laundry every week. Arguably both events have extremely low risk, but at the risk of sounding even more creepy, if I do want to minimize the risk, should I do laundry and get grocery delivery at the same day or separate days in a week?

There are obvious reasons for both argument:

The infection chance, as a function of virus amount in the aerosol or indeed any other surface, has to be (nearly) strictly convex. This is due to both the fact that (a) the probability measure lives in [0,1], an inverse logit type transformation of transformation carries linear functions to convex ones in the left part (unless I anticipate a more than 0.5 chance of infection in one laundry !) and (b) biologically the virus amount has to be meaningful after reaching certain threshold. As a consequence, if the potential virus coming from laundry and grocery delivery are independent (therefore linear additive), separating them into two days strictly decrease the expected chance of actual infection by Jensen inequality:

\[Pr(\mathrm{infection} | \mathrm{virus~amount~} A + \mathrm{virus~amount~} B ) > Pr(\mathrm{infection} | \mathrm{virus~amount~} A) + Pr(\mathrm{infection} |\mathrm{virus~amount~} B )\]

On the other hand, a concentred exposure can be sensible too. For one thing, I can save certain PPE (masks and gloves) by conducting events in the same time. Further, there are some concavities here: for example, the amount of elevator usage is concave, as I can arrange to carry the delivery package right after I finish the laundry in the basement. In general the marginal risk elevation as a function of exposure times is likely decreasing. This is like airplane safety, is the the fixed cost (airplane crashing rate during taking-off/landing; elevator use in laundry) predominate the variable cost, another Jensen inequality kicks in and you do want to choose a direct flight rather than two separated connected ones! \(Pr(\mathrm{virus~exposure} | \mathrm{event~}A + \mathrm{event~}B ) < Pr(\mathrm{virus~exposure} | A) + Pr(\mathrm{virus~exposure} | B )\)

I think in the airplane example, the risk is so low that the first factor is negligible (the Jensen inequality can also be expressed by second order Taylor expansion, but a logit type function will be nearly local in the very left end). In the Covid-19 case, however, both factors are weighted in. So there is probably an optimal soultion with randomized decision each week.