Covid-19: Effect on R of Household Size

This short post makes a simple arithmetical point. I’ll be referring to it later.

Imagine a world where everyone has only two types of contact to whom they may, if infected, transmit the virus: (1) close, household contacts; and (2) casual contacts, at work or school, when shopping, on public transport etc, Nothing in between. That’s my model.

Let’s assume that if someone in a household comes down with Covid-19 everyone else in that household gets it as well. Slightly pessimistic, I know, but not totally unrealistic (if anyone has real-world data, please let me know).

I’ll also be using “average” in a slightly vague fashion, again for simplicity!

Now let’s consider household size.

If everyone lives in single-person households, all transmission is through casual contacts. The effect of household transmission on R is precisely zero.

If every household consists of 2 people, though, then, on average, each of those people will have passed the virus on to 0.5 people within the household.

If a household consists of 3 people, all infected, then one of them has passed Covid-19 to two others (or to one of the others who passed it to the third). So, on average, the 3 people have passed it to 0.67 people each, just within the household.

For 4 people it’s 0.75.

So, if the average household size is 4, and everyone within each infected household catches Covid, then household transmission alone makes a contribution of 0.75 to R!

Since the epidemic in a community grows if R > 1, it’s quite clear that household size is absolutely critical to the ability to control the infection rate. With household contacts making such a large contribution to R, very few casual contacts can be allowed if case numbers are to be prevented from increasing exponentially.

This is exactly what is observed, of course, with “large multigenerational households” more common in areas of high infection rates.

Just thought I’d mention it!