Thread by @PausalZ

Herd immunity is a far squishier concept then many seem to be describing in their "shielding" or "stratified herd immunity" plans. Here is the formula for herd immunity threshold for a SIR model

where \beta is the effective contact rate, N is the number of individuals, and r is the inverse of the duration

The threshold says if are above that level the disease will disappear / we expect no outbreaks of disease. However, that threshold is neither sufficient nor necessary

To show this, let's talk about a perfect vaccine. If you get this vaccine you are perfectly protected from the infection and thus cannot transmit it (everything also applies to imperfect vaccines but it's messier)

Blue circles are vaccinated individuals and red are unvaccinated

In all the following networks, the elements of R_0 are held constant. The network structure will change; but the \beta look the same despite that

Our R_0 will be 3, meaning the herd immunity threshold is 0.67 (but everything applies to other R_0 > 1)

Network 1: Random Mixing

When someone talks about a threshold for herd immunity, this is the underlying network of what they are generally talking about (setting aside WAIFW for the moment). The threshold calculation applies normally

Network 2: Mesh Network

Well it looks like a big ole net. We can guarantee no transmission with only 50% vaccinated. Less than the supposed threshold

"Paul, all you have shown is that you can effectively go below the threshold. That doesn't negate the whole concept. We might instead say we need *at least* the threshold"

Bad news my imaginary interlocutor

Network 3: Clustered Network

In a clustered network, we can go above the herd immunity threshold but still have outbreaks. The following network has 75% vaccinated but an outbreak would occur

This network is closer to how human interactions are actually structured. This is also closer to WAIFW setups (but WAIFW won't capture the power-law for degree within clusters that the above network has)

Network 4: Household Network

This last network is variation on the previous clustered network. It instead places people into households. Depending on the vaccination strategy used, we can go above or below the 'threshold'

If your strategy is to vaccinate the between-household contacts, you can be lower than the threshold

If your strategy is instead to vaccinate the persons with household-only contacts then you will have to go above the threshold

Ultimately, the (possibly) unobserved network matters. If your model only assumes some random mixing parameter for broad groups of people, your model is probably way too simple to say anything meaningful about what policies we should actually consider

Because I did a bad job of defining WAIFW: it is the Who Acquires Infection From Whom matrix. Rather than random edges between persons, WAIFW generates edges random at \beta_w for connections in the group and \beta_o for outside

Below is a 3 group WAIFW with different Pr of connections in each group (by color) and between groups. Essentially WAIFW is a stratified random mixing model