WAIFW matrix

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WAIFW stands for “who acquires infection from whom.” It is a matrix that describes how fast infection moves between different groups in a population, such as people of different ages. In an SIR model, the entries help calculate the basic reproduction number (R0) using the next-generation approach.

A simple two-group example:
[ [β11, β12],
[β21, β22] ]

Here, βij is the transmission rate from an infected person in group i to a susceptible person in group j.

Mixing patterns
- Assortative mixing: people tend to mix with others like themselves. Within-group transmission is stronger, so β11 and β22 are larger than β12 and β21. The matrix can be diagonal [β 0; 0 β] or a more general form with higher within-group rates.
- Disassortative mixing: people mix more with other groups than within their own. Here β11 and β22 are smaller than β12 and β21.
- Homogeneous (random) mixing: all entries are the same (ββ; ββ). Transmission is equally likely regardless of group.

Note that βij may not equal βji, so the matrix can be asymmetric.

The social contact hypothesis, proposed in 2006, says transmission rates are proportional to contact rates between groups: βij ∝ cij. This allows using social contact data to estimate WAIFW matrices.

In short, a WAIFW matrix helps model how infections spread between different groups and how different mixing patterns affect transmission.