WAIFW matrix

The ${\displaystyle 2\times 2}$ WAIFW matrix for two groups is expressed as ${\displaystyle {\begin{bmatrix}\beta _{11}&\beta _{12}\\\beta _{21}&\beta _{22}\end{bmatrix}}}$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \beta_{ij}} is the transmission coefficient from an infected member of group ${\displaystyle i}$ and a susceptible member of group ${\displaystyle j}$. Usually specific mixing patterns are assumed.^{[citation needed]}

Assortative mixing occurs when those with certain characteristics are more likely to mix with others with whom they share those characteristics. It could be given by ${\displaystyle {\begin{bmatrix}\beta &0\\0&\beta \end{bmatrix}}}$^[2] or the general ${\displaystyle 2\times 2}$ WAIFW matrix so long as ${\displaystyle \beta _{11},\beta _{22}>\beta _{12},\beta _{21}}$. Disassortative mixing is instead when ${\displaystyle \beta _{11},\beta _{22}<\beta _{12},\beta _{21}}$.

Homogenous mixing, which is also dubbed random mixing, is given by ${\displaystyle {\begin{bmatrix}\beta &\beta \\\beta &\beta \end{bmatrix}}}$.^[3] Transmission is assumed equally likely regardless of group characteristics when a homogenous mixing WAIFW matrix is used. Whereas for heterogenous mixing, transmission rates depend on group characteristics.

It need not be the case that ${\displaystyle \beta _{ij}=\beta _{ji}}$. Examples of asymmetric WAIFW matrices are^[4]

${\displaystyle {\begin{bmatrix}\beta _{1}&\beta _{2}\\\beta _{1}&\beta _{2}\end{bmatrix}}{\begin{bmatrix}\beta _{1}&\beta _{1}\\\beta _{2}&\beta _{2}\end{bmatrix}}{\begin{bmatrix}0&\beta _{1}\\\beta _{2}&0\end{bmatrix}}}$