WAIFW matrix

In infectious disease modelling, a who acquires infection from whom (WAIFW) matrix is a matrix that describes the rate of transmission of infection between different groups in a population, such as people of different ages.^[1] Used with an SIR model, the entries of the WAIFW matrix can be used to calculate the basic reproduction number using the next generation operator approach.^[2]

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 ${\displaystyle \beta _{ij}$ 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]

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

The social contact hypothesis was proposed by Jacco Wallinga [nl], Peter Teunis, and Mirjam Kretzschmar in 2006. The hypothesis states that transmission rates are proportional to contact rates, ${\displaystyle \beta _{ij}\propto c_{ij}$ and allows for social contact data to be used in place of WAIFW matrices.^[5]

• Mathematical modelling of infectious disease
• Next-generation matrix