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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]

Examples

The $2\times 2$ WAIFW matrix for two groups is expressed as ${\begin{bmatrix}\beta _{11}&\beta _{12}\\\beta _{21}&\beta _{22}\end{bmatrix}}$ where $\beta _{ij}$ is the transmission coefficient from an infected member of group $i$ and a susceptible member of group $j$ . Usually specific mixing patterns are assumed.^{[citation needed]}

Assortative mixing

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 ${\begin{bmatrix}\beta &0\\0&\beta \end{bmatrix}}$ ^[2] or the general $2\times 2$ WAIFW matrix so long as $\beta _{11},\beta _{22}>\beta _{12},\beta _{21}$ . Disassortative mixing is instead when $\beta _{11},\beta _{22}<\beta _{12},\beta _{21}$ .

Homogenous mixing

Homogenous mixing, which is also dubbed random mixing, is given by ${\begin{bmatrix}\beta &\beta \\\beta &\beta \end{bmatrix}}$ .^[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.

Asymmetric mixing

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

{\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}}

Social contact hypothesis

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, $\beta _{ij}\propto c_{ij}$ and allows for social contact data to be used in place of WAIFW matrices.^[5]

References

^ Keeling, Matt J.; Rohani, Pejman (2011). Modeling Infectious Diseases in Humans and Animals. Princeton University Press. p. 58. ISBN 978-1-4008-4103-5.
^ ^a ^b Hens, Niel; Shkedy, Ziv; Aerts, Marc; Faes, Christel; Van Damme, Pierre; Beutels, Philippe (2012). Modeling Infectious Disease Parameters Based on Serological and Social Contact Data - A Modern Statistical Perspective. Springer. ISBN 978-1-4614-4071-0.
^ Goeyvaerts, Nele; Hens, Niel; Ogunjimi, Benson; Aerts, Marc; Shkedy, Ziv; Van Damme, Pierre; Beutels, Philippe (2010), "Estimating infectious disease parameters from data on social contacts and serological status", Journal of the Royal Statistical Society, Series C (Applied Statistics), 59 (2), Royal Statistical Society: 255–277, arXiv:0907.4000, doi:10.1111/j.1467-9876.2009.00693.x, S2CID 15947480
^ Vynnyvky, Emilia; White, Richard G. (2010), An Introduction to Infectious Disease Modelling, OUP Oxford, ISBN 978-0-19-856-576-5
^ Wallinga, Jacco; Teunis, Peter; Kretzschmar, Mirjam (2006), "Using Data on Social Contacts to Estimate Age-specific Transmission Parameters for Respiratory-spread Infectious Agents", American Journal of Epidemiology, 164 (10): 936–944, doi:10.1093/aje/kwj317, hdl:10029/6739, PMID 16968863