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]

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.

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, 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.
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), Royal Statistical Society, 59 (2): 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, 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

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[1] Keeling, Matt J.; Rohani, Pejman (2011). Modeling Infectious Diseases in Humans and Animals. Princeton University Press. p. 58. ISBN 978-1-4008-4103-5.

[hens-2] 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.

[3] 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), Royal Statistical Society, 59 (2): 255–277, arXiv:0907.4000, doi:10.1111/j.1467-9876.2009.00693.x, S2CID 15947480

[4] Vynnyvky, Emilia; White, Richard G. (2010), An Introduction to Infectious Disease Modelling, ISBN 978-0-19-856-576-5

[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

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