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Examples

The 2 × 2 {\displaystyle 2\times 2} WAIFW matrix for two groups is expressed as [ β 11 β 12 β 21 β 22 ] {\displaystyle {\begin{bmatrix}\beta _{11}&\beta _{12}\\\beta _{21}&\beta _{22}\end{bmatrix}}} where β i j {\displaystyle \beta _{ij}} is the transmission coefficient from an infected member of group i {\displaystyle i} and a susceptible member of group j {\displaystyle 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 [ β 0 0 β ] {\displaystyle {\begin{bmatrix}\beta &0\\0&\beta \end{bmatrix}}} ³ or the general 2 × 2 {\displaystyle 2\times 2} WAIFW matrix so long as β 11 , β 22 > β 12 , β 21 {\displaystyle \beta _{11},\beta _{22}>\beta _{12},\beta _{21}} . Disassortative mixing is instead when β 11 , β 22 < β 12 , β 21 {\displaystyle \beta _{11},\beta _{22}<\beta _{12},\beta _{21}} .

Homogenous mixing

Homogenous mixing, which is also dubbed random mixing, is given by [ β β β β ] {\displaystyle {\begin{bmatrix}\beta &\beta \\\beta &\beta \end{bmatrix}}} .⁴ 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 β i j = β j i {\displaystyle \beta _{ij}=\beta _{ji}} . Examples of asymmetric WAIFW matrices are⁵

[ β 1 β 2 β 1 β 2 ] [ β 1 β 1 β 2 β 2 ] [ 0 β 1 β 2 0 ] {\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}}}

Social contact hypothesis

For the psychological contact hypothesis, see 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, β i j ∝ c i j {\displaystyle \beta _{ij}\propto c_{ij}} and allows for social contact data to be used in place of WAIFW matrices.⁶

See also

• Mathematical modelling of infectious disease
• Next-generation matrix

References

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. 978-1-4008-4103-5 ↩

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. 978-1-4614-4071-0 ↩

3. 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. 978-1-4614-4071-0 ↩

4. 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 /wiki/Royal_Statistical_Society ↩

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

6. 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 /wiki/Doi_(identifier) ↩