View source for WAIFW matrix

{{Short description|Tool for modeling spread of disease}}
In [[mathematical modelling in epidemiology|infectious disease modelling]], a '''who acquires infection from whom (WAIFW) matrix''' is a [[Matrix (mathematics)|matrix]] that describes the rate of transmission of infection between different groups in a population, such as people of different ages.<ref>{{cite book |last1=Keeling |first1=Matt J. |last2=Rohani |first2=Pejman |date=2011 |title=Modeling Infectious Diseases in Humans and Animals |publisher=Princeton University Press |page=58 |isbn=978-1-4008-4103-5}}</ref> Used with an [[Compartmental models in epidemiology|SIR model]], the entries of the WAIFW matrix can be used to calculate the [[basic reproduction number]] using the [[next-generation matrix|next generation operator]] approach.<ref name = "hens">{{cite book | title = Modeling Infectious Disease Parameters Based on Serological and Social Contact Data - A Modern Statistical Perspective | publisher = Springer | first1 = Niel | last1 = Hens | first2 = Ziv | last2 = Shkedy | first3 = Marc | last3 = Aerts | first4 = Christel | last4 = Faes | first5 = Pierre | last5 = Van Damme | first6 = Philippe | last6 = Beutels | year = 2012 | isbn = 978-1-4614-4071-0}}</ref>

== Examples ==

The <math>2 \times 2</math> WAIFW matrix for two groups is expressed as <math>\begin{bmatrix} \beta_{11} & \beta_{12} \\ \beta_{21} & \beta_{22} \end{bmatrix}</math> where <math>\beta_{ij}</math> is the transmission coefficient from an infected member of group <math>i</math> and a susceptible member of group <math>j</math>. Usually specific mixing patterns are assumed.{{cn|date=April 2023}}

=== 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
<math>\begin{bmatrix} \beta & 0 \\ 0 & \beta \end{bmatrix}</math><ref name = "hens" /> or the general <math>2 \times 2</math> WAIFW matrix so long as <math>\beta_{11}, \beta_{22} > \beta_{12}, \beta_{21}</math>. Disassortative mixing is instead when <math>\beta_{11}, \beta_{22} < \beta_{12}, \beta_{21}</math>.

=== Homogenous mixing ===
Homogenous mixing, which is also dubbed random mixing, is given by
<math>\begin{bmatrix} \beta & \beta \\ \beta & \beta \end{bmatrix}</math>.<ref>{{citation | first1 = Nele | last1 = Goeyvaerts | first2 = Niel | last2 = Hens | first3 = Benson | last3 = Ogunjimi | first4 = Marc | last4 = Aerts | first5 = Ziv | last5 = Shkedy | first6 = Pierre | last6 = Van Damme | first7 = Philippe | last7 = Beutels | date = 2010 | title = Estimating infectious disease parameters from data on social contacts and serological status | journal = Journal of the Royal Statistical Society, Series C (Applied Statistics) | volume = 59 | issue = 2 | pages = 255–277 | publisher = [[Royal Statistical Society]] | doi = 10.1111/j.1467-9876.2009.00693.x| arxiv = 0907.4000 | s2cid = 15947480 }}</ref> 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 <math>\beta_{ij} = \beta_{ji}</math>. Examples of asymmetric WAIFW matrices are<ref>{{citation | title = An Introduction to Infectious Disease Modelling | first1 = Emilia | last1 = Vynnyvky | first2 = Richard G. | last2 = White | date = 2010 | publisher = OUP Oxford | isbn = 978-0-19-856-576-5}}</ref>

: <math>\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}</math>

== Social contact hypothesis ==

{{for|the psychological contact hypothesis|Contact hypothesis}}

The social contact hypothesis was proposed by {{ill|Jacco Wallinga|nl}}, Peter Teunis, and Mirjam Kretzschmar in 2006. The hypothesis states that transmission rates are proportional to contact rates, <math>\beta_{ij} \propto c_{ij}</math> and allows for social contact data to be used in place of WAIFW matrices.<ref>{{citation | title = Using Data on Social Contacts to Estimate Age-specific Transmission Parameters for Respiratory-spread Infectious Agents | first1 = Jacco | last1 = Wallinga | first2 = Peter | last2 = Teunis | first3 = Mirjam | last3 = Kretzschmar | journal = American Journal of Epidemiology | volume = 164 | number = 10 | year = 2006 | pages = 936–944 | doi = 10.1093/aje/kwj317| pmid = 16968863 | hdl = 10029/6739 | hdl-access = free }}</ref>

== See also ==
* [[Mathematical modelling of infectious disease]]
* [[Next-generation matrix]]

== References ==
{{reflist}}

{{Concepts in infectious disease}}
{{Computer modeling}}
{{Matrix classes}}

[[Category:Matrices]]
[[Category:Epidemiology]]
[[Category:Mathematical and theoretical biology]]
[[Category:Medical statistics]]


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