In infectious disease modelling, a who acquires infection from whom (WAIFW) matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

that describes the rate of transmission of infection between different groups in a population, such as people of different ages. Used with an

SIR model Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious ...

, the entries of the WAIFW matrix can be used to calculate the

basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...

using the next generation operator approach.

Examples

The

2 \times 2

WAIFW matrix for two groups is expressed as

\begin \beta_ & \beta_ \\ \beta_ & \beta_ \end

where

\beta_

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 \beta & 0 \\ 0 & \beta \end

or the general

2 \times 2

WAIFW matrix so long as

\beta_, \beta_ > \beta_, \beta_

. Disassortative mixing is instead when

\beta_, \beta_ < \beta_, \beta_

Homogenous mixing

Homogenous mixing, which is also dubbed random mixing, is given by

\begin \beta & \beta \\ \beta & \beta \end

. 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_ = \beta_

. Examples of asymmetric WAIFW matrices are :

\begin \beta_1 & \beta_2 \\ \beta_1 & \beta_2 \end 
\begin \beta_1 & \beta_1 \\ \beta_2 & \beta_2 \end
\begin 0 & \beta_1 \\ \beta_2 & 0 \end

Social contact hypothesis

The social contact hypothesis was proposed by , Peter Teunis, and Mirjam Kretzschmar in 2006. The hypothesis states that transmission rates are proportional to contact rates,

\beta_ \propto c_

and allows for social contact data to be used in place of WAIFW matrices.

References