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.^{[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

This infectious disease article is a stub. You can help Wikipedia by expanding it.

[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), 59 (2), Royal Statistical Society: 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, OUP Oxford, 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

[1]

[2]

[3]

[4]

[5]

v t e Scientific modelling
Biological	Cellular model Chemical process modeling Ecosystem model Infectious disease model Metabolic network modelling Modelling biological systems Protein structure prediction
Environmental	Atmospheric model Chemical transport model Climate model Geologic modelling Groundwater model Hydrological model Hydrological transport model Modular Ocean Model Wildfire modeling
Sustainability	Energy modeling Integrated assessment modelling Population model
Social	Biopsychosocial model Business process modelling Catastrophe modeling Construction and management simulation Crime mapping Economic model Input–output model
Related topics	Data visualization List of computer simulation software Mathematical modeling Systems theory Systems thinking Visual analytics

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices