Next-generation matrix explained

In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. In population dynamics it is used to compute the basic reproduction number for structured population models. It is also used in multi-type branching models for analogous computations.^[1]

The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann et al. (1990)^[2] and van den Driessche and Watmough (2002).^[3] To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into

n compartments in which there are m<n infected compartments. Let x _{i,} i =1 , 2 , 3 ,\ldots,m be the numbers of infected individuals in the i ^{th} infected compartment at time t. Now, the epidemic model is(d x _{i})/(d t) = F _{i} (x) - V _{i(x)} , where V _{i(x)=} [V ^{+} _{i(x)]}

In the above equations,

F _{i(x)} represents the rate of appearance of new infections in compartment i . V ^{+} _{i} represents the rate of transfer of individuals into compartment i by all other means, and V ^{-} _{i} (x) represents the rate of transfer of individuals out of compartment i .The above model can also be written as(d x)/(d t) = F(x) - V(x)

where

F(x) = \begin{pmatrix} F _{1(x),} & F _{2(x),} & \ldots, & F _{m(x)} \end{pmatrix} ^{T}

and

V(x) = \begin{pmatrix} V _{1(x),} & V _{2} (x), & \ldots, & V _{m(x)} \end{pmatrix} ^{T.}

Let

x _{0} be the disease-free equilibrium. The values of the parts of the Jacobian matrix F(x) and V(x) are:DF(x _{0)} = \begin{pmatrix} F & 0 \\ 0 & 0 \end{pmatrix}

and

DV(x _{0)} = \begin{pmatrix} V & 0 \\ J _{3} & J _{4} \end{pmatrix} respectively.

Here,

F and V are m × m matrices, defined asF = (\partial F _{i})/(\partial x _{j}) (x _{0)} and V = (\partial V _{i})/(\partial x _{j}) (x _{0)} .

Now, the matrix

FV ^{-1} is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of FV ^{-1} with the largest absolute value (the spectral radius of FV ^{-1} . Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.

See also

• Mathematical modelling of infectious disease

Sources

• Book: Zhien . Ma . Jia . Li . Dynamical Modeling and analysis of Epidemics . World Scientific . 2009 . 978-981-279-749-0 . 225820441.
• Book: O. . Diekmann . J. A. P. . Heesterbeek . Mathematical Epidemiology of Infectious Disease . John Wiley & Son . 2000.
• J. M. . Heffernan . R. J. . Smith . L. M. . Wahl . Perspectives on the basic reproductive ratio . J. R. Soc. Interface . 2005 . 2. 4. 281–93. 10.1098/rsif.2005.0042. 16849186 . 1578275.

Notes and References

1. Book: Mode, Charles J., 1927-. Multitype branching processes; theory and applications. 1971. American Elsevier Pub. Co. 0-444-00086-0. New York. 120182.
2. 10.1007/BF00178324. On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology. 28. 4. 365–382. 1990. Diekmann . O.. Heesterbeek . J. A. P.. Metz . J. A. J. . 22275430. 2117040. 1874/8051. free.
3. 10.1016/S0025-5564(02)00108-6. 12387915. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 180. 1–2. 29–48. 2002. van den Driessche . P. . Pauline van den Driessche . Watmough . J.. 17313221.