Next-generation matrix

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.^[1] It is also used in multi-type branching models for analogous computations.^[2]

The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann et al. (1990)^[3] and van den Driessche and Watmough (2002).^[4] To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into ${\displaystyle n}${\displaystyle n} compartments in which there are ${\displaystyle m<n}${\displaystyle m<n} infected compartments. Let ${\displaystyle x_{i},i=1,2,3,\ldots ,m}${\displaystyle x_{i},i=1,2,3,\ldots ,m} be the numbers of infected individuals in the ${\displaystyle i^{th}}${\displaystyle i^{th}} infected compartment at time t. Now, the epidemic model is^{[citation needed ]}

${\displaystyle {\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)}${\displaystyle {\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)}, where ${\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}${\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}

In the above equations, ${\displaystyle F_{i}(x)}${\displaystyle F_{i}(x)} represents the rate of appearance of new infections in compartment ${\displaystyle i}${\displaystyle i}. ${\displaystyle V_{i}^{+}}${\displaystyle V_{i}^{+}} represents the rate of transfer of individuals into compartment ${\displaystyle i}${\displaystyle i} by all other means, and ${\displaystyle V_{i}^{-}(x)}${\displaystyle V_{i}^{-}(x)} represents the rate of transfer of individuals out of compartment ${\displaystyle i}${\displaystyle i}. The above model can also be written as

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=F(x)-V(x)}${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=F(x)-V(x)}

where

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

and

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

Let ${\displaystyle x_{0}}${\displaystyle x_{0}} be the disease-free equilibrium. The values of the parts of the Jacobian matrix ${\displaystyle F(x)}${\displaystyle F(x)} and ${\displaystyle V(x)}${\displaystyle V(x)} are:

${\displaystyle DF(x_{0})={\begin{pmatrix}F&0\0円&0\end{pmatrix}}}${\displaystyle DF(x_{0})={\begin{pmatrix}F&0\0円&0\end{pmatrix}}}

and

${\displaystyle DV(x_{0})={\begin{pmatrix}V&0\\J_{3}&J_{4}\end{pmatrix}}}${\displaystyle DV(x_{0})={\begin{pmatrix}V&0\\J_{3}&J_{4}\end{pmatrix}}}

respectively.

Here, ${\displaystyle F}${\displaystyle F} and ${\displaystyle V}${\displaystyle V} are m×ばつ m matrices, defined as ${\displaystyle F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})}${\displaystyle F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})} and ${\displaystyle V={\frac {\partial V_{i}}{\partial x_{j}}}(x_{0})}${\displaystyle V={\frac {\partial V_{i}}{\partial x_{j}}}(x_{0})}.

Now, the matrix ${\displaystyle FV^{-1}}${\displaystyle FV^{-1}} is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of ${\displaystyle FV^{-1}}${\displaystyle FV^{-1}} with the largest absolute value (the spectral radius of ${\displaystyle FV^{-1}}${\displaystyle 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.^[5]

• Mathematical modelling of infectious disease

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

• Matrices (mathematics)
• Epidemiology

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