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 $n$ 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^{[citation needed]}

{\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}=F_{i}(x)-V_{i}(x)

, where

V_{i}(x)=[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

{\frac {\mathrm {d} x}{\mathrm {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 as $F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})$ and $V={\frac {\partial V_{i}}{\partial x_{j}}}(x_{0})$ .

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.^[5]

References

^ Zhao, Xiao-Qiang (2017), "The Theory of Basic Reproduction Ratios", Dynamical Systems in Population Biology, CMS Books in Mathematics, Springer International Publishing, pp. 285–315, doi:10.1007/978-3-319-56433-3_11, ISBN 978-3-319-56432-6
^ Mode, Charles J., 1927- (1971). Multitype branching processes; theory and applications. New York: American Elsevier Pub. Co. ISBN 0-444-00086-0. OCLC 120182.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^ Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J. (1990). "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. doi:10.1007/BF00178324. hdl:1874/8051. PMID 2117040. S2CID 22275430.
^ van den Driessche, P.; Watmough, J. (2002). "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission". Mathematical Biosciences. 180 (1–2): 29–48. doi:10.1016/S0025-5564(02)00108-6. PMID 12387915. S2CID 17313221.
^ von Csefalvay, Chris (2023), "Simple compartmental models", Computational Modeling of Infectious Disease, Elsevier, pp. 19–91, doi:10.1016/b978-0-32-395389-4.00011-6, ISBN 978-0-323-95389-4, retrieved 2023-02-28

Sources

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.

[1] Zhao, Xiao-Qiang (2017), "The Theory of Basic Reproduction Ratios", Dynamical Systems in Population Biology, CMS Books in Mathematics, Springer International Publishing, pp. 285–315, doi:10.1007/978-3-319-56433-3_11, ISBN 978-3-319-56432-6

[2] Mode, Charles J., 1927- (1971). Multitype branching processes; theory and applications. New York: American Elsevier Pub. Co. ISBN 0-444-00086-0. OCLC 120182.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)

[3] Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J. (1990). "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. doi:10.1007/BF00178324. hdl:1874/8051. PMID 2117040. S2CID 22275430.

[4] van den Driessche, P.; Watmough, J. (2002). "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission". Mathematical Biosciences. 180 (1–2): 29–48. doi:10.1016/S0025-5564(02)00108-6. PMID 12387915. S2CID 17313221.

[5] von Csefalvay, Chris (2023), "Simple compartmental models", Computational Modeling of Infectious Disease, Elsevier, pp. 19–91, doi:10.1016/b978-0-32-395389-4.00011-6, ISBN 978-0-323-95389-4, retrieved 2023-02-28

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See also

References

Sources