Next-generation matrix (original) (raw)

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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 {\displaystyle n} {\displaystyle n} compartments in which there are m < n {\displaystyle m<n} {\displaystyle m<n} infected compartments. Let x i , i = 1 , 2 , 3 , … , m {\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 i t h {\displaystyle i^{th}} {\displaystyle i^{th}} infected compartment at time t. Now, the epidemic model is[_citation needed_]

d x i d t = F i ( x ) − V i ( x ) {\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 V i ( x ) = [ V i − ( x ) − V i + ( x ) ] {\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]} {\displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}

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

d x d t = F ( x ) − V ( x ) {\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

F ( x ) = ( F 1 ( x ) , F 2 ( x ) , … , F m ( x ) ) T {\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

V ( x ) = ( V 1 ( x ) , V 2 ( x ) , … , V m ( x ) ) T . {\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 x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} be the disease-free equilibrium. The values of the parts of the Jacobian matrix F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} and V ( x ) {\displaystyle V(x)} {\displaystyle V(x)} are:

D F ( x 0 ) = ( F 0 0 0 ) {\displaystyle DF(x_{0})={\begin{pmatrix}F&0\\0&0\end{pmatrix}}} {\displaystyle DF(x_{0})={\begin{pmatrix}F&0\\0&0\end{pmatrix}}}

and

D V ( x 0 ) = ( V 0 J 3 J 4 ) {\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, F {\displaystyle F} {\displaystyle F} and V {\displaystyle V} {\displaystyle V} are m × m matrices, defined as F = ∂ F i ∂ x j ( x 0 ) {\displaystyle F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})} {\displaystyle F={\frac {\partial F_{i}}{\partial x_{j}}}(x_{0})} and V = ∂ V i ∂ x j ( x 0 ) {\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 F V − 1 {\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 F V − 1 {\displaystyle FV^{-1}} {\displaystyle FV^{-1}} with the largest absolute value (the spectral radius of F V − 1 {\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]

  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](/wiki/Template:Cite%5Fbook "Template: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