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.<ref>{{Citation|last=Zhao|first=Xiao-Qiang|chapter=The Theory of Basic Reproduction Ratios|date=2017|title=Dynamical Systems in Population Biology|series=CMS Books in Mathematics|pages=285–315|publisher=Springer International Publishing|doi=10.1007/978-3-319-56433-3_11|isbn=978-3-319-56432-6}}</ref> It is also used in multi-type branching models for analogous computations.<ref>{{Cite book|last=Mode, Charles J., 1927-|title=Multitype branching processes; theory and applications|date=1971|publisher=American Elsevier Pub. Co|isbn=0-444-00086-0|location=New York|oclc=120182}}</ref>
The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann ''et al.'' (1990)<ref>{{Cite journal | doi = 10.1007/BF00178324| title = On the definition and the computation of the basic reproduction ratio R<sub>0</sub> in models for infectious diseases in heterogeneous populations| journal = Journal of Mathematical Biology| volume = 28| issue = 4| pages = 365–382| year = 1990| last1 = Diekmann | first1 = O.| last2 = Heesterbeek | first2 = J. A. P.| last3 = Metz | first3 = J. A. J. | s2cid = 22275430| pmid=2117040| hdl = 1874/8051| hdl-access = free}}</ref> and van den Driessche and Watmough (2002).<ref>{{Cite journal | doi = 10.1016/S0025-5564(02)00108-6| pmid = 12387915| title = Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission| journal = Mathematical Biosciences| volume = 180| issue = 1–2| pages = 29–48| year = 2002| last1 = van den Driessche | first1 = P. | author1-link = Pauline van den Driessche | last2 = Watmough | first2 = J.| s2cid = 17313221}}</ref> To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into <math> n </math> compartments in which there are <math> m<n </math> infected compartments. Let <math> x_i, i=1,2,3,\ldots,m </math> be the numbers of infected individuals in the <math> i^{th}</math> infected compartment at time ''t''. Now, the epidemic model is{{cn|date=May 2023}}
:<math> \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x)</math>, where <math> V_i(x)= [V^-_i(x)-V^+_i(x)] </math>
In the above equations, <math> F_i(x)</math> represents the rate of appearance of new infections in compartment <math> i </math>. <math>V^+_i</math> represents the rate of transfer of individuals into compartment <math> i </math> by all other means, and <math>V^-_i (x)</math> represents the rate of transfer of individuals out of compartment <math> i </math>. The above model can also be written as
:<math>\frac{\mathrm{d} x}{\mathrm{d}t}= F(x)-V(x)</math>
where
: <math> F(x) = \begin{pmatrix} F_1(x), & F_2(x), & \ldots, & F_m(x) \end{pmatrix}^T </math>
and
: <math> V(x) = \begin{pmatrix} V_1(x), & V_2 (x), & \ldots, & V_m(x) \end{pmatrix}^T. </math>
Let <math> x_0 </math> be the disease-free equilibrium. The values of the parts of the Jacobian matrix <math> F(x) </math> and <math> V(x) </math> are:
: <math>DF(x_0) = \begin{pmatrix} F & 0 \\ 0 & 0 \end{pmatrix} </math>
and
: <math> DV(x_0) = \begin{pmatrix} V & 0 \\ J_3 & J_4 \end{pmatrix} </math> respectively.
Here, <math>F</math> and <math> V </math> are ''m'' × ''m'' matrices, defined as <math> F= \frac{\partial F_i}{\partial x_j}(x_0) </math> and <math> V=\frac{\partial V_i}{\partial x_j}(x_0) </math>.
Now, the matrix <math> FV^{-1}</math> is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of <math> FV^{-1} </math> with the largest absolute value (the spectral radius of <math> FV^{-1}</math>). Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.<ref>{{Citation |last=von Csefalvay |first=Chris |title=Simple compartmental models |date=2023 |url=https://linkinghub.elsevier.com/retrieve/pii/B9780323953894000116 |work=Computational Modeling of Infectious Disease |pages=19–91 |publisher=Elsevier |language=en |doi=10.1016/b978-0-32-395389-4.00011-6 |isbn=978-0-323-95389-4 |access-date=2023-02-28|url-access=subscription }}</ref>
==See also== *Mathematical modelling of infectious disease
==References== {{Reflist}}
==Sources== * {{cite book |first1=Zhien |last1=Ma |first2=Jia |last2=Li |title=Dynamical Modeling and analysis of Epidemics |publisher=World Scientific |year=2009 |isbn=978-981-279-749-0 |oclc=225820441}} * {{cite book |first1=O. |last1=Diekmann |first2=J. A. P. |last2=Heesterbeek |title=Mathematical Epidemiology of Infectious Disease |publisher=John Wiley & Son |year=2000}} * {{cite journal |first1=J. M. |last1=Heffernan |first2=R. J. |last2=Smith |first3=L. M. |last3=Wahl |title=Perspectives on the basic reproductive ratio |journal=J. R. Soc. Interface |year=2005 |volume= 2|issue= 4|pages= 281–93|doi= 10.1098/rsif.2005.0042|pmid=16849186 |pmc=1578275}}
Category:Matrices (mathematics) Category:Epidemiology