[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/next-generation-matrix-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/next-generation-matrix-wikipedia\/","headline":"Next-generation matrix – Wikipedia","name":"Next-generation matrix – Wikipedia","description":"From Wikipedia, the free encyclopedia In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a","datePublished":"2018-09-25","dateModified":"2018-09-25","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a601995d55609f2d9f5e233e36fbe9ea26011b3b","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a601995d55609f2d9f5e233e36fbe9ea26011b3b","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/next-generation-matrix-wikipedia\/","wordCount":5051,"articleBody":"From Wikipedia, the free encyclopediaIn 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} compartments in which there are mVi(x){displaystyle {frac {mathrm {d} x_{i}}{mathrm {d} t}}=F_{i}(x)-V_{i}(x)}, where  Vi(x)=[Vi\u2212(x)\u2212Vi+(x)]{displaystyle V_{i}(x)=[V_{i}^{-}(x)-V_{i}^{+}(x)]}In the above equations, Fi(x){displaystyle F_{i}(x)} represents the rate of appearance of new infections in compartment i{displaystyle i}.  Vi+{displaystyle V_{i}^{+}} represents the rate of transfer of individuals into compartment i{displaystyle i} by all other means, and Vi\u2212(x){displaystyle V_{i}^{-}(x)} represents the rate of transfer of individuals out of compartment i{displaystyle i}.The above model can also be written asdxdt=F(x)\u2212V(x){displaystyle {frac {mathrm {d} x}{mathrm {d} t}}=F(x)-V(x)}whereF(x)=(F1(x),F2(x),\u2026,Fm(x))T{displaystyle F(x)={begin{pmatrix}F_{1}(x),&F_{2}(x),&ldots ,&F_{m}(x)end{pmatrix}}^{T}}andV(x)=(V1(x),V2(x),\u2026,Vm(x))T.{displaystyle V(x)={begin{pmatrix}V_{1}(x),&V_{2}(x),&ldots ,&V_{m}(x)end{pmatrix}}^{T}.}Let x0{displaystyle x_{0}} be the disease-free equilibrium. The values of the parts of the Jacobian matrix F(x){displaystyle F(x)} and V(x){displaystyle V(x)} are:DF(x0)=(F000){displaystyle DF(x_{0})={begin{pmatrix}F&0\\0&0end{pmatrix}}}andDV(x0)=(V0J3J4){displaystyle DV(x_{0})={begin{pmatrix}V&0\\J_{3}&J_{4}end{pmatrix}}}respectively.Here, F{displaystyle F} and V{displaystyle V} are m\u00a0\u00d7\u00a0m  matrices, defined asF=\u2202Fi\u2202xj(x0){displaystyle F={frac {partial F_{i}}{partial x_{j}}}(x_{0})} and V=\u2202Vi\u2202xj(x0){displaystyle V={frac {partial V_{i}}{partial x_{j}}}(x_{0})}.Now, the matrix FV\u22121{displaystyle FV^{-1}} is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of FV\u22121{displaystyle FV^{-1}} with the largest absolute value (the spectral radius of  FV\u22121{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]See also[edit]References[edit]^ Zhao, Xiao-Qiang (2017), “The Theory of Basic Reproduction Ratios”, Dynamical Systems in Population Biology, CMS Books in Mathematics, Springer International Publishing, pp.\u00a0285\u2013315, doi:10.1007\/978-3-319-56433-3_11, ISBN\u00a0978-3-319-56432-6^ Mode, Charles J., 1927- (1971). Multitype branching processes; theory and applications. New York: American Elsevier Pub. Co. ISBN\u00a00-444-00086-0. OCLC\u00a0120182.{{cite book}}:  CS1 maint: multiple 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\u2013382. doi:10.1007\/BF00178324. hdl:1874\/8051. PMID\u00a02117040. S2CID\u00a022275430.^ van den Driessche, P.; Watmough, J. (2002). “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”. Mathematical Biosciences. 180 (1\u20132): 29\u201348. doi:10.1016\/S0025-5564(02)00108-6. PMID\u00a012387915. S2CID\u00a017313221.^ von Csefalvay, Chris (2023), “Simple compartmental models”, Computational Modeling of Infectious Disease, Elsevier, pp.\u00a019\u201391, doi:10.1016\/b978-0-32-395389-4.00011-6, ISBN\u00a0978-0-323-95389-4, retrieved 2023-02-28Sources[edit]  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/next-generation-matrix-wikipedia\/#breadcrumbitem","name":"Next-generation matrix – Wikipedia"}}]}]