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 [math]\displaystyle{ n }[/math] compartments in which there are [math]\displaystyle{ m\lt n }[/math] infected compartments. Let [math]\displaystyle{ x_i, i=1,2,3,\ldots,m }[/math] be the numbers of infected individuals in the [math]\displaystyle{ i^{th} }[/math] infected compartment at time t. Now, the epidemic model is[citation needed]
- [math]\displaystyle{ \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x) }[/math], where [math]\displaystyle{ V_i(x)= [V^-_i(x)-V^+_i(x)] }[/math]
In the above equations, [math]\displaystyle{ F_i(x) }[/math] represents the rate of appearance of new infections in compartment [math]\displaystyle{ i }[/math]. [math]\displaystyle{ V^+_i }[/math] represents the rate of transfer of individuals into compartment [math]\displaystyle{ i }[/math] by all other means, and [math]\displaystyle{ V^-_i (x) }[/math] represents the rate of transfer of individuals out of compartment [math]\displaystyle{ i }[/math]. The above model can also be written as
- [math]\displaystyle{ \frac{\mathrm{d} x}{\mathrm{d}t}= F(x)-V(x) }[/math]
where
- [math]\displaystyle{ F(x) = \begin{pmatrix} F_1(x), & F_2(x), & \ldots, & F_m(x) \end{pmatrix}^T }[/math]
and
- [math]\displaystyle{ V(x) = \begin{pmatrix} V_1(x), & V_2 (x), & \ldots, & V_m(x) \end{pmatrix}^T. }[/math]
Let [math]\displaystyle{ x_0 }[/math] be the disease-free equilibrium. The values of the parts of the Jacobian matrix [math]\displaystyle{ F(x) }[/math] and [math]\displaystyle{ V(x) }[/math] are:
- [math]\displaystyle{ DF(x_0) = \begin{pmatrix} F & 0 \\ 0 & 0 \end{pmatrix} }[/math]
and
- [math]\displaystyle{ DV(x_0) = \begin{pmatrix} V & 0 \\ J_3 & J_4 \end{pmatrix} }[/math]
respectively.
Here, [math]\displaystyle{ F }[/math] and [math]\displaystyle{ V }[/math] are m × m matrices, defined as [math]\displaystyle{ F= \frac{\partial F_i}{\partial x_j}(x_0) }[/math] and [math]\displaystyle{ V=\frac{\partial V_i}{\partial x_j}(x_0) }[/math].
Now, the matrix [math]\displaystyle{ 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]\displaystyle{ FV^{-1} }[/math] with the largest absolute value (the spectral radius of [math]\displaystyle{ 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.[5]
See also
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.
- ↑ 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. PMID 2117040.
- ↑ 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.
- ↑ von Csefalvay, Chris (2023), "Simple compartmental models" (in en), 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, https://linkinghub.elsevier.com/retrieve/pii/B9780323953894000116, 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. PMID 16849186.
 |  |
