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 compartments in which there are infected compartments. Let be the numbers of infected individuals in the infected compartment at time t. Now, the epidemic model is

, where

In the above equations, represents the rate of appearance of new infections in compartment . represents the rate of transfer of individuals into compartment by all other means, and represents the rate of transfer of individuals out of compartment . The above model can also be written as

where

and

Let be the disease-free equilibrium. The values of the Jacobian matrices and are:

and

respectively.

Here, and are m × m matrices, defined as and .

Now, the matrix is known as the next-generation matrix. The largest modulus of the eigenvalues or spectral radius of is the basic reproduction number of the model.

See also

References

  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)
  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.

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