Next-generation matrix

In epidemiology, the next-generation matrix is a method used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. This method is given by Diekmann et al. (1990)^[1] and van den Driessche and Watmough (2002).^[2] To calculate the basic reproduction number by using a next-generation matrix, the whole population is divided into $n$ compartments in which there are $m<n$ infected compartments. Let $x_i, i=1,2,3,\ldots,m$ be the numbers of infected individuals in the $i^{th}$ infected compartment at time t. Now, the epidemic model is

\frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x)

, where

V_i(x)= [V^-_i(x)-V^+_i(x)]

In the above equations, $F_i(x)$ represents the rate of appearance of new infections in compartment $i$ . $V^+_i$ represents the rate of transfer of individuals into compartment $i$ by all other means, and $V^-_i (x)$ represents the rate of transfer of individuals out of compartment $i$ . The above model can also be written as

\frac{\mathrm{d} x_i}{\mathrm{d}t}= F(x)-V(x)

where

F(x) = \begin{pmatrix} F_1(x), & F_2(x), & \ldots, & F_n(x) \end{pmatrix}^T

and

V(x) = \begin{pmatrix} V_1(x), & V_2 (x), & \ldots, & V_n(x) \end{pmatrix}^T.

Let $x_{0}$ be the disease-free equilibrium. The values of the Jacobian matrices $F(x)$ and $V(x)$ are:

DF(x_0) = \begin{pmatrix} F & 0 \\ 0 & 0 \end{pmatrix}

and

DV(x_0) = \begin{pmatrix} V & 0 \\ J_3 & J_4 \end{pmatrix}

respectively.

Here, $F$ and $V$ are m × m matrices, defined as $F= \frac{\partial F_i}{\partial x_j}(x_0)$ and $V=\frac{\partial V_i}{\partial x_j}(x_0)$ .

Now, the matrix $FV^{-1}$ is known as the next-generation matrix. The largest eigenvalue or spectral radius of $FV^{-1}$ is the basic reproduction number of the model.

References

↑ 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.

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.
Hefferenan, J. M.; Smith, R. J.; Wahl, L. M. (2005). "Prospective on the basic reproductive ratio". J. R. Soc. Interface.

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Next-generation matrix

See also

References

Sources