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 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.

See also

References

  1. ↑ 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.
  2. ↑ 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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