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. It is also used in multi-type branching models for analogous computations.

The method to compute the basic reproduction ratio using the next-generation matrix is given by Diekmann et al. (1990) and van den Driessche and Watmough (2002). 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 $x_i,i=1,2,3,\dots ,m$ be the numbers of infected individuals in the $i^{th}$ infected compartment at time t. Now, the epidemic model isTemplate:Citation needed

$\frac{\mathrm{d}{x}_i}{\mathrm{d}t}=F_i(x)-V_i(x)$, 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

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

where

$F(x)={\left(\begin{array}{cccc}{F}_1(x),& F_2(x),& \dots ,& F_m(x)\end{array}\right)}^T$

and

$V(x)={\left(\begin{array}{cccc}{V}_1(x),& V_2(x),& \dots ,& V_m(x)\end{array}\right)}^T.$

Let $x_0$ be the disease-free equilibrium. The values of the parts of the Jacobian matrix $F(x)$ and $V(x)$ are:

$DF(x_0)=\left(\begin{array}{cc}F& 0\\ 0& 0\end{array}\right)$

and

$DV(x_0)=\left(\begin{array}{cc}V& 0\\ J_3& J_4\end{array}\right)$

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 $F{V}^{-1}$ is known as the next-generation matrix. The basic reproduction number of the model is then given by the eigenvalue of $F{V}^{-1}$ with the largest absolute value (the spectral radius of $F{V}^{-1}$). Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.

See also

• Mathematical modelling of infectious disease

Sources

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