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 $$ x_i, i=1,2,3,\ldots,m $$ 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


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

where


 * $$ F(x) = \begin{pmatrix}

F_1(x), & F_2(x), & \ldots, & F_m(x) \end{pmatrix}^T $$

and


 * $$ V(x) = \begin{pmatrix}

V_1(x), & V_2 (x), & \ldots, & V_m(x) \end{pmatrix}^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) = \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 basic reproduction number of the model is then given by the eigenvalue of $$ FV^{-1} $$ with the largest absolute value (the spectral radius of $$ FV^{-1}$$. Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.