epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evide ...

, the next-generation matrix is used to derive the

basic reproduction number In epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted R_0 (pronounced ''R nought'' or ''R zero''), of an infection is the expected number of ...

, for a

compartmental model A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the ''compartments'' of a system. Sometimes, the physical system that we try to model in equations is too complex, so ...

of the spread of

infectious disease An infection is the invasion of tissues by pathogens, their multiplication, and the reaction of host tissues to the infectious agent and the toxins they produce. An infectious disease, also known as a transmissible disease or communicable d ...

s. In

population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...

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 infected compartments. Let x_i, i=1,2,3,\ldots,m be the numbers of infected individuals in the i^infected compartment at time ''t''. Now, the

epidemic model Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, ...

is :

\frac= F_i (x)-V_i(x)

, where

V_i(x)=^-_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= F(x)-V(x)

where :

F(x) = \begin
     F_1(x), & F_2(x), & \ldots, & F_m(x)
    \end^T

and :

V(x) = \begin
     V_1(x), & V_2 (x), & \ldots, & V_m(x)
    \end^T.

Let

x_0

be the disease-free equilibrium. The values of the parts of the

Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...

F(x)

and

V(x)

are: :

DF(x_0) = \begin
    F & 0 \\
    0 & 0
  \end

and :

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

respectively. Here,

F

and

V

are ''m'' × ''m'' matrices, defined as

F= \frac(x_0)

and

V=\frac(x_0)

. Now, the matrix

FV^

is known as the next-generation matrix. The basic reproduction number of the model is then given by the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

FV^

with the largest absolute value (the

spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spect ...

FV^

. Next generation matrices can be computationally evaluated from observational data, which is often the most productive approach where there are large numbers of compartments.

References

Sources

* * * {{cite journal , first1=J. M. , last1=Heffernan , first2=R. J. , last2=Smith , first3=L. M. , last3=Wahl , title=Perspectives on the basic reproductive ratio , journal=J. R. Soc. Interface , year=2005 , volume= 2, issue= 4, pages= 281–93, doi= 10.1098/rsif.2005.0042, pmid=16849186 , pmc=1578275 Matrices Epidemiology

See also

References

Sources