Chapter 10.EA, Problem 1EA

Find the second-order contact matrix $PQ$ mentioned in the text.

To determine

To solve:

The second-order contact matrix $PQ$ mentioned in the text.

Answer to Problem 1EA

The second-order contact matrix is $\left[\begin{array}{cccccc}1& 2& 0& 2& 1& 1\\ 0& 1& 0& 1& 0& 0\\ 1& 1& 0& 1& 2& 1\end{array}\right]$.

Explanation of Solution

Given:

$P=\left[\begin{array}{ccccc}1& 0& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 1& 0& 0& 0\end{array}\right]$ is the first-order contacts between first and second groups and $Q=\left[\begin{array}{cccccc}1& 1& 0& 1& 1& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 1& 0& 0& 0& 1& 0\end{array}\right]$ is the first-order contacts between second and third group.

Approach:

The product matrix $PQ$ indicates the second-order contact between persons in the first and third groups.

Calculation:

Let $a_{ij}$ denote the element in the $i$ th row and $j$ th column of the matrix $PQ$. $a_{11}=p_{11}{q}_{11}+p_{12}{q}_{21}+p_{13}{q}_{31}+p_{14}{q}_{41}+p_{15}{q}_{51}$ ( The $p_{ij}$ are entries in $P$, and the $q_{ij}$ are entries in $Q$ ).

Similarly,

$a_{12}=2$ ; $a_{13}=0$; $a_{14}=2$ ; $a_{15}=1$ ; $a_{16}=1$ ;

$a_{21}=0$ ; $a_{22}=1$; $a_{23}=0$ ; $a_{24}=1$ ; $a_{25}=0$ ; $a_{26}=0$;

$a_{31}=1$ ; $a_{32}=1$; $a_{33}=0$ ; $a_{34}=1$ ; $a_{35}=2$ ; $a_{36}=1$;

The $PQ$ matrix is represented as:

$PQ=\left[\begin{array}{cccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}& a_{16}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}& a_{26}\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}& a_{36}\end{array}\right]$

$PQ=\left[\begin{array}{cccccc}1& 2& 0& 2& 1& 1\\ 0& 1& 0& 1& 0& 0\\ 1& 1& 0& 1& 2& 1\end{array}\right]$