Chapter 2, Problem 1EA

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

$P=\left[\begin{array}{l}\text{1 0 0 1 0}\\ \text{0 0 1 1 0}\\ \text{1 1 0 0 0}\end{array}\right]$

and $Q=\left[\begin{array}{l}\text{1 1 0 1 1 1}\\ \text{0 0 0 0 1 0}\\ \text{0 0 0 0 0 0}\\ \text{0 1 0 1 0 0}\\ \text{1 0 0 0 1 0}\end{array}\right]$

To determine

The second-order contact matrix $PQ$.

Answer to Problem 1EA

Solution: The second-order contact matrix $PQ={\left[\begin{array}{cccccc}1& 2& 0& 2& 1& 1\\ 0& 1& 0& 1& 0& 0\\ 1& 1& 0& 1& 2& 1\end{array}\right]}_{3\times 6}$.

Explanation of Solution

Given:

First-order contact between the first and the second group is given by

$P=\left[\begin{array}{ccccc}1& 0& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 1& 0& 0& 0\end{array}\right]$

Also, the first-order contact between the second and the third group is given by

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

Explanation:

A second-order contact is an indirect contact between the people of the first and third groups through some people in the second group. It is represented by the matrix $PQ$. $PQ$ is given by

$\begin{array}{c}PQ={\left[\begin{array}{ccccc}1& 0& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 1& 0& 0& 0\end{array}\right]}_{3\times 5}\times {\left[\begin{array}{cccccc}1& 1& 0& 1& 1& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 1& 0& 0\\ 1& 0& 0& 0& 1& 0\end{array}\right]}_{5\times 6}\\ ={\left[\begin{array}{cccccc}1& 2& 0& 2& 1& 1\\ 0& 1& 0& 1& 0& 0\\ 1& 1& 0& 1& 2& 1\end{array}\right]}_{3\times 6}\end{array}$

This shows that the second-order contact matrix $PQ$ is a $3\times 6$ matrix that describes the rows as the people from the first group and columns as the people from the third group. The element from the first row and second column suggests that there is second-order contact between the first person from the first group and the second person from the third group.

Conclusion:

The second-order contact matrix $PQ={\left[\begin{array}{cccccc}1& 2& 0& 2& 1& 1\\ 0& 1& 0& 1& 0& 0\\ 1& 1& 0& 1& 2& 1\end{array}\right]}_{3\times 6}$.