\textbf{Exercise 4 — Candidates who have followed the specialization course}

In an imaginary isolated village, a new contagious but non-fatal disease has appeared. Scientists discovered that an individual could be in one of three following states:
\begin{itemize}
  \item $S$: ``the individual is healthy, that is, not sick and not infected'',
  \item $I$: ``the individual is a healthy carrier, that is, not sick but infected'',
  \item $M$: ``the individual is sick and infected''.
\end{itemize}

\textbf{Part A}

Scientists estimate that a single individual is at the origin of the disease among the 100 people in the population and that, from one week to the next, an individual changes state according to the following process:
\begin{itemize}
  \item among healthy individuals, the proportion of those who become healthy carriers is equal to $\dfrac{1}{3}$ and the proportion of those who become sick is equal to $\dfrac{1}{3}$,
  \item among healthy carriers, the proportion of those who become sick is equal to $\dfrac{1}{2}$.
\end{itemize}

We denote by $P_n = \begin{pmatrix} s_n & i_n & m_n \end{pmatrix}$ the row matrix giving the probabilistic state after $n$ weeks where $s_n, i_n$ and $m_n$ denote respectively the probability that the individual is healthy, a healthy carrier, or sick in the $n$-th week.

We have $P_0 = (0.99 \quad 0 \quad 0.01)$ and for all natural integer $n$,
$$\left\{\begin{aligned}
s_{n+1} &= \frac{1}{3}s_n \\
i_{n+1} &= \frac{1}{3}s_n + \frac{1}{2}i_n \\
m_{n+1} &= \frac{1}{3}s_n + \frac{1}{2}i_n + m_n
\end{aligned}\right.$$

\begin{enumerate}
  \item Write the matrix $A$ called the transition matrix, such that for all natural integer $n$, $P_{n+1} = P_n \cdot A$.
\end{enumerate}