Attempt
Let $x_i$ be the number of workers to be hired that starts its shift on day $i$. So we want to minimize the function $f( {\bf } ) = \sum_{i=1}^7 x_i$. The constraints:
$$ x_1 \leq 17 $$ $$ x_1 + x_2 \leq 13 $$ $$ x_1 + x_2 + x_3 \leq 15 $$ $$ x_1+x_2+x_3+x_4 \leq 19 $$ $$x_1 + x_2 + x_3 + x_4 + x_5 \leq 14 $$ $$ x_2 +x_3 + x_4 + x_5 + x_6 \leq 16 \; \; \; \text{day 1 needs 2 days off} $$ $$ x_3 + x_4 + x_5 + x_6 + x_7 \leq 11 $$ $$ x_i \geq 0 $$
Is this a correct formulation?
Not really.
Guide:
The nurse who are present on day $1$ are the ones who start from day $4,1,5,6,7$. Hence,
$$x_1 + x_4 + x_5 + x_6 + x_7 \ge 17.$$
Use the same strategies on various days.
Also, we have to impose nonnegative integer constraints and remember to write down the objective function.