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The answer is $1008$. An example of a metro system which cannot be covered with $1008$ lines is when the metro station $S_1$ is connected to all the others stations $S_i,i=2,3,\dots,2019$, and there are no other tunnels. Further, it's enough to show any metro system can be covered with $1009$ lines. In fact, we have a connected graph $G$ and want to cover its vertices with paths. Consider a spanning tree $T$ of $G$. It contains $2018$ edges. We delete all other edges of $G$ which are not edges in $T$. Clearly, for any two edges of $T$, there is a path in $T$ that contains both edges. Thus, we partition the edges of $T$ into pairs and construct $1009$ paths that cover them. Obviously, they also cover all the vertices of $T$ (and so of $G$).

Nice problem even if misunderstood it first (oops) ! Here's a slightly different approach. Answer. $1008$. Counterexample for the answer. Take a vertex $v$ and connect it to all the remaining ones directly , then any simple path contains at most $2$ vertices other than $v$, at least $1009$ paths are needed. Estimate. We claim that for a connected graph on $n$ vertices ($n$ is odd) it's possible to cover all vertices with $(n-1)/2$ such paths. We use induction, the base case is clear and taking a connected graph on $(n+2)$ vertices and deleting some $2$ leaves from its spanning tree we'll be done by induction hypothesis, so just connect them before removing.