  We analyze a game-theoretic abstraction of epidemic containment played on an
undirected graph $G$: each player is associated with a node in $G$ and can
either acquire protection from a contagious process or risk infection. After
decisions are made, an infection starts at a random node $v$ and propagates
through all unprotected nodes reachable from $v$. It is known that the price of
anarchy (PoA) in $n$-node graphs can be as large as $\Theta(n)$. Our main
result is a tight bound of order $\sqrt{n\Delta}$ on the PoA, where $\Delta$ is
the maximum degree of the graph. We also study additional factors that can
reduce the PoA, such as higher thresholds for contagion and varying the costs
of becoming infected vs. acquiring protection.
