To formulate a network security problem, Mavronicholas et al.  introduced a strategic game on an undirected graph whose nodes are exposed to infection by attackers, and whose edges are protected by a defender. Subsequently, MedSalem et al.  generalized the model so that they have many defenders instead of a single defender. Then in , we introduced a new network game with the roles of players interchanged, and obtained a graph-theoretic characterization of its pure Nash equilibria. In this paper we study mixed Nash equilibria for stochastic strategies in this new game, and then we generalize our network game to an asynchronous game, where two players repeatedly execute simultaneous games. Although the asynchronous game is formally an infinite game, we show that it has a stable solution by reducing it to a finite game.