Modern infrastructures involve a large population of heterogeneous users or agents in a complex network who strategically make dynamic decisions. In this work, we connect evolutionary games with Poisson games to establish an evolutionary Poisson game framework to capture the random, dynamic and heterogeneous interactions of agents in a holistic fashion. We introduce the notion of evolutionary Poisson stable equilibrium for the game, and show its existence and uniqueness. To illustrate the framework, we use epidemic protection problem as a case study to provide cyber security solutions using the developed framework. In addition, we design online algorithms to control large population behaviors to achieve a system-wide objective based on the techniques of stochastic approximation. Numerical examples are used to illustrate and corroborate our results.