This paper proposes an approach to mitigate epidemic spread in a population of strategic agents by encouraging safer behaviors through carefully designed rewards. These rewards, which adapt to the evolving state of the epidemic, are ascribed by a dynamic payoff mechanism we seek to design. We use a modified SIRS model to track how the epidemic progresses in response to the agents' strategic choices. By employing perturbed best response evolutionary dynamics to model the population's strategic behavior, we extend previous related work so as to allow for noise in the agents' perceptions of the rewards and intrinsic costs of the available strategies. Central to our approach is the use of system-theoretic methods and passivity concepts to obtain a Lyapunov function, ensuring the global asymptotic stability of an endemic equilibrium with minimized infection prevalence under budget constraints. We leverage the Lyapunov function to analyze how the epidemic's spread rate is influenced by the time scale of the payoff mechanism's dynamics. Additionally, we derive anytime upper bounds on both the infectious fraction of the population and the instantaneous cost a social planner must incur to control the spread, allowing us to quantify the trade-off between peak infection prevalence and the corresponding cost. For a class of one-parameter perturbed best response models, we propose a method to learn the model's parameter from data.
