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 vary according to the 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 population's 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 use the Lyapunov function to construct anytime upper bounds for the size of the population's infectious fraction. For a class of one-parameter perturbed best response models, we propose a method to learn the model's parameter from data.