Relaxation dynamics of SIR-flocks with random epidemic states

We study the collective dynamics of a multi-particle system with three epidemic states as an internal state. For the collective modeling of active particle system, we adopt modeling spirits from the swarmalator model and the SIR epidemic model for the temporal evolution of particles' position and internal states. Under suitable assumptions on system parameters and non-collision property of initial spatial configuration, we show that the proposed model does not admit finite-time collisions so that the standard Cauchy-Lipschitz theory can be applied for the global well-posedness. For the relaxation dynamics, we provide several sufficient frameworks leading to the relaxation dynamics of the proposed model. The proposed sufficient frameworks are formulated in terms of system parameters and initial configuration. Under such sufficient frameworks, we show that the state configuration relaxes to the fixed constant configuration via the exponentially perturbed gradient system and explicit dynamics of the SIR model. We present explicit lower and upper bounds for the minimal and maximal relative distances.