Threshold analysis of multi-group SIR models with heterogeneous mixing

We analyze the spread of a social interaction agent of finite duration of interest, such as a petition, behavioral trend, or opinion, through a heterogeneous population using a multi-group SIR model. By integrating the model equations, we obtain explicit final-size relations and identify threshold conditions that determine whether propagation can be sustained. For one-, two-, and three-group systems, we show how within-group reinforcement and cross-group influence shape the geometry of the no-propagation boundary. The results provide a clear geometric characterization of heterogeneous diffusion and offer practical guidance for designing interventions that either promote or inhibit spread.