Epidemic model with group mixing: Stability and optimal control based on limited vaccination resources

We investigate the global dynamics of a multi-group SIR epidemic model. By using the Lyapunov-LaSalle principle, a graph-theoretic approach and the uniform persistence theory, the global dynamics can be obtained for both disease-free and endemic equilibria. The relationship of the basic reproduction ratios between the subgroup model and the mixed group model are established. The optimal control strategy of an infectious disease with the mixing of two sub-groups under limited vaccination resources is also studied. The results suggest that the optimal distribution strategies are dynamically different due to the variance of heterogeneity.