Simplicial SEIQS epidemic dynamics: Bistability and backward bifurcation under a mean-field approximation

We investigate the impact of higher-order, or simplicial, interactions on the mean-field dynamics of a susceptible-exposed-infectious-quarantined-susceptible epidemic model on networks, under a degree-homogeneous mean-field approximation. For the corresponding mean-field system, we derive explicit conditions for the existence and local asymptotic stability of disease-free and endemic equilibria, and obtain closed-form expressions for the critical parameters β0, βc, and vc that govern the onset of subthreshold bistability. The analysis shows that the simplicial transmission coefficient does not alter the linear epidemic threshold, but it can create a bistable regime below βc in which the disease-free equilibrium coexists with two endemic equilibria. We then perform a numerical bifurcation and basin analysis combining equilibrium branches, phase diagrams, time-domain simulations, and basin-of-attraction plots. These results reveal how higher-order interactions enlarge the endemic equilibrium and induce a nontrivial separatrix between extinction and persistence. Finally, a Routh-Hurwitz analysis excludes a Hopf bifurcation of the larger endemic equilibrium under the baseline parameters, while numerical scans over wider parameter ranges provide additional support for the absence of Hopf instability in the tested regimes. The findings suggest that simplicial interactions enrich epidemic dynamics primarily through multistability and basin geometry rather than sustained oscillations, although oscillatory behavior in substantially different regimes remains possible.