Conjectures on bistable and endemic behavior in the simplicial SIS model

Let $A$ be the irreducible infection matrix, let $\beta_1$ and $\beta_2$ denote the pairwise and higher-order infection parameters, and let $\Gamma$ be the diagonal recovery-rate matrix. The simplicial SIS model has the disease-free equilibrium $x^*=\mathbbold{0}_n$, and its bistable, endemic, and disease-free domains are determined by the model parameters.

Behaviors in the bistable and endemic domains. For the simplicial SIS model, in the bistable domain and at fixed $\beta_2$, the domain of attraction of the disease-free equilibrium $x^*=\mathbbold{0}_n$ decreases as $\beta_1$ increases. Once

$$\beta_1=\frac{1}{\rho(\Gamma^{-1}A)},$$

a bifurcation occurs and the origin becomes an unstable equilibrium point in the endemic domain. In the endemic domain, the endemic equilibrium is unique and globally stable for any value of $\beta_2$.

These conjectures are motivated by numerical simulations and are consistent with the behavior observed in the scalar model. The stated global uniqueness and stability in the endemic domain, as well as the precise change in the disease-free equilibrium's domain of attraction across the bifurcation threshold, remain open in the source.