Dynamics of SIR model with heterogeneous response to intervention policy

The paper proves that the heterogeneous SIR model with a continuous Preisach operator admits a non-degenerate line segment of endemic equilibria E0E1 lying strictly inside the segment between the constant-R0 equilibria Eint and Enat, using a monotone fixed-point argument in the variable I via functions R̂0(I), R̂1(I), and their convex combination R̂θ(I) (see Proposition 2 and its proof, along with the model setup) . The candidate solution reaches the same conclusion by formulating a scalar fixed-point equation in R and parameterizing the Preisach uncertainty via p+(u), p0(u), and θ. Aside from a minor inequality slip when showing fθ is strictly decreasing in R (readily repaired by writing g(u)=p+(u)+θp0(u)=θ+(1−θ)p+(u)−θp−(u), which is nondecreasing because p+ is nondecreasing and p− is nonincreasing), the solution is correct and consistent with the paper’s result and assumptions. Overall, both arguments establish the same equilibrium continuum, with the candidate using a different but equivalent monotonicity framework.