When Self-Protection Backfires: Adaptive Contact Behavior Expands the Endemic Basin in an Addiction Model with Nonlinear Relapse

We extend the Susceptible–Addicted–Reformed (SAR) model of \cite{sanchez2023}, which exhibits a forward–backward bifurcation driven by nonlinear relapse, by embedding an epi-economic behavioral layer in the spirit of \cite{fenichel2011}. At-risk individuals choose contact levels by solving a finite-horizon dynamic program that balances the utility of social activity against the expected cost of addiction. We prove that the basic reproduction number $R_0$ and the local stability of the addiction-free equilibrium remain unchanged by the behavioral layer. However, the endemic structure is fundamentally altered: the behavioral response collapses exactly to a scalar mixing factor $M$, and the bifurcation curve factorizes as $R_0(a)=R_{\mathrm{c}\mathrm{l}}(a)\mathrm{/}M(a)$. This yields an exact comparison principle: the saddle-node fold shifts to lower $R_0$ (enlarging the endemic basin) if and only if $M\ge 1$ along the branch. For rational self-protective behavior under conditional proportional mixing, we prove $M\ge 1$, so the basin enlarges; the opposite holds under frequency-dependent mixing. Numerical continuation shows that, at baseline parameters, the fold moves left by $\mathrm{\Delta }{R}_0\approx -0.035$ and the critical initial addiction level drops by 2–6 percentage points. Gillespie simulations confirm that the enlarged basin increases the stochastic probability of addiction establishment by up to threefold near threshold. This counterintuitive result that rational self-protection can make addiction easier to establish has direct implications for prevention policy.