Bistability and complex bifurcation diagrams generated by waning and boosting of immunity

The paper derives the characteristic equation W(λ,τ)=P(λ,τ)+Q(λ,τ)e^{-λτ}, defines F(ω,τ)=|P(iω,τ)|^2−|Q(iω,τ)|^2=ω^2(ω^4+a1(τ)ω^2+a0(τ)), and proves that the delay interval admitting positive roots is finite (Proposition 1) and that the endemic equilibrium eventually becomes stable for any ν (Lemma 2). These constructions, including explicit μ(τ), σ(τ), a0(τ), a1(τ), match the candidate’s setup and limits a0∞>0, a1∞ (the paper gives equivalent closed forms) and the ensuing boundedness of the Hopf window J± (see the model statement (1), the definition of μ,σ,P,Q and F, and the asymptotic formulas in (8) ). The paper also notes stability at τ=0 as theoretically expected . The candidate’s proof employs a clean sum-of-squares identity at μ=0 to control F for large τ and a direct Jacobian argument for τ≈0, then infers finiteness of switches from bounded τ and ω together with the phase condition; this differs in technique from the paper’s Beretta–Kuang monotonic-switch framework but reaches the same conclusions (finite τ-window for Hopf, stability for small and large τ, finitely many switches). Minor caveat: the candidate’s argument that only finitely many zeros occur inside a bounded τ-window relies on isolated zeros and bounded k but does not explicitly supply the monotonicity used in the paper to count switches; still, the conclusions align with the paper’s results and setup (J± bounded; finite switch count) .