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<p>Epidemic dynamics often shift abruptly as behavior and policy flip between a regular regime and an alarm regime in response to prevalence. To capture these on–off changes without sacrificing analytical clarity, we cast an SIR model as a Piecewise-Deterministic Markov Process (PDMP): transmission alternates between two levels, and the system switches endogenously into alarm when cases rise and back to regular when they fall. We first establish global well-posedness and prove the strong Markov property. Linearization at the disease-free equilibrium then yields a closed-form control threshold: extinction occurs precisely when the time-averaged transmission under baseline mixing is below the combined removal rate. The resulting gap between removal and averaged transmission governs rates-positive gap implies exponential decay in the mean and almost-sure extinction from small initial prevalence; negative gap implies a strictly positive probability of finite-time growth while hazards remain near baseline. The framework delivers transparent comparative statics: faster deactivation or stronger regular contact makes control harder, whereas a more effective alarm or faster removal makes it easier. A split-step Euler-thinning simulator, weakly consistent with the generator, verifies these results across scenarios and maps a phase diagram in the activation-deactivation plane. Finally, we analyze a real-data case study (216 weekly COVID-19 observations from Morocco (March 2020-April 2024)) using an incidence-dependent PDMP to detect alarm/regular switching, with decoded plateaus aligning with observed waves and offering policy-relevant early warning. The approach isolates actionable levers-activation speed, alarm persistence, and alarm effectiveness; and explains trigger-driven shifts seen in practice, such as hospital-capacity thresholds or institutional case rules, thereby bridging classical compartmental modeling with discontinuous behavioral response.</p>

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A new method to derive baseline-mixing thresholds and extinction criteria in a PDMP-SIR with endogenous alarm

  • 期刊论文

  • 发表日期：

    2026年2月7日

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  • Yassine Sabbar,
  • Abdulwasea Alkhazzan,
  • Jungang Wang,
  • Jie Zhou

摘要

Epidemic dynamics often shift abruptly as behavior and policy flip between a regular regime and an alarm regime in response to prevalence. To capture these on–off changes without sacrificing analytical clarity, we cast an SIR model as a Piecewise-Deterministic Markov Process (PDMP): transmission alternates between two levels, and the system switches endogenously into alarm when cases rise and back to regular when they fall. We first establish global well-posedness and prove the strong Markov property. Linearization at the disease-free equilibrium then yields a closed-form control threshold: extinction occurs precisely when the time-averaged transmission under baseline mixing is below the combined removal rate. The resulting gap between removal and averaged transmission governs rates-positive gap implies exponential decay in the mean and almost-sure extinction from small initial prevalence; negative gap implies a strictly positive probability of finite-time growth while hazards remain near baseline. The framework delivers transparent comparative statics: faster deactivation or stronger regular contact makes control harder, whereas a more effective alarm or faster removal makes it easier. A split-step Euler-thinning simulator, weakly consistent with the generator, verifies these results across scenarios and maps a phase diagram in the activation-deactivation plane. Finally, we analyze a real-data case study (216 weekly COVID-19 observations from Morocco (March 2020-April 2024)) using an incidence-dependent PDMP to detect alarm/regular switching, with decoded plateaus aligning with observed waves and offering policy-relevant early warning. The approach isolates actionable levers-activation speed, alarm persistence, and alarm effectiveness; and explains trigger-driven shifts seen in practice, such as hospital-capacity thresholds or institutional case rules, thereby bridging classical compartmental modeling with discontinuous behavioral response.