Asymptotic Properties of an Age‐Structured Model of Heroin Transmission

ABSTRACT In this work, we treat an age‐structured epidemiological model of heroin transmission that extends the framework of Fang et al. by incorporating a spontaneous recovery mechanism. Specifically, we introduce a new parameter representing the rate at which untreated drug users cease heroin use and revert to the susceptible class without formal treatment intervention. This addition addresses a critical gap in existing models, as spontaneous recovery through social support, education, and self‐motivation is a commonly observed phenomenon in real‐world addiction dynamics. We establish the existence of two equilibria: the drug‐free equilibrium and the drug‐persistent equilibrium . The basic reproduction number is derived as , where the presence of in the denominator demonstrates that spontaneous recovery acts as an additional control mechanism, reducing and facilitating disease eradication. We prove that the drug‐free equilibrium is locally and globally asymptotically stable when , while the drug‐persistent equilibrium is locally asymptotically stable when . Furthermore, we establish uniform persistence of heroin use in the population when . A key contribution of our analysis is the derivation of two critical thresholds for heroin eradication: the critical treatment rate and the critical spontaneous recovery rate . These thresholds reveal that public health interventions can target either formal treatment programs (increasing beyond ) or community‐based initiatives that promote spontaneous recovery (increasing beyond ), providing two complementary strategies for controlling heroin addiction. Numerical simulations validate our theoretical results and illustrate the impact of the spontaneous recovery parameter on the long‐term dynamics of heroin transmission. Our findings provide a quantitative framework for evaluating the effectiveness of nonclinical intervention strategies and optimizing resource allocation between treatment facilities and community support programs.