Substance use disorder (SUD) is a complex disease involving nontrivial biological, psychological, environmental, and social factors. While many mathematical studies have proposed compartmental models for SUD, almost all of these exclusively model new cases as the result of an infectious process, neglecting any SUD that was primarily developed in social isolation. While these decisions were likely made to facilitate mathematical analysis, isolated SUD development is critical for the most common substances of abuse today, including opioid use disorder developed through prescription use and alcoholism developed primarily due to genetic factors or stress, depression, and other psychological factors. In this paper we will demonstrate that even a simple infectious disease model is structurally unstable with respect to a linear perturbation in the infection term — precisely the sort of term necessary to model SUD development in isolation. This implies that models of SUD which exclusively treat problematic substance use as an infectious disease will have misleading dynamics whenever a non-trivial rate of isolated SUD development exists in actuality. As we will show, linearly perturbed SUD models do not have a use disorder-free equilibrium. To investigate management strategies, we implement optimal control techniques with the goal of minimizing the number of SUD cases over time.

中文翻译：

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物质使用障碍广义模型中的结构不稳定性和线性分配控制

物质使用障碍（SUD）是一种复杂的疾病，涉及重要的生物、心理、环境和社会因素。虽然许多数学研究提出了 SUD 的区室模型，但几乎所有这些研究都专门将新病例建模为感染过程的结果，而忽略了主要在社会隔离中发展的任何 SUD。虽然这些决定可能是为了促进数学分析，但孤立的 SUD 发展对于当今最常见的滥用药物至关重要，包括通过处方使用产生的阿片类药物使用障碍和主要由于遗传因素或压力、抑郁和其他心理因素而产生的酗酒。在本文中，我们将证明，即使是简单的传染病模型，在感染项的线性扰动方面也是结构不稳定的，而感染项正是对孤立的 SUD 发展进行建模所必需的项。这意味着，只要现实中存在单独的 SUD 发展率，专门将有问题的物质使用视为传染病的 SUD 模型就会产生误导性动态。正如我们将要展示的，线性扰动 SUD 模型不具有使用无序平衡。为了研究管理策略，我们实施了最优控制技术，目标是随着时间的推移最大限度地减少 SUD 病例的数量。