Structural instability and linear allocation control in generalized models of substance use disorder

Area of Science:

• Epidemiology
• Mathematical Biology
• Public Health

Background:

• Substance use disorder (SUD) is a complex disease influenced by biological, psychological, environmental, and social factors.
• Existing mathematical models for SUD primarily focus on infectious processes, neglecting isolated development pathways.
• Isolated SUD development is crucial for understanding common substance abuse, including prescription opioid use and alcoholism.

Purpose of the Study:

• To demonstrate the structural instability of simple infectious disease models for SUD when considering isolated development.
• To analyze the impact of isolated SUD development on model dynamics and equilibrium.
• To explore optimal control strategies for minimizing SUD cases over time.

Main Methods:

• Linear perturbation analysis of a basic infectious disease model for SUD.
• Investigation of model dynamics with a non-trivial rate of isolated SUD development.
• Implementation of optimal control techniques to assess management strategies.

Main Results:

• Simple infectious disease models for SUD are structurally unstable when incorporating isolated development.
• Linearly perturbed SUD models lack a disorder-free equilibrium, indicating persistent cases.
• Optimal control strategies can be applied to minimize SUD prevalence.

Conclusions:

• Exclusive reliance on infectious disease models for SUD can yield misleading dynamics.
• Accounting for isolated SUD development is essential for accurate modeling and effective interventions.
• Mathematical modeling combined with optimal control offers a framework for developing public health strategies against SUD.