Bifurcation Analysis and Optimal Control of a Nonlinear SEIR‐M Model for Scabies Transmission

Abstract

Scabies is a highly contagious parasitic skin disease that poses persistent public health challenges, particularly in overcrowded and resource‐limited settings. A rigorous mathematical framework is developed to investigate the transmission dynamics of scabies by incorporating both direct human‐to‐human transmission and indirect environmental contamination by mites. A nonlinear SEIR‐M compartmental model is formulated, coupling human population dynamics with an environmental mite reservoir. Qualitative analysis establishes the positivity and boundedness of solutions, ensuring the biological feasibility of the model. The basic reproduction number is derived and shown to govern the global dynamics of the system. Local and global stability analyses of the disease‐free and endemic equilibria are performed, revealing threshold conditions for disease persistence and elimination. Bifurcation analysis demonstrates the occurrence of transcritical, backward, and Hopf bifurcations, highlighting the rich nonlinear dynamics induced by environmental transmission. An optimal control problem is formulated to evaluate intervention strategies involving treatment, public awareness, and environmental sanitation. Necessary optimality conditions are derived using Pontryagin's Maximum Principle and solved numerically via the forward‐backward sweep method. Numerical simulations indicate that combined control strategies substantially reduce infection prevalence and environmental mite concentration compared to isolated interventions. The results provide a mathematically rigorous foundation for understanding scabies dynamics and demonstrate the effectiveness of integrated control strategies for environmentally mediated infectious diseases.