Optimal Control and Bifurcation Analysis of Cholera Model
Keywords:
Cholera, Optimal Control, Stability Analysis, Local Stability, Bifurcation AnalysisAbstract
Cholera presents a significant public health challenge, especially in regions lacking access to clean water and proper sanitation. This study aims to develop an optimal control framework to effectively mitigate the dynamic spread of cholera and inform more efficient intervention strategies. Initially, we establish the well-posedness of the cholera transmission model, demonstrating the positivity and boundedness of solutions within a specified domain. Utilizing the next-generation matrix approach, we compute the basic reproduction number, a critical epidemiological threshold for disease dynamics. Upon obtaining a value below one, we employ the Jacobian matrix, Metzler matrix, and Lyapunov function analysis to verify the local and global stability of the cholera-free equilibrium. Sensitivity analysis highlights the significant impact of certain model parameters, such as the transmission rate and treatment efficacy, on cholera control. Leveraging the Pontryagin minimum principle, we formulate an optimal control problem to derive the most effective combination of prevention and treatment strategies. Numerical simulations illustrate that the optimal control approach, involving the simultaneous implementation of both interventions, surpasses standalone prevention or treatment measures in reducing the disease burden. This study underscores the importance of integrating prevention and treatment strategies for the effective mitigation of cholera outbreaks. The proposed optimal control framework provides a systematic approach for evaluating the impact of various interventions and informing public health policies. Future work will involve model extensions to incorporate spatial heterogeneity, fractional derivatives, and real-world data integration, enhancing the applicability and robustness of the control strategies.
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