Global Dynamics of a Fractional Order Hepatitis B Model with Media Effect Incorporating Optimal Control and Stochastic Stability

A six-compartment fractional model of hepatitis B was developed to analyze the transmission mechanisms of the infection, also taking into account the impact of the media on the population. Its biological validity was confirmed from an epidemiological point of view. The basic reproduction number, R_0 , was determined using the new generation matrix method. A study of the asymptotic behavior around the equilibrium points showed that the model undergoes a transcritical bifurcation when R_0 = 1, thus establishing R_0 as a critical threshold influencing the evolution of the epidemic. Fractional-order optimal control was applied, incorporating time-evolving prevention and treatment strategies. The results, validated by simulations using realistic parameters, revealed that the fractional approach offers superior performance. The application of prevention and treatment measures led to a progressive reduction in the number of cases and a decrease in the epidemic peak. The analysis was extended to a stochastic framework, incorporating white noise, to study the stochastic stability of the endemic equilibrium point. In addition, sensitivity indices were used to identify key parameters influencing R_0,, facilitating the development of disease control strategies. Numerical simulations corroborated the theoretical results obtained.