Analysis and achievement for fractional optimal control of Hepatitis B with Caputo operator

Abstract

Deciphering the effect of long memory on dynamic behavior is important for controlling disease transmission. Employing fractional order calculus, here, we propose a model with Caputo operator to emphasize the regulation effect of the long memory on the optimal control of Hepatitis B. Considering four different health classes of populations, susceptible, acute infections, chronic infections, and recovered, we try to elucidate the existence of the optimal solution of the fractional optimal control problems (FOCP) by applying a strategy of isolation, treatment, and vaccination. Initially, we discuss the locally asymptotically stability of the equilibrium solutions of the fractional differential model. Sensitivity analysis and numerical simulations are also carried out to verify how parameter changes affect the system dynamics, that is, the threshold effect depending on the basic reproduction number, which provides inspiration for constructing the controlled system. After that, the necessary conditions for the optimality of the Hepatitis B system are derived to focus on the Caputo fractional operator with a singular kernel. An effective numerical simulation algorithm based on trapezoidal approximation is used to validate the control effect in the sense of the transient response by comparing the difference between the fractional-order and integer-order derivatives, and the numeric results can vividly show the attenuation effect of memory on control. This study may provide a powerful rationale for uncovering the importance of the long-memory effects in disease control.

Keywords

• Fractional optimal control
• Hepatitis B epidemic model
• Stability
• Sensitivity analysis
• Singular kernel