Abstract:
The principal method for modeling the spread of infectious diseases generally involves
the application of ordinary differential equations. Studies have demonstrated
that an effective strategy for refining certain mathematical models is the integration
of fractional-order differential equations. To gain a more profound understanding of
the interactions between the hepatitis B virus (HBV), liver cancer, and immune system
cells, a mathematical model that combined both ordinary and fractional differential
equations was investigated. This model was closely aligned with experimental
data on viral DNA load. The work concentrated on four qualitative scenarios: the
innate immune response, adaptive immune response, cytokine response, and the coexistence
of infection dynamics. Unlike earlier models, liver cells were classified
into distinct stages of infection. For populations of non-pathogenic macrophages in
the presence and absence of malignant cells, the study calculated the invasion probability
for transmission dynamics, represented by the control reproduction number,
Rc. The iterated two-step Adams-Bashforth method was employed for numerical
simulations using the ABC fractional derivative in the Caputo sense, while the Latin
Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) techniques
were utilized for parameter sensitivity analysis. The work identified the key
transmission mechanism of viral load and proposed an optimal therapeutic method
for viral treatment. Model parameters were estimated using nonlinear least squares
fitting of longitudinal data (serum HBV DNA viral load) from existing literature.
Finally, the study compared the classical-order model system with the ABC fractional
differential equations model system to determine which offered superior performance.
Both methods were evaluated using simulation results of the state variables,
revealing that the fractional model provides more detailed results than the
classical model.
