Mathematical Model of Immune Response to Hepatitis B Virus and Liver Cancer Co-Existence Dynamics in the Presence of Treatment

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.