Hepatitis B Virus (HBV) infection and heavy alcohol consumption are the two primary pathogenic causes of liver cirrhosis. In this paper, we proposed a deterministic mathematical model and a logistic equation to investigate the dynamics of liver cirrhosis progression as well as to explain the implications of variations in alcohol consumption on chronic hepatitis B patients, respectively. The intricate interactions between liver cirrhosis, recovery, and treatment dynamics are captured by the model. This study aims to show that alcohol consumption by Hepatitis B-infected individuals accelerates liver cirrhosis progression while treatment of acutely infected individuals reduces it. We proved that a unique solution of the proposed model exists, which is positive and bounded. Using the next-generation matrix approach, two basic reproductive numbers RA0 and RAmax are calculated to identify future recurrence. The equilibrium points are calculated, and both equilibria are proved locally and globally asymptotically stable when R0 is below and above one, respectively. It is shown that bifurcation exists at R0 = 1 and a detailed proof for forward bifurcation is given. Furthermore, we performed the sensitivity analysis of the model parameters on R0. For the confirmation of analytical work, we performed numerical simulations, and the results indicate that the treatment and the inhibitory effects reduce the risk of developing liver cirrhosis in individuals, while heavy alcohol consumption accelerates markedly the liver cirrhosis progression in patients with chronic hepatitis B.

This study presents a Pascal polynomial-based numerical framework for solving inverse heat conduction problems governed by the steady-state Poisson equation. The proposed methodology employs the Pascal Polynomial Collocation Method (PPCM) and its regularized variant (PPCM-T) to reconstruct unknown boundary conditions and source terms in two-dimensional bounded domains. Two complementary strategies are adopted: one directly approximates both the temperature field and the unknown source term using Pascal polynomials, while the other reformulates the inverse problem as a direct fourth-order system by assuming that the unknown source satisfies Laplace's equation. Several benchmark examples defined over rectangular and annular geometries are investigated to assess the method's accuracy and stability. The results demonstrate that both PPCM and PPCM-T achieve excellent agreement with analytical or reference solutions under noise-free data, with systematic error reduction as the number of collocation nodes increases. Under noisy boundary data, PPCM-T exhibits superior robustness due to its built-in regularization, maintaining acceptable accuracy and numerical stability. Overall, the Pascal polynomial-based framework provides a flexible, mesh-free, and computationally efficient approach for addressing inverse elliptic and heat conduction problems, offering strong potential for broader applications in computational mechanics and thermal analysis.