Modeling Hepatitis B and Alcohol Effects on Liver Cirrhosis Progression

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.