Optimal control analysis applied to the transmission dynamics of Hepatitis B to primary liver cancer

Infection with hepatitis virus, especially hepatitis B virus causes an irritation to the liver. Ranked among the top ten diseases with high mortality rate, viral hepatitis poses a great health challenges worldwide, with threat to chronic infection, hepatitis-related liver cancer and cirrhosis. Hence, a mathematical model to study the dynamics of hepatitis B virus infection from progressing into primary liver cancer was developed for analysis. We aimed at obtaining the optimal control strategies needed to reduce the number of new cases of this disease, and also reducing the deterioration rate of people living with this disease from sliding into primary liver cancer. We introduced four distinct control variables at each point in the model, and assumed that all the controls are set of Lebesque Measurable functions. The Pontryagin’s maximum principle (PMP) is employed to establish the optimal effect of these controls on the disease under study. Existence of model solution was established using the appropriate theorem. The model with control strategies was analytically solved using PMP and numerically simulated for each compartments, to establish the effect of the control variables on the dynamics of transmission of this infection within the compartment, and their overall effect on the entire population. Differential transform method (DTM) was later adopted as a semi-analytic scheme to solve the developed model. The series solution of DTM was numerically plotted for each compartment and compared with Runge-Kutta order 4 (RK4) numerical scheme. Analysis of the model with control pinpoint the importance of sensitization and vaccination on the overall dynamics of the infection, while the numerical plot of DTM and RK4 established the efficacy of the adopted semi-analytic method (DTM) to accurately solve the system of equations of the model.