Analysis of Vertical Transmission Dynamics of Infectious Hepatitis B Virus: Mathematical Model Involving Vaccination And Treatment in Turkana County, Kenya

Abstract

Hepatitis B has been a major global health menace for it is a potentially life-threatening liver disease. Close to 0.25 billion persons are living with this infectious disease across the world. It’s transmitted by infected individual to uninfected person either vertically (transmission before or during birth by carrier mother to the baby) or horizontally (transmission when the bodily fluid of an infected person comes into contact with the hepatitis B virus-free person). This can happen through the sharing of non-sterilized injection syringes, tattooing objects and through sexual intercourse. This particular project studied a mathematical model that combined both vaccination and treatment as a means to controlling the hepatitis B virus (HBV). In our mathematical model, equations are derived from the flow chart representing the HBV transmission dynamics. The determination of the disease-free equilibrium state (DFE), the endemic equilibrium state (EE) and the basic reproduction number 𝑅0 were made. The stability of these points were determined and the results show that the disease-free equilibrium was both locally and globally asymptotically stable. In other words, 𝑅0 < 1. The stability analysis of endemic equilibrium point also reveals that the point is locally and globally asymptotically stable, that is, 𝑅0 > 1. The basic reproduction number 𝑅0 is computed using the next generation matrix method. The system of ordinary differential equations (ODEs), which is non-linear are solved by numerical simulation. This was achieved by use of Runge-kutta method of order four with the help of MATLAB software. These results show that either of the method, treatment or vaccination, administered is effective in alleviating the spread of HBV disease. However, when both control strategies are combined, the disease is quickly controlled and ultimately brought to eradication.