Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus

Figure 1

Flow diagram of HBV dynamics under application of vaccine and treatment. 𝑆, 𝐸, 𝐼, 𝐶, and 𝑅 denote five compartment of susceptible, exposed, acute infection, chronic carriers, and immune or recovered class, respectively. The 𝜌(𝐼+𝜃𝐶)𝑆 indicates the horizontal transmission from compartment 𝑆 to 𝐸, whereas 𝜈𝑝₁𝐶 denotes the vertical transmission from 𝐶 to 𝐶 by birth of offspring from a carrier individual. Similarly, 𝜈𝑝₂𝑅 represents the proportion of immune newborn from recovered class. 𝜆₃𝐶 shows individuals’ spontaneous recovery and move from compartment 𝐶 to 𝑅 and 𝑢₂𝐶 denotes proportions of carriers move to recovered class by treatment. 𝜆₄𝑅 denotes a portion that moves from compartment 𝑅 to 𝑆 due to loss of immunity. 𝑝₃𝜆₂𝐼 shows proportion of acute infection individuals who become carries and (1−𝑝₃)𝜆₂𝐼 shows proportion of acute infection individuals, clear HBV, and move from 𝐼 to 𝑅. 𝑢₁𝑆 denotes proportions of susceptible move to recovered class by vaccination and each compartment has its own death rate.