Table 1 Parameters and their biological meaning in model (1.2)

Λ

constant recruitment rate

μ

natural death rate

p

the rate for susceptible individuals to be vaccinated

ρ

the rate for vaccinees to obtain immunity and move into recovered population

η

the proportion of vaccines that lose efficacy

\(\beta_{1}(a)\)

the rate for acute hepatitis B patients infecting susceptible individuals at age a

\(\beta_{2}(a)\)

the rate for acute hepatitis B patients infecting vaccinees at age a

\(\beta_{3}(a)\)

the rate for chronic hepatitis B patients infecting susceptible individuals at age a

\(\beta_{4}(a)\)

the rate for chronic hepatitis B patients infecting vaccinees at age a

\(\gamma_{1}(a)\)

the rate for exposed individuals being acute hepatitis B patients at age a

\(\gamma_{2}(a)\)

the rate for exposed individuals being chronic hepatitis B patients at age a

\(\varepsilon_{1}(a)\)

acute hepatitis B death rate at age a

\(\varepsilon_{2}(a)\)

chronic hepatitis B death rate at age a

\(\delta_{1}(a)\)

the rate for acute hepatitis B patients being recovered population at age a

\(\delta_{2}(a)\)

the rate for chronic hepatitis B patients being recovered population at age a