Modeling the dynamics of acute and chronic hepatits B with optimal control

Scientific Reports
(2023) 13:14980
doi: 10.1038/s41598-023-39582

Deterministic Model

$$ \Large \begin{cases} \frac{dS}{dt} &= (1 - \eta B)\Lambda - (\nu + \mu_0)S - (A + \gamma B)\alpha S, \\ \frac{dA}{dt} &= \alpha SA + \gamma\alpha SB - (\gamma_1 + \beta + \mu_0)A, \\ \frac{dB}{dt} &= \beta A - (\mu_1 + \gamma_2 + \mu_0 - \eta\Lambda)B, \\ \frac{dR}{dt} &= \gamma_2 B - \mu_0 R + \gamma_1 A + \nu S. \end{cases} $$

$\large\Lambda\colon\text{The rate of newborns,}$
$\large\nu\colon\text{The vaccination parameter,}$
$\large\eta\colon\text{The maternally infected,}$
$\large\gamma\colon\text{The reduced transmission rate,}$
$\large\mu_0\colon\text{The proportion of natural death,}$
$\large\mu_1\colon\text{The portion of death due to the disease,}$
$\large\alpha\colon\text{The contact parameter,}$
$\large\gamma_1\colon\text{The recovery rate from acute class,}$
$\large\gamma_2\colon\text{The recovery rate from chronic class,}$
$\large\beta\colon\text{The proportion who move from acute class to chronic.}$

$\Lambda = 0.0121, \quad \eta = 0.5, \quad \mu_0 = 0.01693, \quad \nu = 0.02, \quad \alpha = 0.95$
$\gamma = 0.16, \quad \gamma_1 = 0.05, \quad \beta = 0.23, \quad \gamma_2 = 0.002, \quad \mu_1 = 0.8$

$S_0 = 80, \quad A_0 = 10, \quad B_0 = 5, \quad R_0 = 5$

Controlled Mode

$$ \Large \begin{cases} \frac{dS}{dt} &= (1 - \eta B)\Lambda - (\mu_0 + u_1)S - \alpha S (A + \gamma B), \\ \frac{dA}{dt} &= \alpha S (A + \gamma B) - (u_2 + \mu_0 + \gamma_1 + \beta)A, \\ \frac{dB}{dt} &= \beta A - (\mu_1 + \gamma_2 + \mu_0 + \eta\Lambda + u_2) B, \\ \frac{dR}{dt} &= \gamma_2 B - \mu_0 R + \gamma_1 A + u_1 S + (B + A)u_2. \end{cases} $$

$u_1(t)$: Vaccination effort
$u_2(t)$: Treatment effort

$$ \large 0 \le u_1(t), u_2(t) \le 1 $$

$w_1 = 0.1, \quad w_2 = 0.6, \quad w_3 = 0.001, \quad w_4 = 0.9$

Controlled vs Uncontrolled (Deterministic)

Control Functions

Reproductive Number

$$ \Large R_0 = \frac{\alpha\Lambda (\gamma(\mu_0 + \gamma_1 + \beta + u_2) + \beta)}{(\mu_0 + u_1)(\mu_0 + \gamma_1 + \beta + u_2)(\mu_0 + \mu_1 + \gamma_2 + u_2 - \eta\Lambda)} $$