Optimal Control Studies on Age Structural Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

Abstract: In this study initially, we propose an age structured model and calculate the equilibrium points and basic reproduction number. Later we propose an optimal control problem to understand the roles of treatment in controlling the epidemic. From the Stability analysis we see that the infection free equilibrium remains asymptotically stable whenever $R_0 < 1$ and as $R_0$ crosses unity we have the infected equilibrium to be stable. From the sensitivity analysis parameters $u_{11}$, $b_1$, $\beta_1$, $d_1$ and $\mu$ were found to be sensitive. Findings from the Optimal Control studies suggests that the infection among the adult population(age $\geq 30)$ is least considering the second control $u_{12}$ whereas, when both the controls $u_{11}$ and $u_{12}$ are considered together the infectives is minimum in case of young populations(age $ \leq 30$). The cumulative infected population reduced the maximum when the second control was considered followed by considering both the controls together. The control $u_{12}$ was effective for mild epidemic $(R_0 \in(1, 2))$ whereas control $u_{11}$ was found to be highly effective when epidemic was severe $(R_0 \in(2, 7))$ for the population of age group $(\leq 30)$. Whereas for age group $(\geq 30)$ the control $u_{12}$ was highly effective for the entire range of basic reproduction number. The effect of saturation level in treatment is also explored numerically.