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July  2021, 14(7): 2535-2555. doi: 10.3934/dcdss.2021054

## Optimal control strategy for an age-structured SIR endemic model

 1 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China 2 Department of Mathematics & Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan 3 Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan 4 Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author: Asaf Khan

Received  July 2019 Revised  December 2020 Published  May 2021

In this article, we consider an age-structured SIR endemic model. The model is formulated from the available literature while adding some new assumptions. In order to control the infection, we consider vaccination as a control variable and a control problem is presented for further analysis. The method of weak derivatives and minimizing sequence argument are used for deriving necessary conditions and existence results. The desired criterion is achieved and sample simulations were presented which shows the effectiveness of the control.

Citation: Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan, Gul Zaman. Optimal control strategy for an age-structured SIR endemic model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2535-2555. doi: 10.3934/dcdss.2021054
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##### References:
The plot represent the density of susceptible population $s(t, a)$ without control
The plot shows the behavior of the density of susceptible population $s(t, a)$ with control at time $t$ and age $a$
The plot represent the density of infected population $i(t, a)$ without control
The plot shows the behavior of the density of infected population $i(t, a)$ with control at time $t$ and age $a$
The plot represent the density of recovered population $r(t, a)$ without control
The plot shows the behavior of the density of recovered population $r(t, a)$ with control at time $t$ and age $a$
Behavior of the control variable with respect to time $t$ and age $a$
The curves blue, green and red represents the solution profile of $s(t, a)$ at fixed ages $12$, $28$ and $52$, respectively
The plot shows the solution profile of $s(t, a)$ at fixed time $8$, $24$ and $40$ represented by blue, green and red curves, respectively
The curves blue, green and red represents the solution profile of $i(t, a)$ at fixed ages $12$, $28$ and $52$, respectively
The plot shows the solution profile of $i(t, a)$ at fixed time $8$, $24$ and $40$ represented by blue, green and red curves, respectively
The curves blue, green and red represents the solution profile of $r(t, a)$ at fixed ages $12$, $28$ and $52$, respectively. The solid and dotted curves represent solution profile without and with control, respectively
The plot shows the solution profile of $r(t, a)$ at fixed time $8$, $24$ and $40$ represented by blue, green and red curves, respectively. Whereas, the solid and dotted curves represent solution profile without and with control, respectively
The solid (blue), dotted (red) and dashed (green) represents sample curves of $u(t, a)$ at different ages $12$, $28$ and $52$, respectively
The solid (blue), dotted (red) and dashed (green) represents sample curves of $u(t, a)$ at different time $8$, $24$ and $40$, respectively
Parameters values used in numerical simulation
 Parameters Values References B 0.02 Assumed $\hat{\lambda}(t, a)$ 0.03 [26] $\mu(a)$ $0.01(1+\sin((a-20)\frac{\pi}{40}))$ Assumed $\gamma(a)$ $0.2$ [13] $b(a)$ $\left\{ \begin{array}{lll} 0.2\sin^2((a-15)\frac{\pi}{30}), \; 15  Parameters Values References B 0.02 Assumed$ \hat{\lambda}(t, a) $0.03 [26]$ \mu(a)  0.01(1+\sin((a-20)\frac{\pi}{40})) $Assumed$ \gamma(a)  0.2 $[13]$ b(a)  \left\{ \begin{array}{lll} 0.2\sin^2((a-15)\frac{\pi}{30}), \; 15
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