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

• * Corresponding author: Asaf Khan
• 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.

Mathematics Subject Classification: Primary: 49J20, 49J50; Secondary: 65M06.

 Citation: • • Figure 1.  The plot represent the density of susceptible population $s(t, a)$ without control

Figure 2.  The plot shows the behavior of the density of susceptible population $s(t, a)$ with control at time $t$ and age $a$

Figure 3.  The plot represent the density of infected population $i(t, a)$ without control

Figure 4.  The plot shows the behavior of the density of infected population $i(t, a)$ with control at time $t$ and age $a$

Figure 5.  The plot represent the density of recovered population $r(t, a)$ without control

Figure 6.  The plot shows the behavior of the density of recovered population $r(t, a)$ with control at time $t$ and age $a$

Figure 7.  Behavior of the control variable with respect to time $t$ and age $a$

Figure 8.  The curves blue, green and red represents the solution profile of $s(t, a)$ at fixed ages $12$, $28$ and $52$, respectively

Figure 9.  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

Figure 10.  The curves blue, green and red represents the solution profile of $i(t, a)$ at fixed ages $12$, $28$ and $52$, respectively

Figure 11.  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

Figure 12.  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

Figure 13.  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

Figure 14.  The solid (blue), dotted (red) and dashed (green) represents sample curves of $u(t, a)$ at different ages $12$, $28$ and $52$, respectively

Figure 15.  The solid (blue), dotted (red) and dashed (green) represents sample curves of $u(t, a)$ at different time $8$, $24$ and $40$, respectively

Table 1.  Parameters values used in numerical simulation

 Parameters Values References B 0.02 Assumed $\hat{\lambda}(t, a)$ 0.03  $\mu(a)$ $0.01(1+\sin((a-20)\frac{\pi}{40}))$ Assumed $\gamma(a)$ $0.2$  $b(a)$ \$ \left\{ \begin{array}{lll} 0.2\sin^2((a-15)\frac{\pi}{30}), \; 15
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