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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  July 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2535-2555. doi: 10.3934/dcdss.2021054
##### References:
 [1] A. Alexanderian, M. K. Gobbert, K. R. Fister, H. Gaff, S. Lenhart and E. Schaefer, An age-structured model for the spread of epidemic cholera: Analysis and simulation, Nonlinear Analysis: Real World Applications, 12 (2011), 3483-3498.  doi: 10.1016/j.nonrwa.2011.06.009. [2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Springer, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [3] T. Arbogast and F. A. Milner, A finite difference method for a two-sex model of population dynamics, SIAM Journal of Numerical Analysis, 26 (1989), 1474-1486.  doi: 10.1137/0726086. [4] V. Barbu and M. Iannelli, Optimal control of population dynamics, Journal of Optimization Theory & Applications, 102 (1999), 1-14.  doi: 10.1023/A:1021865709529. [5] S. Bowong, Optimal control of the dynamics of tuberculosis, Nonlinear Dynamics, 61 (2010), 729-748.  doi: 10.1007/s11071-010-9683-9. [6] L.-M. Cai, C. Modnak and J. Wang, An age-structured model for cholera control with vaccination, Applied Mathematics & Computations, 299 (2017), 127-140.  doi: 10.1016/j.amc.2016.11.013. [7] R. D. Demasse, J.-J. Tewa, S. Bowong and Y. Emvudu, Optimal control for an age-structured model for the transmission of hepatitis B, Journal of Mathematical Biology, 73 (2016), 305-333.  doi: 10.1007/s00285-015-0952-6. [8] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling, 22 (2009), 173-211.  doi: 10.1111/j.1939-7445.2008.00033.x. [9] Y. Emvudu, R. D. Demasse and D. Djeudeu, Optimal control using state-dependent Riccati equation of lost of sight in a tuberculosis model, Computational and Applied Mathematics, 32 (2013), 191-210.  doi: 10.1007/s40314-013-0002-1. [10] K. R. Fister and S. Lenhart, Optimal control of a competitive system with age-structured, Journal of Mathematical Analysis & Applications, 291 (2004), 526-537.  doi: 10.1016/j.jmaa.2003.11.031. [11] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron Journal of Differential Equations, 32 (1998), 1-12. [12] G. Grippenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, UK, 1990.  doi: 10.1017/CBO9780511662805. [13] D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3 (2006), 63-69. [14] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [15] M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, GX Dordrecht, The Netherlands, 2017. doi: 10.1007/978-94-024-1146-1. [16] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [17] H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710. [18] A. Khan and G. Zaman, Asymptotic behavior of an age structure SIRS endemic model, Applied and Computational Mathematics, 17 (2018), 185-204. [19] A. Khan and G. Zaman, Global analysis of an age-structured SEIR endemic model, Chaos, Solitons and Fractals, 108 (2018), 154-165.  doi: 10.1016/j.chaos.2018.01.037. [20] A. Khan and G. Zaman, Optimal control strategy of SEIR endemic model with continuous age-structure in the exposed and infectious classes, Optimal Control Applications & Methods, 39 (2018), 1716-1727.  doi: 10.1002/oca.2437. [21] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, Journal of Mathematical Biology, 35 (1997), 775-792.  doi: 10.1007/s002850050076. [22] T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, Journal of Mathematical Analysis & Applications, 402 (2013), 477-492.  doi: 10.1016/j.jmaa.2013.01.044. [23] H. Liu, J. Yu and G. Zhu, Global stability of an age-structured SIR epidemic model with pulse vaccination strategy, Journal of System Sciences & Complexity, 25 (2012), 417-429.  doi: 10.1007/s11424-011-9177-y. [24] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. [25] F. A. Milner and G. Rabbiolo, Rapidly converging numerical algorithms for models of population dynamics, Journal of Mathematical Biology, 30 (1992), 733-753.  doi: 10.1007/BF00173266. [26] R. M. Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, Journal of Mathematical Analysis & Applications, 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035. [27] M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: pros and cons, Mathematical Biosciences & Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008. [28] G. U. Rahman, R. P. Agarwal, L. Liu and A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Analysis: Real World Applications, 43 (2018), 96-120.  doi: 10.1016/j.nonrwa.2018.02.006. [29] N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optimal Control Applications & Methods, 37 (2016), 1355-1374.  doi: 10.1002/oca.2247. [30] M. Thater, K. Chudej and H. J. Pesch, Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth, Mathematical Biosciences & Engineering, 15 (2018), 485-505.  doi: 10.3934/mbe.2018022. [31] G. Zaman and A. Khan, Dynamical aspects of an age-structured SIR endemic model, Computers and Mathematics with Applications, 72 (2016), 1690-1702.  doi: 10.1016/j.camwa.2016.07.027. [32] F.-Q. Zhang, R. Liu and Y. Chen, Optimal harvesting in a periodic food chain model with size structures in predators, Applied Mathematics & Optimization, 75 (2017), 229-251.  doi: 10.1007/s00245-016-9331-y.

show all references

##### References:
 [1] A. Alexanderian, M. K. Gobbert, K. R. Fister, H. Gaff, S. Lenhart and E. Schaefer, An age-structured model for the spread of epidemic cholera: Analysis and simulation, Nonlinear Analysis: Real World Applications, 12 (2011), 3483-3498.  doi: 10.1016/j.nonrwa.2011.06.009. [2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Springer, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [3] T. Arbogast and F. A. Milner, A finite difference method for a two-sex model of population dynamics, SIAM Journal of Numerical Analysis, 26 (1989), 1474-1486.  doi: 10.1137/0726086. [4] V. Barbu and M. Iannelli, Optimal control of population dynamics, Journal of Optimization Theory & Applications, 102 (1999), 1-14.  doi: 10.1023/A:1021865709529. [5] S. Bowong, Optimal control of the dynamics of tuberculosis, Nonlinear Dynamics, 61 (2010), 729-748.  doi: 10.1007/s11071-010-9683-9. [6] L.-M. Cai, C. Modnak and J. Wang, An age-structured model for cholera control with vaccination, Applied Mathematics & Computations, 299 (2017), 127-140.  doi: 10.1016/j.amc.2016.11.013. [7] R. D. Demasse, J.-J. Tewa, S. Bowong and Y. Emvudu, Optimal control for an age-structured model for the transmission of hepatitis B, Journal of Mathematical Biology, 73 (2016), 305-333.  doi: 10.1007/s00285-015-0952-6. [8] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling, 22 (2009), 173-211.  doi: 10.1111/j.1939-7445.2008.00033.x. [9] Y. Emvudu, R. D. Demasse and D. Djeudeu, Optimal control using state-dependent Riccati equation of lost of sight in a tuberculosis model, Computational and Applied Mathematics, 32 (2013), 191-210.  doi: 10.1007/s40314-013-0002-1. [10] K. R. Fister and S. Lenhart, Optimal control of a competitive system with age-structured, Journal of Mathematical Analysis & Applications, 291 (2004), 526-537.  doi: 10.1016/j.jmaa.2003.11.031. [11] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron Journal of Differential Equations, 32 (1998), 1-12. [12] G. Grippenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, UK, 1990.  doi: 10.1017/CBO9780511662805. [13] D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3 (2006), 63-69. [14] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [15] M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics: Models, Methods and Numerics, Springer, GX Dordrecht, The Netherlands, 2017. doi: 10.1007/978-94-024-1146-1. [16] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [17] H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710. [18] A. Khan and G. Zaman, Asymptotic behavior of an age structure SIRS endemic model, Applied and Computational Mathematics, 17 (2018), 185-204. [19] A. Khan and G. Zaman, Global analysis of an age-structured SEIR endemic model, Chaos, Solitons and Fractals, 108 (2018), 154-165.  doi: 10.1016/j.chaos.2018.01.037. [20] A. Khan and G. Zaman, Optimal control strategy of SEIR endemic model with continuous age-structure in the exposed and infectious classes, Optimal Control Applications & Methods, 39 (2018), 1716-1727.  doi: 10.1002/oca.2437. [21] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, Journal of Mathematical Biology, 35 (1997), 775-792.  doi: 10.1007/s002850050076. [22] T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, Journal of Mathematical Analysis & Applications, 402 (2013), 477-492.  doi: 10.1016/j.jmaa.2013.01.044. [23] H. Liu, J. Yu and G. Zhu, Global stability of an age-structured SIR epidemic model with pulse vaccination strategy, Journal of System Sciences & Complexity, 25 (2012), 417-429.  doi: 10.1007/s11424-011-9177-y. [24] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. [25] F. A. Milner and G. Rabbiolo, Rapidly converging numerical algorithms for models of population dynamics, Journal of Mathematical Biology, 30 (1992), 733-753.  doi: 10.1007/BF00173266. [26] R. M. Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, Journal of Mathematical Analysis & Applications, 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035. [27] M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: pros and cons, Mathematical Biosciences & Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008. [28] G. U. Rahman, R. P. Agarwal, L. Liu and A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Analysis: Real World Applications, 43 (2018), 96-120.  doi: 10.1016/j.nonrwa.2018.02.006. [29] N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optimal Control Applications & Methods, 37 (2016), 1355-1374.  doi: 10.1002/oca.2247. [30] M. Thater, K. Chudej and H. J. Pesch, Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth, Mathematical Biosciences & Engineering, 15 (2018), 485-505.  doi: 10.3934/mbe.2018022. [31] G. Zaman and A. Khan, Dynamical aspects of an age-structured SIR endemic model, Computers and Mathematics with Applications, 72 (2016), 1690-1702.  doi: 10.1016/j.camwa.2016.07.027. [32] F.-Q. Zhang, R. Liu and Y. Chen, Optimal harvesting in a periodic food chain model with size structures in predators, Applied Mathematics & Optimization, 75 (2017), 229-251.  doi: 10.1007/s00245-016-9331-y.
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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