Optimal control strategy for an age-structured SIR endemic model

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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

Hassan Tahir ^1, , Asaf Khan ^2,3,, , Anwarud Din ^4, , Amir Khan ^2, and Gul Zaman ^3,

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.

Keywords: Age-structured, sensitivities, adjoint variable, finite difference method, simulation.

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

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

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. Google Scholar
[2]	S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Springer, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	S. Bowong, Optimal control of the dynamics of tuberculosis, Nonlinear Dynamics, 61 (2010), 729-748. doi: 10.1007/s11071-010-9683-9. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	G. Grippenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, UK, 1990. doi: 10.1017/CBO9780511662805. Google Scholar
[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. Google Scholar
[14]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar
[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. Google Scholar
[16]	H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. Google Scholar
[17]	H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23 (2002), 199-213. doi: 10.1002/oca.710. Google Scholar
[18]	A. Khan and G. Zaman, Asymptotic behavior of an age structure SIRS endemic model, Applied and Computational Mathematics, 17 (2018), 185-204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Springer, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	S. Bowong, Optimal control of the dynamics of tuberculosis, Nonlinear Dynamics, 61 (2010), 729-748. doi: 10.1007/s11071-010-9683-9. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	G. Grippenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, UK, 1990. doi: 10.1017/CBO9780511662805. Google Scholar
[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. Google Scholar
[14]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. Google Scholar
[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. Google Scholar
[16]	H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. Google Scholar
[17]	H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23 (2002), 199-213. doi: 10.1002/oca.710. Google Scholar
[18]	A. Khan and G. Zaman, Asymptotic behavior of an age structure SIRS endemic model, Applied and Computational Mathematics, 17 (2018), 185-204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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

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

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

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

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

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

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

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

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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

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

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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

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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

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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

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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

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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

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Table 1. 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<a<40, \\0, \; \; \; \; \; \; \; \; \; \; \hbox{otherwise}. \end{array} \right. $	[1]
$ u_{\max} $	0.70	[6]
$ A_1 $	100	[6]
$ B_1 $	1	Assumed
$ B_2 $	1000	[6]

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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<a<40, \\0, \; \; \; \; \; \; \; \; \; \; \hbox{otherwise}. \end{array} \right. $	[1]
$ u_{\max} $	0.70	[6]
$ A_1 $	100	[6]
$ B_1 $	1	Assumed
$ B_2 $	1000	[6]

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