# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021144
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## Optimal control of the SIR epidemic model based on dynamical systems theory

 Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: This work was partially supported by the JSPS KAKENHI Grant Number JP17H02859

We consider the susceptible-infected-removed (SIR) epidemic model and apply optimal control to it successfully. Here two control inputs are considered, so that the infection rate is decreased and infected individuals are removed. Our approach is to reduce computation of the optimal control input to that of the stable manifold of an invariant manifold in a Hamiltonian system. The situation in which the target equilibrium has a center direction is very different from similar previous work. Some numerical examples in which the computer software AUTO is used to numerically compute the stable manifold are given to demonstrate the usefulness of our approach for the optimal control in the SIR model. Our study suggests how we can decrease the number of infected individuals quickly before a critical situation occurs while keeping social and economic burdens small.

Citation: Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021144
##### References:
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Jacobson, Primer on Optimal Control Theory, Society of Industrial and Applied Mathematics, Philadelphia, 2010. doi: 10.1137/1.9780898718560.  Google Scholar [27] M. Struwe, Variational Methods, 4$^{th}$ edition, Springer-Verlag, Berlin, 2008.  Google Scholar [28] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar [29] M. Xin and H. Pan, Nonlinear optimal control of spacecraft approaching a tumbling target, Aerosp Sci. Technol., 15 (2011), 79-89.  doi: 10.1109/ACC.2009.5160182.  Google Scholar [30] K. Yagasaki, Numerical analysis of the Hamilton-Jacobi-Bellman equations for nonlinear control systems by computation of stable manifolds, work-in-progress. Google Scholar [31] K. Yagasaki, Numerical dynamical systems approach to the Hamilton-Jacobi-Bellman equations for nonlinear control problems, work-in-progress. Google Scholar [32] X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Compt. Model., 47 (2008), 235-245.  doi: 10.1016/j.mcm.2007.04.003.  Google Scholar [33] J. N. Yang, Z. Li and S. Vongchavalitkul, Generalization of optimal control theory: Linear and nonlinear control, J. Eng. Mech., 120 (1994), 266-283.  doi: 10.1061/(ASCE)0733-9399(1994)120:2(266).  Google Scholar

show all references

##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2$^{nd}$ edition, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [2] H. Behncke, Optimal control of deterministic epidemics, Optim. Control Appl. Meth., 21 (2000), 269-285.  doi: 10.1002/oca.678.  Google Scholar [3] L. Bolzoni, E. Bonacini, C. Soresina and M. Groppi, Time-optimal control strategies in SIR epidemic models, Math. Biosci., 292 (2017), 86-96.  doi: 10.1016/j.mbs.2017.07.011.  Google Scholar [4] C. Castilho, Optimal control of an epidemic through educational campaigns, Electron. J. Differential Equations, (2006), 11 pp.  Google Scholar [5] E. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, 2012, Available from: http://cmvl.cs.concordia.ca. Google Scholar [6] P. Di Giamberardino and D. Iacoviello, Optimal control of SIR epidemic model with state dependent switching cost index, Biomed. Signal Process Control, 31 (2017), 377-380.  doi: 10.1016/j.bspc.2016.09.011.  Google Scholar [7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [8] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-based Approach, Princeton University Press, Princeton, 2008.   Google Scholar [9] G. H. M. van der Heijden and K. Yagasaki, Nonintegrability of an extensible conducting rod in a uniform magnetic field, J. Phys. A, 44 (2011), 495101, 11 pp. doi: 10.1088/1751-8113/44/49/495101.  Google Scholar [10] D. Iacoviello and G. Liuzzi, Fixed/free final time SIR epidemic models with multiple controls, Int. J. Simul. Model., 7 (2008), 81-92.  doi: 10.2507/IJSIMM07(2)3.103.  Google Scholar [11] D. Iacoviello and N. Stasio, Optimal control for SIRC epidemic outbreak, Comput. Meth. Prog. Bio., 110 (2013), 333-342.  doi: 10.1016/j.cmpb.2013.01.006.  Google Scholar [12] D. P. Jin and H. Y. Hu, Optimal control of a tethered subsatellite of three degrees of freedom, Nonlinear Dynam., 46 (2006), 161-178.  doi: 10.1007/s11071-006-9021-4.  Google Scholar [13] H. R. Joshi, S. Lenhart, S. Hota and F. Agusto, Optimal control of an SIR model with changing behavior through an education campaign, Electron. J. Differential Equations, (2015), 14 pp.  Google Scholar [14] T. K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, BioSystems, 104 (2011), 127-135.  doi: 10.1016/j.biosystems.2011.02.001.  Google Scholar [15] D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, Englewood Cliffs, NJ, 1970. Google Scholar [16] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.   Google Scholar [17] A. Lahrouz, H. El Mahjour, A. Settati and A. Bernoussi, Dynamics and optimal control of a non-linear epidemic model with relapse and cure, Phys. A, 496 (2018), 299-317.  doi: 10.1016/j.physa.2018.01.007.  Google Scholar [18] F. L. Lewis, D. Vrabie and V. L. Syrmos, Optimal Control, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17. John Wiley and Sons, Hoboken, NJ, 2012. doi: 10.1002/9781118122631.  Google Scholar [19] J. D. Murray, Mathematical Biology I: An Introduction, 3$^{rd}$ edition, Springer-Verlag, New York, 2002.  Google Scholar [20] H. M. Osinga and J. Hauser, The geometry of the solution set of nonlinear optimal control problems, J. Dynam. Differential Equations, 18 (2006), 881-900.  doi: 10.1007/s10884-006-9051-0.  Google Scholar [21] H. S. Rodrigues, M. T. T. Monteiro and F. F. M. Torres, Dynamics of Dengue epidemics when using optimal control, Math. Comp. Model., 52 (2010), 1667-1673.  doi: 10.1016/j.mcm.2010.06.034.  Google Scholar [22] N. Sakamoto and A. J. van der Schaft, Analytical approximation methods for the stabilizing solution of the Hamilton–Jacobi equation, IEEE Trans. Automat. Contr., 53 (2008), 2335-2350.  doi: 10.1109/TAC.2008.2006113.  Google Scholar [23] A. J. van der Schaft, On a state space approach to non-linear $H_\infty$ control, Syst. Control Lett., 16 (1991), 1-8.  doi: 10.1016/0167-6911(91)90022-7.  Google Scholar [24] A. J. van der Schaft, $L_2$-Gain and Passivity Techniques in Nonlinear Control, 2$^{nd}$ edition, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0507-7.  Google Scholar [25] M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403.  doi: 10.1088/0951-7715/22/10/004.  Google Scholar [26] J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, Society of Industrial and Applied Mathematics, Philadelphia, 2010. doi: 10.1137/1.9780898718560.  Google Scholar [27] M. Struwe, Variational Methods, 4$^{th}$ edition, Springer-Verlag, Berlin, 2008.  Google Scholar [28] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar [29] M. Xin and H. Pan, Nonlinear optimal control of spacecraft approaching a tumbling target, Aerosp Sci. Technol., 15 (2011), 79-89.  doi: 10.1109/ACC.2009.5160182.  Google Scholar [30] K. Yagasaki, Numerical analysis of the Hamilton-Jacobi-Bellman equations for nonlinear control systems by computation of stable manifolds, work-in-progress. Google Scholar [31] K. Yagasaki, Numerical dynamical systems approach to the Hamilton-Jacobi-Bellman equations for nonlinear control problems, work-in-progress. Google Scholar [32] X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Compt. Model., 47 (2008), 235-245.  doi: 10.1016/j.mcm.2007.04.003.  Google Scholar [33] J. N. Yang, Z. Li and S. Vongchavalitkul, Generalization of optimal control theory: Linear and nonlinear control, J. Eng. Mech., 120 (1994), 266-283.  doi: 10.1061/(ASCE)0733-9399(1994)120:2(266).  Google Scholar
Phase portrait of (3.5): (a) $c>a/r$; (b) $c<a/r$
Projection of the stable manifold $W^{\mathrm{s}}$ onto the $(S,I)$-plane in case (i) for $(a,r) = (0.4,1)$. The stable manifold is plotted as the blue and red lines for $c = 0.8$ and $0.95$, respectively, and the corresponding stable subspace is plotted as the dashed lines with the same colors. The bullets '$\bullet$' represent the loci of equilibria, and the black dashed lines represent the relations $S+I = 1$, $S = 0.9$ and $S = 0.98$
Controlled trajectory with $S_0 = 0.9$ converging to $(S,I) = (0.8,0)$ in (1.3) for $(a,r) = (0.4,1)$: (a) $S$; (b) $I$; (c) $k_1u_1$; (d) $u_2$. These components are plotted as black, red and blue lines for cases (i), (ii) and (iii), respectively. The value of $I_0$ is approximately $0.0482$ but slightly different in the three cases
Controlled trajectory with $S_0 = 0.98$ converging to $(S,I) = (0.95,0)$ in (1.3) for $(a,r) = (0.4,1)$: (a) $S$; (b) $I$; (c) $k_1u_1$; (d) $u_2$. These components are plotted as black, red and blue lines for cases (i), (ii) and (iii), respectively. The value of $I_0$ is approximately $0.0171$ but slightly different in the three cases
Uncontrolled and controlled trajectories with the same initial point $(S_0,I_0)$ in (1.2) and (1.3) for $(a,r) = (0.4,1)$ in case (i): (a) $(S_0,I_0) = (0.9,0.04720\ldots)$; (b) $(S_0,I_0) = (0.98,0.017268\ldots)$. Uncontrolled and controlled ones are plotted as red and black lines, respectively. The circle '$\circ$' represents the initial point
Target equilibrium $(c,0)$ to which the controlled trajectory starting at $(S_0,I_0)$ converges in (1.3) for $a = 0.4$ and $r = 1$: (a) $S_0 = 0.9$; (b) $S_0 = 0.98$. It is plotted as black, red and blue lines for cases (i), (ii) and (iii), respectively
Value function $V$ for $a = 0.4$ and $r = 1$: (a) $S_0 = 0.9$; (b) $S_0 = 0.98$. It is plotted as black, red and blue lines for cases (i), (ii) and (iii), respectively
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