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March  2022, 12(1): 201-223. doi: 10.3934/mcrf.2021007

## Optimal control of the transmission rate in compartmental epidemics

 Dipartimento di Scienze Matematiche, Informatiche e Fisiche (DMIF), Università di Udine, via delle Scienze 206, 33100 Udine, Italy

Received  September 2020 Revised  December 2020 Published  March 2022 Early access  March 2021

Fund Project: Partially supported by PRID project PRIDEN

We introduce a general system of ordinary differential equations that includes some classical and recent models for the epidemic spread in a closed population without vital dynamic in a finite time horizon. The model is vectorial, in the sense that it accounts for a vector valued state function whose components represent various kinds of exposed/infected subpopulations, with a corresponding vector of control functions possibly different for any subpopulation. In the general setting, we prove well-posedness and positivity of the initial value problem for the system of state equations and the existence of solutions to the optimal control problem of the coefficients of the nonlinear part of the system, under a very general cost functional. We also prove the uniqueness of the optimal solution for a small time horizon when the cost is superlinear in all control variables with possibly different exponents in the interval $(1,2]$. We consider then a linear cost in the control variables and study the singular arcs. Full details are given in the case $n = 1$ and the results are illustrated by the aid of some numerical simulations.

Citation: Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control and Related Fields, 2022, 12 (1) : 201-223. doi: 10.3934/mcrf.2021007
##### References:
 [1] M. R. M. Anderson, Infectious Diseases of Humans, Oxford University Press, London, 1991. [2] H. Behncke, Optimal control of deterministic epidemics, Optimal Control Appl. Methods, 21 (2000), 269-285.  doi: 10.1002/oca.678. [3] W. Bock and Y. Jayathunga, Optimal control and basic reproduction numbers for a compartmental spatial multipatch dengue model, Math. Methods Appl. Sci., 41 (2018), 3231-3245.  doi: 10.1002/mma.4812. [4] J. Bonnans, Frederic, D. Giorgi, V. Grelard, B. Heymann, S. Maindrault, P. Martinon, O. Tissot and J. Liu, Bocop-A collection of examples, Technical report, INRIA, 2017. [5] B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Mathématiques & Applications (Berlin) Mathematics & Applications, vol. 40, Springer-Verlag, Berlin, 2003. [6] A. E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., John Wiley & Sons, New York-London-Sydney, 1975. [7] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3. [8] H. J. A. Diekmann, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation., Wiley, New York, 2000. [9] L. Feng, M. Kumar and L. Mark, An optimal control theory approach to non-pharmaceutical interventions, BMC Infectious Diseases, 10 (2010), 1471-2334. [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, (1998), No. 32, 12 pp. [11] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. [12] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492.  doi: 10.3934/mbe.2009.6.469. [13] S. R. Gani and S. Halawar, Optimal control analysis of deterministic and stochastic epidemic model with media awareness programs, Int. J. Optim. Control. Theor. Appl. IJOCTA, 9 (2019), 24-35.  doi: 10.11121/ijocta.01.2019.00423. [14] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the covid-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7. [15] A. B. Gumel, Modelling strategies for controlling sars outbreaks, Proc. R. Soc. Lond. B., 271 (2004), 2223-2232. [16] J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., 1980. [17] E. Hansen and T. Day, Optimal control of epidemics with limited resources, J. Math. Biol., 62 (2011), 423-451.  doi: 10.1007/s00285-010-0341-0. [18] G. Herzog and R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology, Nonlinear Anal. Real World Appl., 5 (2004), 33-44.  doi: 10.1016/S1468-1218(02)00075-5. [19] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [20] L. O. Jay, Lobatto methods, in Encyclopedia of Applied and Computational Mathematics, Numerical Analysis of Ordinary Differential Equations, Springer-The Language of Science, 2015. doi: 10.1007/978-3-540-70529-1. [21] K. Kandhway and J. Kuri, How to run a campaign: Optimal control of SIS and SIR information epidemics, Appl. Math. Comput., 231 (2014), 79-92.  doi: 10.1016/j.amc.2013.12.164. [22] W. O. Kermack and A. G. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proceedings of the Royal Society of London Series A, 115 (1927), 700-721. [23] T. Kruse and P. Strack, Optimal control of an epidemic through social distancing, (2020), 28 pp. https://ssrn.com/abstract=3581295 [24] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Contin. Dyn. Syst., (2011), 981-990. [25] J. Lee, J. Kim and H.-D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, J. Theoret. Biol., 317 (2013), 310-320.  doi: 10.1016/j.jtbi.2012.10.032. [26] S. Maharaj and A. Kleczkowski, Controlling epidemic spread by social distancing: Do it well or not at all, BMC Public Health, 12 (2012), Art. No. 679. doi: 10.1186/1471-2458-12-679. [27] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [28] O. Sharomi and T. Malik, Optimal control in epidemiology, Ann. Oper. Res., 251 (2017), 55-71.  doi: 10.1007/s10479-015-1834-4. [29] I. S. Team Commands, Bocop: an open source toolbox for optimal control, 2017. http://bocop.org [30] C. Tsay, F. Lejarza, M. A. Stadtherr and M. Baldea., Modeling, state estimation, and optimal control for the us covid-19 outbreak, Sci. Rep., 10 (2020), Art. No. 10711. doi: 10.1038/s41598-020-67459-8. [31] X. Yan and Z. Yun, Control of epidemics by quarantine and isolation strategies in highly mobile populations, Int. J. Inf. Syst. Sci., 5 (2009), 271-286. [32] X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Modelling, 47 (2008), 235-245.  doi: 10.1016/j.mcm.2007.04.003.

show all references

##### References:
 [1] M. R. M. Anderson, Infectious Diseases of Humans, Oxford University Press, London, 1991. [2] H. Behncke, Optimal control of deterministic epidemics, Optimal Control Appl. Methods, 21 (2000), 269-285.  doi: 10.1002/oca.678. [3] W. Bock and Y. Jayathunga, Optimal control and basic reproduction numbers for a compartmental spatial multipatch dengue model, Math. Methods Appl. Sci., 41 (2018), 3231-3245.  doi: 10.1002/mma.4812. [4] J. Bonnans, Frederic, D. Giorgi, V. Grelard, B. Heymann, S. Maindrault, P. Martinon, O. Tissot and J. Liu, Bocop-A collection of examples, Technical report, INRIA, 2017. [5] B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Mathématiques & Applications (Berlin) Mathematics & Applications, vol. 40, Springer-Verlag, Berlin, 2003. [6] A. E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., John Wiley & Sons, New York-London-Sydney, 1975. [7] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3. [8] H. J. A. Diekmann, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation., Wiley, New York, 2000. [9] L. Feng, M. Kumar and L. Mark, An optimal control theory approach to non-pharmaceutical interventions, BMC Infectious Diseases, 10 (2010), 1471-2334. [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, (1998), No. 32, 12 pp. [11] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. [12] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492.  doi: 10.3934/mbe.2009.6.469. [13] S. R. Gani and S. Halawar, Optimal control analysis of deterministic and stochastic epidemic model with media awareness programs, Int. J. Optim. Control. Theor. Appl. IJOCTA, 9 (2019), 24-35.  doi: 10.11121/ijocta.01.2019.00423. [14] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the covid-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7. [15] A. B. Gumel, Modelling strategies for controlling sars outbreaks, Proc. R. Soc. Lond. B., 271 (2004), 2223-2232. [16] J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, N. Y., 1980. [17] E. Hansen and T. Day, Optimal control of epidemics with limited resources, J. Math. Biol., 62 (2011), 423-451.  doi: 10.1007/s00285-010-0341-0. [18] G. Herzog and R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology, Nonlinear Anal. Real World Appl., 5 (2004), 33-44.  doi: 10.1016/S1468-1218(02)00075-5. [19] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [20] L. O. Jay, Lobatto methods, in Encyclopedia of Applied and Computational Mathematics, Numerical Analysis of Ordinary Differential Equations, Springer-The Language of Science, 2015. doi: 10.1007/978-3-540-70529-1. [21] K. Kandhway and J. Kuri, How to run a campaign: Optimal control of SIS and SIR information epidemics, Appl. Math. Comput., 231 (2014), 79-92.  doi: 10.1016/j.amc.2013.12.164. [22] W. O. Kermack and A. G. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proceedings of the Royal Society of London Series A, 115 (1927), 700-721. [23] T. Kruse and P. Strack, Optimal control of an epidemic through social distancing, (2020), 28 pp. https://ssrn.com/abstract=3581295 [24] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Contin. Dyn. Syst., (2011), 981-990. [25] J. Lee, J. Kim and H.-D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, J. Theoret. Biol., 317 (2013), 310-320.  doi: 10.1016/j.jtbi.2012.10.032. [26] S. Maharaj and A. Kleczkowski, Controlling epidemic spread by social distancing: Do it well or not at all, BMC Public Health, 12 (2012), Art. No. 679. doi: 10.1186/1471-2458-12-679. [27] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [28] O. Sharomi and T. Malik, Optimal control in epidemiology, Ann. Oper. Res., 251 (2017), 55-71.  doi: 10.1007/s10479-015-1834-4. [29] I. S. Team Commands, Bocop: an open source toolbox for optimal control, 2017. http://bocop.org [30] C. Tsay, F. Lejarza, M. A. Stadtherr and M. Baldea., Modeling, state estimation, and optimal control for the us covid-19 outbreak, Sci. Rep., 10 (2020), Art. No. 10711. doi: 10.1038/s41598-020-67459-8. [31] X. Yan and Z. Yun, Control of epidemics by quarantine and isolation strategies in highly mobile populations, Int. J. Inf. Syst. Sci., 5 (2009), 271-286. [32] X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Modelling, 47 (2008), 235-245.  doi: 10.1016/j.mcm.2007.04.003.
$J_{LL}$ with ${\bar{u}} = 0.1$
$J_{QQ}$ with ${\bar{u}} = 0.08$
$J_{QQ}$ with ${\bar{u}} = 0.04$
$J_{QL}$ with ${\bar{u}} = 0.1$
$J_{QL}$ with ${\bar{u}} = 0.08$
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