doi: 10.3934/dcdsb.2022029
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination

1. 

Department of Intelligent and Control Systems, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka 820-8502, Japan

2. 

Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803-4918, USA

3. 

Inria EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France

* Corresponding author: Hiroshi Ito

Received  June 2021 Revised  January 2022 Early access February 2022

Fund Project: The work of H. Ito was supported by JSPS KAKENHI Grant Number JP20K04536. The work of M. Malisoff was supported by NSF Grant 1711299

We provide a new global strict Lyapunov function construction for a susceptible, infected, and recovered (or SIR) disease dynamics that includes quarantine of infected individuals and mass vaccination. We use the Lyapunov function to design feedback controls to asymptotically stabilize a desired endemic equilibrium, and to prove input-to-state stability for the dynamics with a suitable restriction on the disturbances. Our simulations illustrate the potential of our feedback controls to reduce peak levels of infected individuals.

Citation: Hiroshi Ito, Michael Malisoff, Frédéric Mazenc. Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022029
References:
[1]

N. Ahmed, A. Raza, M. Rafiq, A. Ahmadian, N. Batool and S. Salahshour, Numerical and bifurcation analysis of SIQR model, Chaos Solitons Fractals, 150 (2021), Paper No. 111133, 19 pp. doi: 10.1016/j.chaos.2021.111133.

[2]

C. Beck, F. Bullo, G. Como, K. Drakopoulos, D. Nguyen, C. Nowzari, V. Preciado and S. Sundaram, Special Section on mathematical modeling, analysis, and control of epidemics, SIAM Journal on Control and Optimization, to appear.

[3]

I. Bhogaraju, M. Farasat, M. Malisoff and M. Krstic, Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties, Systems Control Lett, 152 (2021), Paper No. 104933, 9 pp. doi: 10.1016/j.sysconle.2021.104933.

[4]

F. CacaceF. Conte and A. Germani, Output transformations and separation results for feedback linearisable delay systems, Internat. J. Control, 91 (2018), 797-812.  doi: 10.1080/00207179.2017.1293848.

[5]

F. Cacace and A. Germani, Output feedback control of linear systems with input, state and output delays by chains of predictors, Automatica, 85 (2017), 455-461.  doi: 10.1016/j.automatica.2017.08.013.

[6]

N. Crokidakis, Modeling the early evolution of the COVID-19 in Brazil: Results from a SIQR model, Internat. J. Modern Phys. C, 31 (2020), 2050135, 7 pp. doi: 10.1142/S0129183120501351.

[7]

A. GumelS. RuanT. DayJ. WatmoughF. BrauerP. van den DriesscheD. GabrielsonC. BowmanM. AlexanderS. ArdalJ. Wu and B. Sahai, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society B, 271 (2004), 2223-2232.  doi: 10.1098/rspb.2004.2800.

[8]

H. HethcoteM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.

[9]

H. Ito, Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5171-5196.  doi: 10.3934/dcdsb.2020338.

[10]

H. Ito, A strict smooth Lyapunov function and input-to-state stability of SIR model, In American Control Conference, (2021), 4799–4804. doi: 10.23919/ACC50511.2021.9482900.

[11]

H. Ito, A construction of strict Lyapunov functions for a bilinear balancing model, IFAC-PapersOnLine, 54 (2021), 161-166.  doi: 10.1016/j.ifacol.2021.10.346.

[12]

H. Ito, Vaccination with input-to-state stability for SIR model of epidemics, In Proceedings of the 60th IEEE Conference on Decision and Control, (2021), 2808–2813. doi: 10.1109/CDC45484.2021.9683439.

[13]

R. KatzE. Fridman and A. Selivanov, Boundary delayed observer-controller design for reaction-diffusion systems, IEEE Trans. Automat. Control, 66 (2021), 275-282.  doi: 10.1109/TAC.2020.2973803.

[14] M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008. 
[15]

W. Kermack and A. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Academy of London Series A, 115 (1927), 700-721. 

[16]

H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, 2002.

[17]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology, 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75.

[18]

A. Korobeinikov and G. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.

[19]

I. LocatelliB. Trochsel and V. Rousson, Estimating the basic reproduction number for COVID-19 in Western Europe, PLOS ONE, 16 (2021), 1-9.  doi: 10.1371/journal.pone.0248731.

[20]

Y. MaJ.-B. Liu and H. Li, Global dynamics of an SIQR model with vaccination and elimination hybrid strategies, Mathematics, 6 (2018), 328.  doi: 10.3390/math6120328.

[21]

M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag, London, 2009. doi: 10.1007/978-1-84882-535-2.

[22]

M. MattioniS. Monaco and D. Normand-Cyrot, Nonlinear discrete-time systems with delayed control: A reduction, Systems Control Lett., 114 (2018), 31-37.  doi: 10.1016/j.sysconle.2018.02.007.

[23]

M. NadiniL. ZinoA. Rizzo and M. Porfiri, A multi-agent model to study epidemic spreading and vaccination strategies in an urban-like environment, Applied Network Science, 5 (2020), 68.  doi: 10.1007/s41109-020-00299-7.

[24]

T. Odagaki, Analysis of the outbreak of COVID-19 in Japan by SIQR model, Infectious Disease Modelling, 5 (2020), 691-698.  doi: 10.1016/j.idm.2020.08.013.

[25]

A. Selivanov and E. Fridman, Predictor-based networked control under uncertain transmission delays, Automatica, 70 (2016), 101-108.  doi: 10.1016/j.automatica.2016.03.032.

[26]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642.

[27]

E. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.

[28]

K. SyL. White and B. Nichols, Population density and basic reproductive number of COVID-19 across United States counties, PLOS ONE, 16 (2021), 1-11.  doi: 10.1371/journal.pone.0249271.

[29]

C. Tian, Q. Zhang and L. Zhang, Global stability in a networked SIR epidemic model, Appl. Math. Lett., 107 (2020), 106444, 6 pp. doi: 10.1016/j.aml.2020.106444.

[30]

C. YouY. DengW. HuJ. SunQ. LinF. ZhouC. PangY. ZhangZ. Chen and X. Zhou, Estimation of the time-varying reproduction number of COVID-19 outbreak in China, International Journal of Hygiene and Environmental Health, 228 (2020), 113555. 

[31]

B. Zhou, Construction of strict Lyapunov-Krasovskii functionals for time-varying time-delay systems, Automatica, 107 (2019), 382-397.  doi: 10.1016/j.automatica.2019.05.058.

[32]

B. Zhou, Y. Tian and J. Lam, On construction of Lyapunov functions for scalar linear time-varying systems, Systems Control Lett., 135 (2020), 104591 10 pp. doi: 10.1016/j.sysconle.2019.104591.

[33]

L. ZinoA. Rizzo and M. Porfiri, On assessing control actions for epidemic models on temporal network, IEEE Control Syst. Lett., 4 (2020), 797-802. 

show all references

References:
[1]

N. Ahmed, A. Raza, M. Rafiq, A. Ahmadian, N. Batool and S. Salahshour, Numerical and bifurcation analysis of SIQR model, Chaos Solitons Fractals, 150 (2021), Paper No. 111133, 19 pp. doi: 10.1016/j.chaos.2021.111133.

[2]

C. Beck, F. Bullo, G. Como, K. Drakopoulos, D. Nguyen, C. Nowzari, V. Preciado and S. Sundaram, Special Section on mathematical modeling, analysis, and control of epidemics, SIAM Journal on Control and Optimization, to appear.

[3]

I. Bhogaraju, M. Farasat, M. Malisoff and M. Krstic, Sequential predictors for delay-compensating feedback stabilization of bilinear systems with uncertainties, Systems Control Lett, 152 (2021), Paper No. 104933, 9 pp. doi: 10.1016/j.sysconle.2021.104933.

[4]

F. CacaceF. Conte and A. Germani, Output transformations and separation results for feedback linearisable delay systems, Internat. J. Control, 91 (2018), 797-812.  doi: 10.1080/00207179.2017.1293848.

[5]

F. Cacace and A. Germani, Output feedback control of linear systems with input, state and output delays by chains of predictors, Automatica, 85 (2017), 455-461.  doi: 10.1016/j.automatica.2017.08.013.

[6]

N. Crokidakis, Modeling the early evolution of the COVID-19 in Brazil: Results from a SIQR model, Internat. J. Modern Phys. C, 31 (2020), 2050135, 7 pp. doi: 10.1142/S0129183120501351.

[7]

A. GumelS. RuanT. DayJ. WatmoughF. BrauerP. van den DriesscheD. GabrielsonC. BowmanM. AlexanderS. ArdalJ. Wu and B. Sahai, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society B, 271 (2004), 2223-2232.  doi: 10.1098/rspb.2004.2800.

[8]

H. HethcoteM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.

[9]

H. Ito, Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 5171-5196.  doi: 10.3934/dcdsb.2020338.

[10]

H. Ito, A strict smooth Lyapunov function and input-to-state stability of SIR model, In American Control Conference, (2021), 4799–4804. doi: 10.23919/ACC50511.2021.9482900.

[11]

H. Ito, A construction of strict Lyapunov functions for a bilinear balancing model, IFAC-PapersOnLine, 54 (2021), 161-166.  doi: 10.1016/j.ifacol.2021.10.346.

[12]

H. Ito, Vaccination with input-to-state stability for SIR model of epidemics, In Proceedings of the 60th IEEE Conference on Decision and Control, (2021), 2808–2813. doi: 10.1109/CDC45484.2021.9683439.

[13]

R. KatzE. Fridman and A. Selivanov, Boundary delayed observer-controller design for reaction-diffusion systems, IEEE Trans. Automat. Control, 66 (2021), 275-282.  doi: 10.1109/TAC.2020.2973803.

[14] M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008. 
[15]

W. Kermack and A. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Academy of London Series A, 115 (1927), 700-721. 

[16]

H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, 2002.

[17]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology, 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75.

[18]

A. Korobeinikov and G. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1.

[19]

I. LocatelliB. Trochsel and V. Rousson, Estimating the basic reproduction number for COVID-19 in Western Europe, PLOS ONE, 16 (2021), 1-9.  doi: 10.1371/journal.pone.0248731.

[20]

Y. MaJ.-B. Liu and H. Li, Global dynamics of an SIQR model with vaccination and elimination hybrid strategies, Mathematics, 6 (2018), 328.  doi: 10.3390/math6120328.

[21]

M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag, London, 2009. doi: 10.1007/978-1-84882-535-2.

[22]

M. MattioniS. Monaco and D. Normand-Cyrot, Nonlinear discrete-time systems with delayed control: A reduction, Systems Control Lett., 114 (2018), 31-37.  doi: 10.1016/j.sysconle.2018.02.007.

[23]

M. NadiniL. ZinoA. Rizzo and M. Porfiri, A multi-agent model to study epidemic spreading and vaccination strategies in an urban-like environment, Applied Network Science, 5 (2020), 68.  doi: 10.1007/s41109-020-00299-7.

[24]

T. Odagaki, Analysis of the outbreak of COVID-19 in Japan by SIQR model, Infectious Disease Modelling, 5 (2020), 691-698.  doi: 10.1016/j.idm.2020.08.013.

[25]

A. Selivanov and E. Fridman, Predictor-based networked control under uncertain transmission delays, Automatica, 70 (2016), 101-108.  doi: 10.1016/j.automatica.2016.03.032.

[26]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642.

[27]

E. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.

[28]

K. SyL. White and B. Nichols, Population density and basic reproductive number of COVID-19 across United States counties, PLOS ONE, 16 (2021), 1-11.  doi: 10.1371/journal.pone.0249271.

[29]

C. Tian, Q. Zhang and L. Zhang, Global stability in a networked SIR epidemic model, Appl. Math. Lett., 107 (2020), 106444, 6 pp. doi: 10.1016/j.aml.2020.106444.

[30]

C. YouY. DengW. HuJ. SunQ. LinF. ZhouC. PangY. ZhangZ. Chen and X. Zhou, Estimation of the time-varying reproduction number of COVID-19 outbreak in China, International Journal of Hygiene and Environmental Health, 228 (2020), 113555. 

[31]

B. Zhou, Construction of strict Lyapunov-Krasovskii functionals for time-varying time-delay systems, Automatica, 107 (2019), 382-397.  doi: 10.1016/j.automatica.2019.05.058.

[32]

B. Zhou, Y. Tian and J. Lam, On construction of Lyapunov functions for scalar linear time-varying systems, Systems Control Lett., 135 (2020), 104591 10 pp. doi: 10.1016/j.sysconle.2019.104591.

[33]

L. ZinoA. Rizzo and M. Porfiri, On assessing control actions for epidemic models on temporal network, IEEE Control Syst. Lett., 4 (2020), 797-802. 

Figure 1.  Populations of (1) with $ u = 0 $ and $ \hat{\rho} = 0.0001 $
Figure 2.  Populations of (1) under the control (76) with $ \omega = 0.01 $, $ c = 1 $, and $ \hat{\rho} = 0.0001 $
Figure 3.  Populations of (1) under the control (76) with $ \omega = 0.1 $, $ c = 0.1 $, and $ \hat{\rho} = 0.0001 $
Figure 4.  Populations of (1) under the control (76) with $ \omega = 0.1 $, $ c = 0.1 $, and $ \hat{\rho} = 0.0001 $
Figure 5.  Infected population of (1) with three different controls in the presence of perturbation $ \epsilon(t) $
[1]

Z. Feng. Final and peak epidemic sizes for SEIR models with quarantine and isolation. Mathematical Biosciences & Engineering, 2007, 4 (4) : 675-686. doi: 10.3934/mbe.2007.4.675

[2]

Eunha Shim. A note on epidemic models with infective immigrants and vaccination. Mathematical Biosciences & Engineering, 2006, 3 (3) : 557-566. doi: 10.3934/mbe.2006.3.557

[3]

Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635

[4]

Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595

[5]

Alfonso Ruiz Herrera. Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2533-2548. doi: 10.3934/cpaa.2020111

[6]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603

[7]

Connell McCluskey. Lyapunov functions for disease models with immigration of infected hosts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4479-4491. doi: 10.3934/dcdsb.2020296

[8]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

[9]

Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi. Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models. Mathematical Biosciences & Engineering, 2015, 12 (1) : 117-133. doi: 10.3934/mbe.2015.12.117

[10]

Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369

[11]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i

[12]

Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145

[13]

Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045

[14]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059

[15]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[16]

Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291

[17]

Fred Brauer. Some simple epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 1-15. doi: 10.3934/mbe.2006.3.1

[18]

Fred Brauer, Zhilan Feng, Carlos Castillo-Chávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 1-15. doi: 10.3934/mbe.2010.7.1

[19]

Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006

[20]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999

2021 Impact Factor: 1.497

Article outline

Figures and Tables

[Back to Top]