Article Contents
Article Contents

# Feedback control of isolation and contact for SIQR epidemic model via strict Lyapunov function

• *Corresponding author: Hiroshi Ito

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 derive feedback control laws for isolation, contact regulation, and vaccination for infectious diseases, using a strict Lyapunov function. We use an SIQR epidemic model describing transmission, isolation via quarantine, and vaccination for diseases to which immunity is long-lasting. Assuming that mass vaccination is not available to completely eliminate the disease in a time horizon of interest, we provide feedback control laws that drive the disease to an endemic equilibrium. We prove the input-to-state stability (or ISS) robustness property on the entire state space, when the immigration perturbation is viewed as the uncertainty. We use an ISS Lyapunov function to derive the feedback control laws. A key ingredient in our analysis is that all compartment variables are present not only in the Lyapunov function, but also in a negative definite upper bound on its time derivative. We illustrate the efficacy of our method through simulations, and we discuss the usefulness of parameters in the controls. Since the control laws are feedback, their values are updated based on data acquired in real time. We also discuss the degradation caused by the delayed data acquisition occurring in practical implementations, and we derive bounds on the delays under which the ISS property is ensured when delays are present.

Mathematics Subject Classification: Primary: 93D30, 93C10, 93D09; Secondary: 92D25, 34D23.

 Citation:

• Figure 1.  Comparison of uncontrolled and controlled populations of (1)

Figure 2.  Populations of (1) with constant vaccination $\omega_V = 0$

Figure 3.  Infected plus isolated population and accumulated total of vaccinated individuals

Figure 4.  Populations of (1) with isolation coefficient $c_{\lozenge} = 0.18\overline{c}_{\lozenge}$

Figure 5.  Total of infected individuals of (1)

Figure 6.  Control input of (1) for contact rate

Figure 7.  Populations of (1) with constant contact $\omega_C = 0$

Figure 8.  Populations of (1) controlled by (11) in (3) with $2$ day delay

Figure 9.  Populations of (1) controlled by (11) in (3) with $7$ day delay

Figure 10.  Populations of (1) controlled by (11) in (3) with $\omega_V = 0.000045$

Figure 11.  Semi-log plot with number of vaccinated individuals per day for two different $\omega_V$'s

Figure 12.  Effects of using piecewise constant control which updates values of $u_V$, $u_I$, and $u_C$ in (11) used in the controls (3) every $14$ days and keeps the control values constant between updates

Table 1.  Parameters, Functions, and Sets from Sections 2-3

 Symbols Meanings $S(t)$ number of susceptible individuals in (1) $I(t)$ number of infected individuals in (1) $Q(t)$ number of individuals isolated after infection in (1) $R(t)$ number of recovered individuals in (1) $B$ immigration rate in (1) $\epsilon(t)$ immigration perturbation in (1) $\mathcal P$ perturbation set $(- B, \infty)$ from (2) $\beta(t)$ transmission and contact rate in (1) $\mu$ nonassociative mortality in (1) $\gamma$ recovery rate in (1) $\rho(t)$ vaccination rate in (1) $\nu(t)$ rate of isolation in (1) $\tau$ reciprocal of average isolation time in (1) $\hat \rho$ nominal vaccination rate in (3) $u_V(t)$ vaccination rate control from (11) $\hat \nu$ nominal isolation rate in (3) $u_I(t)$ isolation rate control from (11) $\hat \beta$ nominal transmission and contact in (3) $u_C(t)$ transmission and contact control from (11) $\underline \beta$ tuning constant in $u_C(t)$ from (11c) $X_*$ equilibrium $(S_*, I_*, Q_*, R_*)$ from (6) $\lambda$ and $\chi$ $\lambda=\gamma+\hat\nu+\mu$ and $\chi=\hat\rho+\mu$ used in equilibrium (6) $\mathcal D$ set of feasible states $(0, \infty)^4$ for system (1) $H_i$, $i=1, 2, 3$ functions (13) used in controls (11) $c_\lozenge$ and $c$ constants $c_{\lozenge}\in(0, 2\overline{c}_{\lozenge})$ and $c>0$ from (13) $\overline{c}_{\lozenge}$ bound (12) related to tuning constant $c_\lozenge$ $(\tilde S, \tilde \xi, \tilde Q, \tilde R)$ error states $(S-S_\star, \ln(I)-\ln(I_\star), Q-Q_\star, R-R_*)$ of (15) $\psi_\star$ constant $\lambda I_\star$ in (15) $\tilde {\mathcal D}$ feasible states $(-S_\star, \infty)\times \mathbb{R}\times(-Q_\star, \infty)\times (-R_\star, \infty)$ of (15) $\tilde{\mathcal{P}}$ perturbation set $[-\psi_\star/4, \psi_\star/4]\cap(-B, \infty)$ from Theorem 3.1
•  [1] 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), 104933, 9 pp. doi: 10.1016/j.sysconle.2021.104933. [2] F. Cacace and A. Germani, Output feedback control of linear systems with input, state and output delays by chains of predictors, Automatica J. IFAC, 85 (2017), 455-461.  doi: 10.1016/j.automatica.2017.08.013. [3] R. Carli, G. Cavone, N. Epicoco, P. Scarabaggio and M. Dotoli, Model predictive control to mitigate the COVID-19 outbreak in a multi-region scenario, Annual Reviews in Control, 50 (2020), 373-393.  doi: 10.1016/j.arcontrol.2020.09.005. [4] Y. Ding and H. Schellhorn, Optimal control of the SIR model with constrained policy, with an application to COVID-19, Mathematical Biosciences, 344 (2022), 108758, 15 pp. doi: 10.1016/j.mbs.2021.108758. [5] P. E. M. Fine, Herd immunity: History, theory, practice, Epidemiologic Reviews, 15 (1993), 265-302.  doi: 10.1093/oxfordjournals.epirev.a036121. [6] R. A. Freeman and P. V. Kokotović, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques, Birkhäuser, Boston, 1996. doi: 10.1007/978-0-8176-4759-9. [7] A. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. Alexander, S. Ardal, J. Wu and B. Sahai, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London Series B, 271 (2004), 2223-2232.  doi: 10.1098/rspb.2004.2800. [8] H. Hethcote, M. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences, 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 construction of strict Lyapunov functions for a bilinear balancing model, IFAC-PapersOnLine, 54 (2021), 161-166.  doi: 10.1016/j.ifacol.2021.10.346. [11] H. Ito, Vaccination with input-to-state stability for SIR model of epidemics, Proceedings of the 60th IEEE Conference on Decision and Control, (2021), 2812-2817. doi: 10.1109/CDC45484.2021.9683439. [12] H. Ito, M. Malisoff and F. Mazenc, Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination, Discrete and Continuous Dynamical Systems-B, 27 (2022), 6969-6988.  doi: 10.3934/dcdsb.2022029. [13] Japanse Ministry of Health, Labour and Welfare, https://www.mhlw.go.jp/english/, (2021)., [14] M. J. 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, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bulletin of Mathematical Biology, 30 (2006), 615-626.  doi: 10.1007/s11538-005-9037-9. [19] A. Korobeinikov and G. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Applied Mathematics Letters, 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1. [20] R. Kovacevic, N. Stilianakis and V. Veliov, A distributed optimal control model applied to COVID-19 pandemic, SIAM Journal on Control and Optimization, 60 (2022), 221-245.  doi: 10.1137/20M1373840. [21] H. Liu and X. Tian, Data-driven optimal control of a SEIR model for COVID-19, Communications on Pure and Applied Analysis, (2022), to appear. [22] Q. Luo, R. Weightman, S. McQuade, M. Diaz, E. Trelat, W. Barbour, D. Work, S. Samaranayake and B. Piccoli, Optimization of vaccination for COVID-19 in the midst of a pandemic, Networks and Heterogeneous Media, 17 (2022), 443-466.  doi: 10.3934/nhm.2022016. [23] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag, London, Ltd., London, 2009. doi: 10.1007/978-1-84882-535-2. [24] S. T. McQuade, R. Weightman, N. Merrill, A. Yadav, E. Trelat, S. Allred and B. Piccoli, Control of COVID-19 outbreak using an extended SEIR model, Mathematical Models and Methods in Applied Sciences, 31 (2021), 2399-2424.  doi: 10.1142/S0218202521500512. [25] M. M. Morato, S. B. Bastos, D. O. Cajueiro and J. E. Normey-Rico, An optimal predictive control strategy for COVID-19 (SARS-CoV-2) social distancing policies in Brazil, Annual Reviews in Control, 50 (2020), 417-431.  doi: 10.1016/j.arcontrol.2020.07.001. [26] 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. [27] R. Parino, L. Zino, G. Calafiore and A. Rizzo, A model predictive control approach to optimally devise a two-dose vaccination rollout: A case study on COVID-19 in Italy, International Journal of Robust and Nonlinear Control, (2021), to appear. [28] P. Pepe, A nonlinear version of Halanay's inequality for the uniform convergence to the origin, Math. Control Relat. Fields, 12 (2022), 789-811.  doi: 10.3934/mcrf.2021045. [29] P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems Control Lett., 55 (2006), 1006-1014.  doi: 10.1016/j.sysconle.2006.06.013. [30] T. Perkins and G. España, Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions, Bull. Math. Biol., 82 (2020), Paper No. 118, 24 pp. doi: 10.1007/s11538-020-00795-y. [31] A. Selivanov and E. Fridman, Predictor-based networked control under uncertain transmission delays, Automatica J. IFAC, 70 (2016), 101-108.  doi: 10.1016/j.automatica.2016.03.032. [32] R. Sepulchre, M. Janković and P. Kokotovic, Constructive Nonlinear Control, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1997. doi: 10.1007/978-1-4471-0967-9. [33] J. Sereno, A. Anderson, A. Ferramosca, E. Hernandez-Vargas and A. Gonzalez, Minimizing the epidemic final size while containing the infected peak prevalence in SIR systems, Automatica J. IFAC, 144 (2022), Paper No. 110496. doi: 10.1016/j.automatica.2022.110496. [34] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018. [35] E. D. Sontag and Y. Wang, On characterizations of input-to-state stability property, Systems Control Lett., 24 (1995), 351-359.  doi: 10.1016/0167-6911(94)00050-6. [36] Statistics Bureau of Japan, https://www.stat.go.jp/english/data/jinsui/tsuki/index.ht, (2021)., [37] S. Tang, Y. Xiao and D. Clancy, New modelling approach concerning integrated disease control and cost-effectivity, Nonlinear Analysis, 63 (2005), 439-471.  doi: 10.1016/j.na.2005.05.029. [38] C. Tian, Q. Zhang and L. Zhang, Global stability in a networked SIR epidemic model, Applied Mathematics Letters, 107 (2020), 106444, 6 pp. doi: 10.1016/j.aml.2020.106444. [39] W. W. C. Topley and G. S. Wilson, The spread of bacterial infection. The problem of herd-immunity, Journal of Hygiene, 21 (1923), 243-249.  doi: 10.1017/S0022172400031478. [40] G. G. Walter and M. Contreras, Compartmental Modeling with Networks, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1590-5.

Figures(12)

Tables(1)