# American Institute of Mathematical Sciences

September  2021, 26(9): 5171-5196. doi: 10.3934/dcdsb.2020338

## Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases

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

Received  May 2020 Revised  September 2020 Published  September 2021 Early access  November 2020

Fund Project: The author is supported by JSPS KAKENHI Grant Number 20K04536

This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectories. Essential obstacles and keys for the construction of Lyapunov functions are elucidated. The proposed Lyapunov functions which have strictly negative derivative allow us to not only establish ISS, but also get rid of the use of LaSalle's invariance principle and popular simplifying assumptions.

Citation: Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338
##### References:
 [1] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Springer, Berlin, 2005. doi: 10.1007/b139028. [2] D. Bichara, A. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292.  doi: 10.1007/s12190-013-0693-x. [3] A. Chaillet, D. Angeli and H. Ito, Combining iISS and ISS with respect to small inputs: The Strong iISS property, IEEE Trans. Automat. Contr., 59 (2014), 2518-2524.  doi: 10.1109/TAC.2014.2304375. [4] Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age, Math. Biosci. Eng., 11 (2014), 449-469.  doi: 10.3934/mbe.2014.11.449. [5] S. Dashkovskiy, H. Ito and F. Wirth, On a small-gain theorem for ISS networks in dissipative Lyapunov form, European J. Contr., 17 (2011), 357-365.  doi: 10.3166/ejc.17.357-365. [6] K. Dietz, Epidemics and Rumours: A survey, J. Roy. Stat. Soc. A, 130 (1976), 505-528.  doi: 10.2307/2982521. [7] G. Dirr, H. Ito, A. Rantzer and B. S. Rüffer, Separable Lyapunov functions: Constructions and limitations, Discrete and Continuous Dynamical Systems - B, 20 (2015), 2497-2526.  doi: 10.3934/dcdsb.2015.20.2497. [8] Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Disc. Cont. Dynam. Sys. B, 15 (2011), 61-74.  doi: 10.3934/dcdsb.2011.15.61. [9] A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73.  doi: 10.1051/mmnp:2008011. [10] 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. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [12] D. J. Hill and P. J. Moylan, Stability results for nonlinear feedback systems, Automatica, 13 (1977), 377-382.  doi: 10.1016/0005-1098(77)90020-6. [13] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Trans. Autom. Control, 51 (2006), 1626-1643.  doi: 10.1109/TAC.2006.882930. [14] H. Ito, Interpreting models of infectious diseases in terms of integral input-to-state stability, submitted, a preprint is available at arXiv: 2004.02552. [15] Z. P. Jiang, I. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32 (1996), 1211-1215.  doi: 10.1016/0005-1098(96)00051-9. [16] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton Univ. Press, Princeton, 2008. [17] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics., Proc. R. Soc. Lond., A115 (1927), 700-721. [18] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Upper Saddle River, 2002. [19] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75. [20] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bulletin Math. Biol., 68 (2006), 615-626.  doi: 10.1007/s11538-005-9037-9. [21] A. Korobeinikov and G. C. 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. [22] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5. [23] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag, London, 2009. doi: 10.1007/978-1-84882-535-2. [24] A. N. Michel, On the status of stability of interconnected systems, IEEE Trans. Automat. Contr., 28 (1983), 639-653.  doi: 10.1109/TAC.1983.1103292. [25] A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach, SIAM J. Control Optim., 53 (2015), 3364-3382.  doi: 10.1137/14097269X. [26] A. Mironchenko and H. Ito, Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Control Relat. Fields, 6 (2016), 447-466.  doi: 10.3934/mcrf.2016011. [27] Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya and Y. Takeuchi, Stability of epidemic models with waning immunity, SUT J. Mathematics, 50 (2014), 205-245. [28] S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448.  doi: 10.1016/j.aml.2009.11.014. [29] 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. [30] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [31] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018. [32] E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  doi: 10.1016/0167-6911(89)90028-5. [33] E. D. Sontag, Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.  doi: 10.1016/S0167-6911(98)00003-6. [34] E. D. Sontag and Y. Wang, On characterizations of input-to-state stability property, Syst. Control Lett., 24 (1995), 351-359.  doi: 10.1016/0167-6911(94)00050-6. [35] 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.

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##### References:
 [1] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Springer, Berlin, 2005. doi: 10.1007/b139028. [2] D. Bichara, A. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292.  doi: 10.1007/s12190-013-0693-x. [3] A. Chaillet, D. Angeli and H. Ito, Combining iISS and ISS with respect to small inputs: The Strong iISS property, IEEE Trans. Automat. Contr., 59 (2014), 2518-2524.  doi: 10.1109/TAC.2014.2304375. [4] Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age, Math. Biosci. Eng., 11 (2014), 449-469.  doi: 10.3934/mbe.2014.11.449. [5] S. Dashkovskiy, H. Ito and F. Wirth, On a small-gain theorem for ISS networks in dissipative Lyapunov form, European J. Contr., 17 (2011), 357-365.  doi: 10.3166/ejc.17.357-365. [6] K. Dietz, Epidemics and Rumours: A survey, J. Roy. Stat. Soc. A, 130 (1976), 505-528.  doi: 10.2307/2982521. [7] G. Dirr, H. Ito, A. Rantzer and B. S. Rüffer, Separable Lyapunov functions: Constructions and limitations, Discrete and Continuous Dynamical Systems - B, 20 (2015), 2497-2526.  doi: 10.3934/dcdsb.2015.20.2497. [8] Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Disc. Cont. Dynam. Sys. B, 15 (2011), 61-74.  doi: 10.3934/dcdsb.2011.15.61. [9] A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73.  doi: 10.1051/mmnp:2008011. [10] 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. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [12] D. J. Hill and P. J. Moylan, Stability results for nonlinear feedback systems, Automatica, 13 (1977), 377-382.  doi: 10.1016/0005-1098(77)90020-6. [13] H. Ito, State-dependent scaling problems and stability of interconnected iISS and ISS systems, IEEE Trans. Autom. Control, 51 (2006), 1626-1643.  doi: 10.1109/TAC.2006.882930. [14] H. Ito, Interpreting models of infectious diseases in terms of integral input-to-state stability, submitted, a preprint is available at arXiv: 2004.02552. [15] Z. P. Jiang, I. Mareels and Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica, 32 (1996), 1211-1215.  doi: 10.1016/0005-1098(96)00051-9. [16] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton Univ. Press, Princeton, 2008. [17] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics., Proc. R. Soc. Lond., A115 (1927), 700-721. [18] H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Upper Saddle River, 2002. [19] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75. [20] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bulletin Math. Biol., 68 (2006), 615-626.  doi: 10.1007/s11538-005-9037-9. [21] A. Korobeinikov and G. C. 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. [22] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5. [23] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag, London, 2009. doi: 10.1007/978-1-84882-535-2. [24] A. N. Michel, On the status of stability of interconnected systems, IEEE Trans. Automat. Contr., 28 (1983), 639-653.  doi: 10.1109/TAC.1983.1103292. [25] A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach, SIAM J. Control Optim., 53 (2015), 3364-3382.  doi: 10.1137/14097269X. [26] A. Mironchenko and H. Ito, Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Control Relat. Fields, 6 (2016), 447-466.  doi: 10.3934/mcrf.2016011. [27] Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya and Y. Takeuchi, Stability of epidemic models with waning immunity, SUT J. Mathematics, 50 (2014), 205-245. [28] S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448.  doi: 10.1016/j.aml.2009.11.014. [29] 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. [30] E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [31] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018. [32] E. D. Sontag, A 'universal' construction of Artstein's theorem on nonlinear stabilization, Systems Control Lett., 13 (1989), 117-123.  doi: 10.1016/0167-6911(89)90028-5. [33] E. D. Sontag, Comments on integral variants of ISS, Syst. Control Lett., 34 (1998), 93-100.  doi: 10.1016/S0167-6911(98)00003-6. [34] E. D. Sontag and Y. Wang, On characterizations of input-to-state stability property, Syst. Control Lett., 24 (1995), 351-359.  doi: 10.1016/0167-6911(94)00050-6. [35] 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.
Level sets of the ISS Lyapunov function (22) for the disease-free equilibrium with $\hat{B} = 3$ (Dash lines); The arrows are segments of trajectories of (8) for $B(t) = \hat{B}$; The dotted line is $S = \hat{x}_1$
Level sets of the ISS Lyapunov function (37) for the endemic equilibrium with $\hat{B} = 17$ (Dash lines); The arrows are segments of trajectories of (8) for $B(t) = \hat{B}$; The dotted lines are $S = \hat{x}_1$, $I = \hat{x}_2$ and $SI = \hat{x}_1\hat{x}_2$; The lower left area along $S$-axis cannot be filled with sublevel sets of any Lyapunov functions
Obstacles in constructing a strict Lyapunov function in terms of level sets: The lines and the arrows are segments of level sets and trajectories, respectively
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