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doi: 10.3934/dcdsb.2020338

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

Hiroshi Ito

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

Received  May 2020 Revised  September 2020 Published  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.

Keywords: Epidemic models, input-to-state stability, Lyapunov functions, ordinary differential equations.
Mathematics Subject Classification: Primary: 93D30, 93C10, 93D09; Secondary: 92D25, 34D23.
Citation: Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Dietz, Epidemics and Rumours: A survey, J. Roy. Stat. Soc. A, 130 (1976), 505-528.  doi: 10.2307/2982521.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton Univ. Press, Princeton, 2008.   Google Scholar
[17]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics., Proc. R. Soc. Lond., A115 (1927), 700-721.   Google Scholar

[18]

H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Upper Saddle River, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Springer, Berlin, 2005. doi: 10.1007/b139028.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Dietz, Epidemics and Rumours: A survey, J. Roy. Stat. Soc. A, 130 (1976), 505-528.  doi: 10.2307/2982521.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton Univ. Press, Princeton, 2008.   Google Scholar
[17]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics., Proc. R. Soc. Lond., A115 (1927), 700-721.   Google Scholar

[18]

H. K. Khalil, Nonlinear Systems, 3rd edition, Prentice-Hall, Upper Saddle River, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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 $
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Figure 2.  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
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Figure 3.  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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