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November  2014, 19(9): 2865-2887. doi: 10.3934/dcdsb.2014.19.2865

Stability criteria for SIS epidemiological models under switching policies

1. 

Laboratoire de Modélisation, Information et Systèmes (MIS), University of Picardie Jules Verne, 80025 Amiens, France

2. 

Departement Werktuigkunde, KU Leuven, Leuven, Belgium

3. 

Hamilton Institute, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland

4. 

Dept. of Computer Science and Mathematics, University of Passau, 94030 Passau, Germany

Received  June 2013 Revised  April 2014 Published  September 2014

We study the spread of disease in an SIS model for a structured population. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.
Citation: Mustapha Ait Rami, Vahid S. Bokharaie, Oliver Mason, Fabian R. Wirth. Stability criteria for SIS epidemiological models under switching policies. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2865-2887. doi: 10.3934/dcdsb.2014.19.2865
References:
[1]

M. Ait Rami, V. S. Bokharaie, O. Mason and F. Wirth, Extremal norms for positive linear inclusions, in Proc. 20th Int. Symposium on Mathematical Theory of Networks and Systems, MTNS 2012, Melbourne, Australia, 2012.

[2]

Z. Artstein, Averaging of time-varying differential equations revisited, Journal of Differential Equations, 243 (2007), 146-167. doi: 10.1016/j.jde.2007.01.022.

[3]

N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2012), 1729-1739. doi: 10.1007/s00285-012-0611-0.

[4]

N. T. J. Bailey, The Mathematical Theory of Epidemics, Griffin, London, 1957.

[5]

F. Bauer, J. Stoer and C. Witzgall, Absolute and monotonic norms, Numerische Mathematik, 3 (1961), 257-264. doi: 10.1007/BF01386026.

[6]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, SIAM, Philadelphia, PA, USA, 1987. doi: 10.1137/1.9781611971262.

[7]

T. Björk, Finite dimensional optimal filters for a class of Ito-processes with jumping parameters, Stochastics, 4 (1980), 167-183. doi: 10.1080/17442508008833160.

[8]

V. S. Bokharaie, O. Mason and F. Wirth, Spread of epidemics in time-dependent networks, Proc. 19th Int. Symposium on Mathematical Theory of Networks and Systems, MTNS 2010.

[9]

I. Chueshov, Monotone Random Systems, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[10]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, 178 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[12]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Mathematical Biosciences, 16 (1973), 75-101. doi: 10.1016/0025-5564(73)90046-1.

[13]

P. De Leenheer, Stabiliteit, Regeling en Stabilisatie van Positieve Systemen, PhD thesis, University of Gent, 2000.

[14]

V. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt Berlin, 1995.

[15]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, 1, Springer-Verlag, Berlin, 2003.

[16]

L. Fainshil, M. Margaliot and P. Chigansky, On the stability of positive linear switched systems under arbitrary switching laws, IEEE Transactions on Automatic Control, 54 (2009), 897-899. doi: 10.1109/TAC.2008.2010974.

[17]

A. Fall, A. Iggidr, G. Sallet and J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-68. doi: 10.1051/mmnp:2008011.

[18]

A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516. doi: 10.1016/j.jmaa.2012.05.029.

[19]

L. Gurvits, R. Shorten and O. Mason, On the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 52 (2007), 1099-1103. doi: 10.1109/TAC.2007.899057.

[20]

H. W. Hethcote and J. A. York, Gonorrhea Transmission and Control, 56 of Lectures Notes in Biomathematics, Springer-Verlag, New York, NY, 1984. doi: 10.1007/978-3-662-07544-9.

[21]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, USA, 1985. doi: 10.1017/CBO9780511810817.

[22]

I. Kats and N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mec., 24 (1960), 1225-1246. doi: 10.1016/0021-8928(60)90103-9.

[23]

V. S. Kozyakin, Algebraic unsolvability of problem of absolute stability of desynchronized systems, Autom. Rem. Control, 51 (1990), 754-759.

[24]

N. Krasovskii and E. Lidskii, Analytical design of controllers in systems with random attributes, Automation and Remote Control, 22 (1961), 1021-1025.

[25]

A. Lajmanovic and J. Yorke, A deterministic model for gonorrhea in nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

[26]

D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, MA, USA, 2003. doi: 10.1007/978-1-4612-0017-8.

[27]

X. Liu and X.-Q. Zhao, A periodic epidemic model with age-structure in a patchy environment, SIAM Journal of Applied Mathematics, 71 (2011), 1896-1917. doi: 10.1137/100813610.

[28]

N. Lloyd, Degree Theory, Cambridge University Press, 1978.

[29]

M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, NY, 1990.

[30]

O. Mason and F. Wirth, Extremal norms for positive linear inclusions, Linear Algebra and its Applications, 444 (2014), 100-113. doi: 10.1016/j.laa.2013.11.020.

[31]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.

[32]

J. Norris, Markov Chains, Cambridge University Press, Cambridge, 2008.

[33]

S. Pröll, Stability of Switched Epidemiological Models, Master's thesis, Institute for Mathematics, University of Würzburg, Würzburg, Germany, 2013.

[34]

C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, Journal of Mathematical Biology, 64 (2012), 933-949. doi: 10.1007/s00285-011-0440-6.

[35]

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability theory for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X.

[36]

H. A. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, USA, 1995.

[37]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, USA, 2011.

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[39]

F. Wirth, The generalized spectral radius and extremal norms, Lin. Alg. Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3.

[40]

World Health Organization (WHO), The global burden of disease: 2004 update, http://www.who.int/healthinfo/global_burden_disease/2004_report_update/en/index.html, 2008, Last Retrieved: 05 December 2011.

show all references

References:
[1]

M. Ait Rami, V. S. Bokharaie, O. Mason and F. Wirth, Extremal norms for positive linear inclusions, in Proc. 20th Int. Symposium on Mathematical Theory of Networks and Systems, MTNS 2012, Melbourne, Australia, 2012.

[2]

Z. Artstein, Averaging of time-varying differential equations revisited, Journal of Differential Equations, 243 (2007), 146-167. doi: 10.1016/j.jde.2007.01.022.

[3]

N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2012), 1729-1739. doi: 10.1007/s00285-012-0611-0.

[4]

N. T. J. Bailey, The Mathematical Theory of Epidemics, Griffin, London, 1957.

[5]

F. Bauer, J. Stoer and C. Witzgall, Absolute and monotonic norms, Numerische Mathematik, 3 (1961), 257-264. doi: 10.1007/BF01386026.

[6]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, SIAM, Philadelphia, PA, USA, 1987. doi: 10.1137/1.9781611971262.

[7]

T. Björk, Finite dimensional optimal filters for a class of Ito-processes with jumping parameters, Stochastics, 4 (1980), 167-183. doi: 10.1080/17442508008833160.

[8]

V. S. Bokharaie, O. Mason and F. Wirth, Spread of epidemics in time-dependent networks, Proc. 19th Int. Symposium on Mathematical Theory of Networks and Systems, MTNS 2010.

[9]

I. Chueshov, Monotone Random Systems, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[10]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, 178 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.

[11]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[12]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Mathematical Biosciences, 16 (1973), 75-101. doi: 10.1016/0025-5564(73)90046-1.

[13]

P. De Leenheer, Stabiliteit, Regeling en Stabilisatie van Positieve Systemen, PhD thesis, University of Gent, 2000.

[14]

V. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt Berlin, 1995.

[15]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, 1, Springer-Verlag, Berlin, 2003.

[16]

L. Fainshil, M. Margaliot and P. Chigansky, On the stability of positive linear switched systems under arbitrary switching laws, IEEE Transactions on Automatic Control, 54 (2009), 897-899. doi: 10.1109/TAC.2008.2010974.

[17]

A. Fall, A. Iggidr, G. Sallet and J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-68. doi: 10.1051/mmnp:2008011.

[18]

A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516. doi: 10.1016/j.jmaa.2012.05.029.

[19]

L. Gurvits, R. Shorten and O. Mason, On the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 52 (2007), 1099-1103. doi: 10.1109/TAC.2007.899057.

[20]

H. W. Hethcote and J. A. York, Gonorrhea Transmission and Control, 56 of Lectures Notes in Biomathematics, Springer-Verlag, New York, NY, 1984. doi: 10.1007/978-3-662-07544-9.

[21]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, USA, 1985. doi: 10.1017/CBO9780511810817.

[22]

I. Kats and N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mec., 24 (1960), 1225-1246. doi: 10.1016/0021-8928(60)90103-9.

[23]

V. S. Kozyakin, Algebraic unsolvability of problem of absolute stability of desynchronized systems, Autom. Rem. Control, 51 (1990), 754-759.

[24]

N. Krasovskii and E. Lidskii, Analytical design of controllers in systems with random attributes, Automation and Remote Control, 22 (1961), 1021-1025.

[25]

A. Lajmanovic and J. Yorke, A deterministic model for gonorrhea in nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

[26]

D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, MA, USA, 2003. doi: 10.1007/978-1-4612-0017-8.

[27]

X. Liu and X.-Q. Zhao, A periodic epidemic model with age-structure in a patchy environment, SIAM Journal of Applied Mathematics, 71 (2011), 1896-1917. doi: 10.1137/100813610.

[28]

N. Lloyd, Degree Theory, Cambridge University Press, 1978.

[29]

M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, NY, 1990.

[30]

O. Mason and F. Wirth, Extremal norms for positive linear inclusions, Linear Algebra and its Applications, 444 (2014), 100-113. doi: 10.1016/j.laa.2013.11.020.

[31]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.

[32]

J. Norris, Markov Chains, Cambridge University Press, Cambridge, 2008.

[33]

S. Pröll, Stability of Switched Epidemiological Models, Master's thesis, Institute for Mathematics, University of Würzburg, Würzburg, Germany, 2013.

[34]

C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, Journal of Mathematical Biology, 64 (2012), 933-949. doi: 10.1007/s00285-011-0440-6.

[35]

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability theory for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X.

[36]

H. A. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, USA, 1995.

[37]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, USA, 2011.

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[39]

F. Wirth, The generalized spectral radius and extremal norms, Lin. Alg. Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3.

[40]

World Health Organization (WHO), The global burden of disease: 2004 update, http://www.who.int/healthinfo/global_burden_disease/2004_report_update/en/index.html, 2008, Last Retrieved: 05 December 2011.

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