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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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