Global dynamics of a piece-wise epidemic model with switching vaccination strategy

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November 2014, 19(9): 2915-2940. doi: 10.3934/dcdsb.2014.19.2915

Global dynamics of a piece-wise epidemic model with switching vaccination strategy

Aili Wang ^1, , Yanni Xiao ^1, and Robert A. Cheke ^2,

1.	Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China
2.	Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent ME44TB, United Kingdom

Received December 2013 Revised May 2014 Published September 2014

Abstract
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A piece-wise epidemic model of a switching vaccination program, implemented once the number of people exposed to a disease-causing virus reaches a critical level, is proposed. In addition, variation or uncertainties in interventions are examined with a perturbed system version of the model. We also analyzed the global dynamic behaviors of both the original piece-wise system and the perturbed version theoretically, using generalized Jacobian theory, Lyapunov constants for a non-smooth vector field and a generalization of Dulac's criterion. The main results show that, as the critical value varies, there are three possibilities for stabilization of the piece-wise system: (i) at the disease-free equilibrium; (ii) at the endemic states for the two subsystems or (iii) at a generalized equilibrium which is a novel global attractor for non-smooth systems. The perturbed system exhibits new global attractors including a pseudo-focus of parabolic-parabolic (PP) type, a pseudo-equilibrium and a crossing cycle surrounding a sliding mode region. Our findings demonstrate that an infectious disease can be eradicated either by increasing the vaccination rate or by stabilizing the number of infected individuals at a previously given level, conditional upon a suitable critical level and the parameter values.

Keywords: Piece-wise epidemic model, generalized equilibrium, global dynamics, limit cycle., perturbed system, vaccination.

Mathematics Subject Classification: Primary: 92D30, 92B05; Secondary: 34C0.

Citation: Aili Wang, Yanni Xiao, Robert A. Cheke. Global dynamics of a piece-wise epidemic model with switching vaccination strategy. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2915-2940. doi: 10.3934/dcdsb.2014.19.2915

References:

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[2]	M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Springer, New York, 2008.
[3]	, China Ministry of Health and joint UN programme on HIV/AIDS WHO, Estimates for the HIV/AIDS Epidemic in China, 2009.
[4]	F. Clarke, Y. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
[5]	A. B. Claudio, P. D. S. Paulo and A. T. Marco, A singular approach to discontinuous vector fields on the plane, J. Diff. Equa., 231 (2006), 633-655. doi: 10.1016/j.jde.2006.08.017.
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[7]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.
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[10]	, A/H1N1 vaccination program extends to all Beijingers,, November 7, (2009), 2009.
[11]	, Guangdong starts A/H1N1 vaccination for migrant workers,, January 6, (2010), 2010.
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[13]	G. R. Jiang and Q. G. Yang, Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination, Appl. Math. Comput., 215 (2009), 1035-1046. doi: 10.1016/j.amc.2009.06.032.
[14]	R. I. Leine, Bifurcations of equilibria in non-smooth continuous systems, Phys. D, 223 (2006), 121-137. doi: 10.1016/j.physd.2006.08.021.
[15]	R. I. Leine and D. H. van Campen, Bifurcation phenomena in non-smooth dynamical systems, Eur. J. Mech. A Solids, 25 (2006), 595-616. doi: 10.1016/j.euromechsol.2006.04.004.
[16]	D. Liberzon, Switching in Systems and Control, Springer-Verlag, New York, 1973. doi: 10.1007/978-1-4612-0017-8.
[17]	J. Melin, Does distribution theory contain means for extending Poincare-Bendixson theory, J. Math. Anal. Appl., 303 (2005), 81-89. doi: 10.1016/j.jmaa.2004.06.069.
[18]	M. E. M. Meza, A. Bhaya, E. K. Kaszkurewicz, D. A. Silveira and M. I. Costa, Threshold policies control for predator-prey systems using a control Liapunov function approach, Theor. Popul. Biol., 67 (2005), 273-284. doi: 10.1016/j.tpb.2005.01.005.
[19]	M. E. M. Meza, M. I. S. Costa, A. Bhaya and E. Kaszkurewicz, Threshold policies in the control of predator-prey models, Preprints of the 15th Triennial World Congress (IFAC), Barcelona, Spain, (2002), 1-6.
[20]	L. F. Nie, Z. D. Teng and A. Torres, Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination, Nonl. Anal. RWA., 13 (2012), 1621-1629. doi: 10.1016/j.nonrwa.2011.11.019.
[21]	A. d'. Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Bios., 179 (2002), 57-72. doi: 10.1016/S0025-5564(02)00095-0.
[22]	L. Sanchez, Convergence to equilibria in the Lorenz system via monotone methods, J. Diff. Equa., 217 (2005), 341-362. doi: 10.1016/j.jde.2004.08.005.
[23]	S. Y. Tang and J. H. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonl. Anal. TMA., 76 (2013), 165-180. doi: 10.1016/j.na.2012.08.013.
[24]	S. Y. Tang, J. H. Liang, Y. N. Xiao and R. A. Cheke, Sliding bifurcation of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061-1080. doi: 10.1137/110847020.
[25]	S. Y. Tang, Y. N. Xiao and et.al., Community-based measures for mitigating the 2009 H1N1 pandemic in China, PLoS ONE, 5 (2010), 1-11(e10911). doi: 10.1371/journal.pone.0010911.
[26]	V. I. Utkin, Sliding Modes and Their Applications in Variable Structure Systems, Mir, Moscow, 1978.
[27]	V. I. Utkin, Sliding Modes in Control and Optimization, Springer, Berlin, 1992. doi: 10.1007/978-3-642-84379-2.
[28]	A. L. Wang and Y. N. Xiao, Sliding bifurcation and global dynamics of a Filippov epidemic model with vaccination, Internat. J. Bifur. Chaos, 23 (2013). doi: 10.1142/S0218127413501447.
[29]	W. D. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.
[30]	Y. N. Xiao and S. Y. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonl. Anal. RWA., 11 (2010), 4154-4163. doi: 10.1016/j.nonrwa.2010.05.002.
[31]	Y. N. Xiao, X. X. Xu and S. Y. Tang, Sliding mode control of outbreaks of emerging infectious diseases, Bull. Math. Biol., 74 (2012), 2403-2422. doi: 10.1007/s11538-012-9758-5.
[32]	Y. N. Xiao, T. T. Zhao and S. Y. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445-461. doi: 10.3934/mbe.2013.10.445.
[33]	T. R. Zhang and W. D. Wang, Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235. doi: 10.1016/j.apm.2012.02.012.

show all references

References:

[1]	J. Arino, C. C. Mccluskey and P. V. D. Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276. doi: 10.1137/S0036139902413829.
[2]	M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Springer, New York, 2008.
[3]	, China Ministry of Health and joint UN programme on HIV/AIDS WHO, Estimates for the HIV/AIDS Epidemic in China, 2009.
[4]	F. Clarke, Y. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
[5]	A. B. Claudio, P. D. S. Paulo and A. T. Marco, A singular approach to discontinuous vector fields on the plane, J. Diff. Equa., 231 (2006), 633-655. doi: 10.1016/j.jde.2006.08.017.
[6]	B. Coll, A. Gasull and R. Prohens, Degenerate hopf bifurcations in discontinuous planar system, J. Math. Anal. Appl., 253 (2001), 671-690. doi: 10.1006/jmaa.2000.7188.
[7]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.
[8]	D. Greenhalgh, Q. J. A. Khan and F. I. Lewis, Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity, Nonl. Anal. TMA., 63 (2005), e779-e788. doi: 10.1016/j.na.2004.12.018.
[9]	M. A. Han and W. N. Zhang, On hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002.
[10]	, A/H1N1 vaccination program extends to all Beijingers,, November 7, (2009), 2009.
[11]	, Guangdong starts A/H1N1 vaccination for migrant workers,, January 6, (2010), 2010.
[12]	J. Hui and L. S. Chen, Impulsive vaccination of SIR epidemic models with nonlinear incidence rates, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 595-605. doi: 10.3934/dcdsb.2004.4.595.
[13]	G. R. Jiang and Q. G. Yang, Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination, Appl. Math. Comput., 215 (2009), 1035-1046. doi: 10.1016/j.amc.2009.06.032.
[14]	R. I. Leine, Bifurcations of equilibria in non-smooth continuous systems, Phys. D, 223 (2006), 121-137. doi: 10.1016/j.physd.2006.08.021.
[15]	R. I. Leine and D. H. van Campen, Bifurcation phenomena in non-smooth dynamical systems, Eur. J. Mech. A Solids, 25 (2006), 595-616. doi: 10.1016/j.euromechsol.2006.04.004.
[16]	D. Liberzon, Switching in Systems and Control, Springer-Verlag, New York, 1973. doi: 10.1007/978-1-4612-0017-8.
[17]	J. Melin, Does distribution theory contain means for extending Poincare-Bendixson theory, J. Math. Anal. Appl., 303 (2005), 81-89. doi: 10.1016/j.jmaa.2004.06.069.
[18]	M. E. M. Meza, A. Bhaya, E. K. Kaszkurewicz, D. A. Silveira and M. I. Costa, Threshold policies control for predator-prey systems using a control Liapunov function approach, Theor. Popul. Biol., 67 (2005), 273-284. doi: 10.1016/j.tpb.2005.01.005.
[19]	M. E. M. Meza, M. I. S. Costa, A. Bhaya and E. Kaszkurewicz, Threshold policies in the control of predator-prey models, Preprints of the 15th Triennial World Congress (IFAC), Barcelona, Spain, (2002), 1-6.
[20]	L. F. Nie, Z. D. Teng and A. Torres, Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination, Nonl. Anal. RWA., 13 (2012), 1621-1629. doi: 10.1016/j.nonrwa.2011.11.019.
[21]	A. d'. Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Bios., 179 (2002), 57-72. doi: 10.1016/S0025-5564(02)00095-0.
[22]	L. Sanchez, Convergence to equilibria in the Lorenz system via monotone methods, J. Diff. Equa., 217 (2005), 341-362. doi: 10.1016/j.jde.2004.08.005.
[23]	S. Y. Tang and J. H. Liang, Global qualitative analysis of a non-smooth Gause predator-prey model with a refuge, Nonl. Anal. TMA., 76 (2013), 165-180. doi: 10.1016/j.na.2012.08.013.
[24]	S. Y. Tang, J. H. Liang, Y. N. Xiao and R. A. Cheke, Sliding bifurcation of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061-1080. doi: 10.1137/110847020.
[25]	S. Y. Tang, Y. N. Xiao and et.al., Community-based measures for mitigating the 2009 H1N1 pandemic in China, PLoS ONE, 5 (2010), 1-11(e10911). doi: 10.1371/journal.pone.0010911.
[26]	V. I. Utkin, Sliding Modes and Their Applications in Variable Structure Systems, Mir, Moscow, 1978.
[27]	V. I. Utkin, Sliding Modes in Control and Optimization, Springer, Berlin, 1992. doi: 10.1007/978-3-642-84379-2.
[28]	A. L. Wang and Y. N. Xiao, Sliding bifurcation and global dynamics of a Filippov epidemic model with vaccination, Internat. J. Bifur. Chaos, 23 (2013). doi: 10.1142/S0218127413501447.
[29]	W. D. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.
[30]	Y. N. Xiao and S. Y. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonl. Anal. RWA., 11 (2010), 4154-4163. doi: 10.1016/j.nonrwa.2010.05.002.
[31]	Y. N. Xiao, X. X. Xu and S. Y. Tang, Sliding mode control of outbreaks of emerging infectious diseases, Bull. Math. Biol., 74 (2012), 2403-2422. doi: 10.1007/s11538-012-9758-5.
[32]	Y. N. Xiao, T. T. Zhao and S. Y. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445-461. doi: 10.3934/mbe.2013.10.445.
[33]	T. R. Zhang and W. D. Wang, Hopf bifurcation and bistability of a nutrient-phytoplankton-zooplankton model, Appl. Math. Model., 36 (2012), 6225-6235. doi: 10.1016/j.apm.2012.02.012.

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