-
Previous Article
On the steady state of a shadow system to the SKT competition model
- DCDS-B Home
- This Issue
-
Next Article
Stochastically perturbed sliding motion in piecewise-smooth systems
Global dynamics of a piece-wise epidemic model with switching vaccination strategy
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 |
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. |
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. |
[1] |
Boris Kruglikov, Martin Rypdal. A piece-wise affine contracting map with positive entropy. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 393-394. doi: 10.3934/dcds.2006.16.393 |
[2] |
Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999 |
[3] |
Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 |
[4] |
Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076 |
[5] |
Martin Luther Mann Manyombe, Joseph Mbang, Jean Lubuma, Berge Tsanou. Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers. Mathematical Biosciences & Engineering, 2016, 13 (4) : 813-840. doi: 10.3934/mbe.2016019 |
[6] |
Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114 |
[7] |
Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 |
[8] |
Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010 |
[9] |
Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 |
[10] |
Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059 |
[11] |
Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977 |
[12] |
Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060 |
[13] |
Alfonso Ruiz Herrera. Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2533-2548. doi: 10.3934/cpaa.2020111 |
[14] |
Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure and Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027 |
[15] |
Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 |
[16] |
Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893 |
[17] |
Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292 |
[18] |
Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77 |
[19] |
Aiyong Chen, Chi Zhang, Wentao Huang. Limit speed of traveling wave solutions for the perturbed generalized KdV equation. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022048 |
[20] |
Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic and Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]