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Backward bifurcation and global stability in an epidemic model with treatment and vaccination
1. | College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China, China |
2. | Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China |
3. | Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi |
References:
[1] |
L. J. S. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci.,163 (2000) 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
J. Arino, C. C. McCluskey and P. van den 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. |
[3] |
F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004) 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[4] |
M. Boven, F. Mooi, J. Schellekens, H. de Melker and M. Kretzschmar, Pathogen adaptation under imperfect vaccination: implications for pertussis, Proc. R. Soc. Lond. B, 272 (2005) 1617-1624. |
[5] |
B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012) 709-720.
doi: 10.1016/j.jmaa.2011.07.006. |
[6] |
C. Castillon-Charez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu, Mathematical approaches for emerging and reemerging infectious diseases: An introduction, Springer-Verlag, New York, 2001, pp. 269.
doi: 10.1007/978-1-4613-0065-6. |
[7] |
C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Math. Biosci. Eng., 1 (2004) 361-404.
doi: 10.3934/mbe.2004.1.361. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. |
[9] |
A. B. Gumel, C. C. McCluskey and J. Watmough, An SVEIR modelfor assessing potential impact of an imperfect anti-SARS vaccine, Math. Biosci. Eng., 3 (2006) 485-512.
doi: 10.3934/mbe.2006.3.485. |
[10] |
H. W. Hethcote, Oscillations in an endemic model for pertussis, Can. Appl. Math. Quart., 6 (1998) 61-88. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000) 599-653.
doi: 10.1137/S0036144500371907. |
[12] |
K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997) 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[13] |
Z. Hu, S. Liu and H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Anal. Real World Appl., 9 (2008) 2302-2312.
doi: 10.1016/j.nonrwa.2007.08.009. |
[14] |
Z. Hu, W. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012) 12-20.
doi: 10.1016/j.mbs.2012.03.010. |
[15] |
J. Hui and D. Zhu, Global stability and periodicity on SIS epidemic models with backward bifurcation, Comput. Math. Appl., 50 (2005) 1271-1290.
doi: 10.1016/j.camwa.2005.06.003. |
[16] |
T. K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Biosyst., 111 (2013) 37-50.
doi: 10.1016/j.biosystems.2012.10.003. |
[17] |
M. Y. Li and J. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083.
doi: 10.1137/S0036141094266449. |
[18] |
M. Y. Li and J. Muldowney, On R.A. Smith's autonomous convergence theorem, Rocky Mountain J. Math., 25 (1995) 365-379.
doi: 10.1216/rmjm/1181072289. |
[19] |
M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1993) 27-39.
doi: 10.1006/jdeq.1993.1097. |
[20] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Appl. Math. Comput., 210 (2009) 141-150.
doi: 10.1016/j.amc.2008.12.085. |
[21] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIS epidemic model with treatment, Chaos Solitons Fractals, 42 (2009) 2822-2832.
doi: 10.1016/j.chaos.2009.04.024. |
[22] |
X. Z. Li, J. Wang and M. Ghosh, Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination, Appl. Math. Modelling, 34 (2010) 437-450.
doi: 10.1016/j.apm.2009.06.002. |
[23] |
S. M. Moghadas, Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. Math. Comput., 149 (2004) 689-702.
doi: 10.1016/S0096-3003(03)00171-1. |
[24] |
S. M. Moghadas, Modelling the effect of imperfect vaccines on disease epidemiology, Discr. Cont. Dyn. Syst. Ser. B, 4 (2004) 999-1012.
doi: 10.3934/dcdsb.2004.4.999. |
[25] |
X. Mei and J. Huang, Differential Geometry, $4^{th}$ edition, Higher Education Press, Beijing, 2008. |
[26] |
I. Nasell, On the time to extinction in recurrent epidemics, J. R. Stat. Soc. Ser. B, 61 (1999) 309-330.
doi: 10.1111/1467-9868.00178. |
[27] |
I. Nasell, Stochastic model of some endemic infections, Math. Biosci., 179 (2002) 1-9.
doi: 10.1016/S0025-5564(02)00098-6. |
[28] |
H. Shu and L. Wang, Role of CD4+ T-cell proliferation in HIV infection under antiretroviral therapy, J. Math. Anal. Appl., 394 (2012) 529-544.
doi: 10.1016/j.jmaa.2012.05.027. |
[29] |
Y. Tang and W. Li, Global analysis of an epidemic model with a constant removal rate, Math. Comput. Modelling, 45 (2007) 834-843.
doi: 10.1016/j.mcm.2006.08.003. |
[30] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002) 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[31] |
P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000) 525-540.
doi: 10.1007/s002850000032. |
[32] |
W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006) 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[33] |
W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004) 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
[34] |
J. Wang, S. Liu, B. Zheng and Y. Takeuchi, Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012) 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
[35] |
H. Wan and J. Cui, A model for the transmission of malaria, Discr. Cont. Dyn. Syst. Ser. B, 227 (2009) 479-496.
doi: 10.3934/dcdsb.2009.11.479. |
[36] |
W. Yang, C. Sun and J. Arino, Global analysis for a general epidemiological model with vaccination and varying population, J. Math. Anal. Appl., 372 (2010) 208-223.
doi: 10.1016/j.jmaa.2010.07.017. |
[37] |
X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008) 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
[38] |
X. Zhang and X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal. Real World Appl., 10 (2009) 565-575.
doi: 10.1016/j.nonrwa.2007.10.011. |
[39] |
L. Zhou and M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012) 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |
[40] |
X. Zhou and J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simul., 16 (2011) 4438-4450.
doi: 10.1016/j.cnsns.2011.03.026. |
show all references
References:
[1] |
L. J. S. Allen and A. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci.,163 (2000) 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
J. Arino, C. C. McCluskey and P. van den 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. |
[3] |
F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004) 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[4] |
M. Boven, F. Mooi, J. Schellekens, H. de Melker and M. Kretzschmar, Pathogen adaptation under imperfect vaccination: implications for pertussis, Proc. R. Soc. Lond. B, 272 (2005) 1617-1624. |
[5] |
B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012) 709-720.
doi: 10.1016/j.jmaa.2011.07.006. |
[6] |
C. Castillon-Charez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu, Mathematical approaches for emerging and reemerging infectious diseases: An introduction, Springer-Verlag, New York, 2001, pp. 269.
doi: 10.1007/978-1-4613-0065-6. |
[7] |
C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Math. Biosci. Eng., 1 (2004) 361-404.
doi: 10.3934/mbe.2004.1.361. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. |
[9] |
A. B. Gumel, C. C. McCluskey and J. Watmough, An SVEIR modelfor assessing potential impact of an imperfect anti-SARS vaccine, Math. Biosci. Eng., 3 (2006) 485-512.
doi: 10.3934/mbe.2006.3.485. |
[10] |
H. W. Hethcote, Oscillations in an endemic model for pertussis, Can. Appl. Math. Quart., 6 (1998) 61-88. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000) 599-653.
doi: 10.1137/S0036144500371907. |
[12] |
K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997) 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[13] |
Z. Hu, S. Liu and H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Anal. Real World Appl., 9 (2008) 2302-2312.
doi: 10.1016/j.nonrwa.2007.08.009. |
[14] |
Z. Hu, W. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012) 12-20.
doi: 10.1016/j.mbs.2012.03.010. |
[15] |
J. Hui and D. Zhu, Global stability and periodicity on SIS epidemic models with backward bifurcation, Comput. Math. Appl., 50 (2005) 1271-1290.
doi: 10.1016/j.camwa.2005.06.003. |
[16] |
T. K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Biosyst., 111 (2013) 37-50.
doi: 10.1016/j.biosystems.2012.10.003. |
[17] |
M. Y. Li and J. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083.
doi: 10.1137/S0036141094266449. |
[18] |
M. Y. Li and J. Muldowney, On R.A. Smith's autonomous convergence theorem, Rocky Mountain J. Math., 25 (1995) 365-379.
doi: 10.1216/rmjm/1181072289. |
[19] |
M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1993) 27-39.
doi: 10.1006/jdeq.1993.1097. |
[20] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Appl. Math. Comput., 210 (2009) 141-150.
doi: 10.1016/j.amc.2008.12.085. |
[21] |
X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIS epidemic model with treatment, Chaos Solitons Fractals, 42 (2009) 2822-2832.
doi: 10.1016/j.chaos.2009.04.024. |
[22] |
X. Z. Li, J. Wang and M. Ghosh, Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination, Appl. Math. Modelling, 34 (2010) 437-450.
doi: 10.1016/j.apm.2009.06.002. |
[23] |
S. M. Moghadas, Analysis of an epidemic model with bistable equilibria using the Poincaré index, Appl. Math. Comput., 149 (2004) 689-702.
doi: 10.1016/S0096-3003(03)00171-1. |
[24] |
S. M. Moghadas, Modelling the effect of imperfect vaccines on disease epidemiology, Discr. Cont. Dyn. Syst. Ser. B, 4 (2004) 999-1012.
doi: 10.3934/dcdsb.2004.4.999. |
[25] |
X. Mei and J. Huang, Differential Geometry, $4^{th}$ edition, Higher Education Press, Beijing, 2008. |
[26] |
I. Nasell, On the time to extinction in recurrent epidemics, J. R. Stat. Soc. Ser. B, 61 (1999) 309-330.
doi: 10.1111/1467-9868.00178. |
[27] |
I. Nasell, Stochastic model of some endemic infections, Math. Biosci., 179 (2002) 1-9.
doi: 10.1016/S0025-5564(02)00098-6. |
[28] |
H. Shu and L. Wang, Role of CD4+ T-cell proliferation in HIV infection under antiretroviral therapy, J. Math. Anal. Appl., 394 (2012) 529-544.
doi: 10.1016/j.jmaa.2012.05.027. |
[29] |
Y. Tang and W. Li, Global analysis of an epidemic model with a constant removal rate, Math. Comput. Modelling, 45 (2007) 834-843.
doi: 10.1016/j.mcm.2006.08.003. |
[30] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002) 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[31] |
P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000) 525-540.
doi: 10.1007/s002850000032. |
[32] |
W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006) 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[33] |
W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004) 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
[34] |
J. Wang, S. Liu, B. Zheng and Y. Takeuchi, Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012) 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
[35] |
H. Wan and J. Cui, A model for the transmission of malaria, Discr. Cont. Dyn. Syst. Ser. B, 227 (2009) 479-496.
doi: 10.3934/dcdsb.2009.11.479. |
[36] |
W. Yang, C. Sun and J. Arino, Global analysis for a general epidemiological model with vaccination and varying population, J. Math. Anal. Appl., 372 (2010) 208-223.
doi: 10.1016/j.jmaa.2010.07.017. |
[37] |
X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008) 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
[38] |
X. Zhang and X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal. Real World Appl., 10 (2009) 565-575.
doi: 10.1016/j.nonrwa.2007.10.011. |
[39] |
L. Zhou and M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012) 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |
[40] |
X. Zhou and J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simul., 16 (2011) 4438-4450.
doi: 10.1016/j.cnsns.2011.03.026. |
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