# American Institute of Mathematical Sciences

June  2014, 19(4): 999-1025. doi: 10.3934/dcdsb.2014.19.999

## 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

Received  June 2013 Revised  January 2014 Published  April 2014

In this paper, we consider a class of epidemic models described by five nonlinear ordinary differential equations. The population is divided into susceptible, vaccinated, exposed, infectious, and recovered subclasses. One main feature of this kind of models is that treatment and vaccination are introduced to control and prevent infectious diseases. The existence and local stability of the endemic equilibria are studied. The occurrence of backward bifurcation is established by using center manifold theory. Moveover, global dynamics are studied by applying the geometric approach. We would like to mention that in the case of bistability, global results are difficult to obtain since there is no compact absorbing set. It is the first time that higher (greater than or equal to four) dimensional systems are discussed. We give sufficient conditions in terms of the system parameters by extending the method in Arino et al. [2]. Numerical simulations are also provided to support our theoretical results. By carrying out sensitivity analysis of the basic reproduction number in terms of some parameters, some effective measures to control infectious diseases are analyzed.
Citation: 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
##### 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.
 [1] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [2] Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445 [3] Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119 [4] Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81 [5] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [6] Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297 [7] 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 [8] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 [9] Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105 [10] Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056 [11] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [12] Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333 [13] Jinliang Wang, Gang Huang, Yasuhiro Takeuchi, Shengqiang Liu. Sveir epidemiological model with varying infectivity and distributed delays. Mathematical Biosciences & Engineering, 2011, 8 (3) : 875-888. doi: 10.3934/mbe.2011.8.875 [14] Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069 [15] 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 [16] Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057 [17] Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016 [18] Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173 [19] Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033 [20] Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863

2021 Impact Factor: 1.497