# 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 & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
##### References:
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Appl., 385 (2012) 709-720. doi: 10.1016/j.jmaa.2011.07.006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983.  Google Scholar [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.  Google Scholar [10] H. W. Hethcote, Oscillations in an endemic model for pertussis, Can. Appl. Math. Quart., 6 (1998) 61-88.  Google Scholar [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000) 599-653. doi: 10.1137/S0036144500371907.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1993) 27-39. doi: 10.1006/jdeq.1993.1097.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] X. Mei and J. Huang, Differential Geometry, $4^{th}$ edition, Higher Education Press, Beijing, 2008. Google Scholar [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.  Google Scholar [27] I. Nasell, Stochastic model of some endemic infections, Math. Biosci., 179 (2002) 1-9. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983.  Google Scholar [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.  Google Scholar [10] H. W. Hethcote, Oscillations in an endemic model for pertussis, Can. Appl. Math. Quart., 6 (1998) 61-88.  Google Scholar [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000) 599-653. doi: 10.1137/S0036144500371907.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] M. Y. Li and J. Muldowney, On Bendixson's criterion, J. Differential Equations, 106 (1993) 27-39. doi: 10.1006/jdeq.1993.1097.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] X. Mei and J. Huang, Differential Geometry, $4^{th}$ edition, Higher Education Press, Beijing, 2008. Google Scholar [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.  Google Scholar [27] I. Nasell, Stochastic model of some endemic infections, Math. Biosci., 179 (2002) 1-9. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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