American Institute of Mathematical Sciences

Advanced
2013, 10(5&6): 1651-1668. doi: 10.3934/mbe.2013.10.1651

Different types of backward bifurcations due to density-dependent treatments

Baojun Song 1, , Wen Du 2, and  Jie Lou 2,

1. 

Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043
2. 

Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China

Received  October 2012 Revised  May 2013 Published  August 2013

A set of deterministic SIS models with density-dependent treatments are studied to understand the disease dynamics when different treatment strategies are applied. Qualitative analyses are carried out in terms of general treatment functions. It has become customary that a backward bifurcation leads to bistable dynamics. However, this study finds that finds that bistability may not be an option at all; the disease-free equilibrium could be globally stable when there is a backward bifurcation. Furthermore, when a backward bifurcation occurs, the fashion of bistability could be the coexistence of either dual stable equilibria or the disease-free equilibrium and a stable limit cycle. We also extend the formula for mean infection period from density-independent treatments to density-dependent ones. Finally, the modeling results are applied to the transmission of gonorrhea in China, suggesting that these gonorrhea patients may not seek medical treatments in a timely manner.
Keywords: density-dependent, gonorrhea., Treatment, bifurcation.
Mathematics Subject Classification: Primary: 92D30, 37G10, 34D20; Secondary: 34C23, 92B0.
Citation: Baojun Song, Wen Du, Jie Lou. Different types of backward bifurcations due to density-dependent treatments. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1651-1668. doi: 10.3934/mbe.2013.10.1651
References:
[1]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, 40 (2012).  doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[2]

H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography,, Discrete Dynamics in Nature and Society, 2011 (6539).  doi: 10.1155/2011/653937.  Google Scholar

[3]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosc. Eng., 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[4]

X. S. Chen, X. D. Gong, G. J. Liang and G. C. Zhang, Epidemiologic trends of sexually transmitted diseases in China,, Sex Transm. Dis., 27 (2000), 138.  doi: 10.1097/00007435-200003000-00003.  Google Scholar

[5]

J. Cui, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation,, J. Theor. Biol., 254 (2008), 275.  doi: 10.1016/j.jtbi.2008.05.015.  Google Scholar

[6]

H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control,", Lecture Notes in Biomathematics, (1984).   Google Scholar

[7]

Z. Hu, S. Liu and H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate,, Nonlinear Analysis, 9 (2008), 2302.  doi: 10.1016/j.nonrwa.2007.08.009.  Google Scholar

[8]

X. Li, W. Li and Mini Ghosh, Stability and bifurcation of an epidemic model with nonlinear incidence and treatment,, Applied Mathematics and Computation, 210 (2009), 141.  doi: 10.1016/j.amc.2008.12.085.  Google Scholar

[9]

B. R. Morin, L. Medina-Rios, E. T. Camacho and C. Castillo-Chavez, Static behavioral effects on gonorrhea transmission dynamics in a MSM population,, J. Theor. Biol., 267 (2012), 35.  doi: 10.1016/j.jtbi.2010.07.027.  Google Scholar

[10]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[11]

W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[12]

X. Zhang and X. Liu, Bifurcation of an epidemic model with saturated treatment function,, J. Math. Anal. Appl., 348 (2008), 433.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[13]

Centers for Disease Control and Prevention, Gonorrhea-CDC fact sheet,, June 2012. Available from: \url{http://www.cdc.gov/std/gonorrhea/STDFact-gonorrhea-detailed.htm}., (2012).   Google Scholar

[14]

, China Yearbook., Available from: \url{http://www.yearbook.cn/}., ().   Google Scholar

[15]

, Ministry of Health of the People's Republic of China., Available from: \url{http://www.moh.gov.cn/publicfiles/business/htmlfiles/mohjbyfkzj/s2907/index.htm}., ().   Google Scholar

show all references

References:
[1]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Second edition, 40 (2012).  doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[2]

H. Cao, Y. Zhou and B. Song, Complex dynamics of discrete SEIS models with simple demography,, Discrete Dynamics in Nature and Society, 2011 (6539).  doi: 10.1155/2011/653937.  Google Scholar

[3]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosc. Eng., 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[4]

X. S. Chen, X. D. Gong, G. J. Liang and G. C. Zhang, Epidemiologic trends of sexually transmitted diseases in China,, Sex Transm. Dis., 27 (2000), 138.  doi: 10.1097/00007435-200003000-00003.  Google Scholar

[5]

J. Cui, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation,, J. Theor. Biol., 254 (2008), 275.  doi: 10.1016/j.jtbi.2008.05.015.  Google Scholar

[6]

H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control,", Lecture Notes in Biomathematics, (1984).   Google Scholar

[7]

Z. Hu, S. Liu and H. Wang, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate,, Nonlinear Analysis, 9 (2008), 2302.  doi: 10.1016/j.nonrwa.2007.08.009.  Google Scholar

[8]

X. Li, W. Li and Mini Ghosh, Stability and bifurcation of an epidemic model with nonlinear incidence and treatment,, Applied Mathematics and Computation, 210 (2009), 141.  doi: 10.1016/j.amc.2008.12.085.  Google Scholar

[9]

B. R. Morin, L. Medina-Rios, E. T. Camacho and C. Castillo-Chavez, Static behavioral effects on gonorrhea transmission dynamics in a MSM population,, J. Theor. Biol., 267 (2012), 35.  doi: 10.1016/j.jtbi.2010.07.027.  Google Scholar

[10]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[11]

W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[12]

X. Zhang and X. Liu, Bifurcation of an epidemic model with saturated treatment function,, J. Math. Anal. Appl., 348 (2008), 433.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[13]

Centers for Disease Control and Prevention, Gonorrhea-CDC fact sheet,, June 2012. Available from: \url{http://www.cdc.gov/std/gonorrhea/STDFact-gonorrhea-detailed.htm}., (2012).   Google Scholar

[14]

, China Yearbook., Available from: \url{http://www.yearbook.cn/}., ().   Google Scholar

[15]

, Ministry of Health of the People's Republic of China., Available from: \url{http://www.moh.gov.cn/publicfiles/business/htmlfiles/mohjbyfkzj/s2907/index.htm}., ().   Google Scholar

[1]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207
[2]

Jacques A. L. Silva, Flávia T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5 (4) : 843-857. doi: 10.3934/mbe.2008.5.843
[3]

Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008
[4]

J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054
[5]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313
[6]

Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20
[7]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175
[8]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[9]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400
[10]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[11]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[12]

Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157
[13]

Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359
[14]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
[15]

Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481
[16]

Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647
[17]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209
[18]

Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062
[19]

Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150
[20]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences