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

Different types of backward bifurcations due to density-dependent treatments

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.
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

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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

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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

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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

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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

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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

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

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, 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

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