# American Institute of Mathematical Sciences

• Previous Article
Modeling eating behaviors: The role of environment and positive food association learning via a Ratatouille effect
• MBE Home
• This Issue
• Next Article
Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation
2016, 13(4): 813-840. doi: 10.3934/mbe.2016019

## Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers

 1 Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812 Yaounde, Cameroon, Cameroon 2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa, South Africa

Received  August 2015 Revised  January 2016 Published  May 2016

In this paper, an epidemic model is investigated for infectious diseases that can be transmitted through both the infectious individuals and the asymptomatic carriers (i.e., infected individuals who are contagious but do not show any disease symptoms). We propose a dose-structured vaccination model with multiple transmission pathways. Based on the range of the explicitly computed basic reproduction number, we prove the global stability of the disease-free when this threshold number is less or equal to the unity. Moreover, whenever it is greater than one, the existence of the unique endemic equilibrium is shown and its global stability is established for the case where the changes of displaying the disease symptoms are independent of the vulnerable classes. Further, the model is shown to exhibit a transcritical bifurcation with the unit basic reproduction number being the bifurcation parameter. The impacts of the asymptomatic carriers and the effectiveness of vaccination on the disease transmission are discussed through through the local and the global sensitivity analyses of the basic reproduction number. Finally, a case study of hepatitis B virus disease (HBV) is considered, with the numerical simulations presented to support the analytical results. They further suggest that, in high HBV prevalence countries, the combination of effective vaccination (i.e. $\geq 3$ doses of HepB vaccine), the diagnosis of asymptomatic carriers and the treatment of symptomatic carriers may have a much greater positive impact on the disease control.
Citation: Martin Luther Mann Manyombe, Joseph Mbang, Jean Lubuma, Berge Tsanou. Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers. Mathematical Biosciences & Engineering, 2016, 13 (4) : 813-840. doi: 10.3934/mbe.2016019
##### References:
 [1] H. Abboubakar, J. C. Kamgang, L. N. Nkamba, D. Tieudjo and L. Emini, Modeling the dynamics of arboviral diseases with vaccination perspective,, Biomath, 4 (2015).  doi: 10.11145/j.biomath.2015.07.241.  Google Scholar [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991).   Google Scholar [3] 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.  doi: 10.1137/S0036139902413829.  Google Scholar [4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Springer, (2001).  doi: 10.1007/978-1-4757-3516-1.  Google Scholar [5] C. Castillo-Chavez and B. Song, Dynamical model of tuberclosis and their applications,, Math.Biosci.Eng, 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar [6] 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 [7] C. P. Farrington, On vaccine efficacy and reproduction numbers,, Math. Biosci., 185 (2003), 89.  doi: 10.1016/S0025-5564(03)00061-0.  Google Scholar [8] G. Francois, M. Kew, P. Van Damme, M. J. Mphahlele and A. Meheus, Mutant hepatitis B viruses: A matter of academic interest only or a problem with far-reaching implications,, Vaccine, 19 (2001), 3799.  doi: 10.1016/S0264-410X(01)00108-6.  Google Scholar [9] M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modelling the spread of carrier-dependent infectious diseases with environmental effect,, Appl. Math. Comput., 152 (2004), 385.  doi: 10.1016/S0096-3003(03)00564-2.  Google Scholar [10] J. Gjorgjieva, K. Smith, G. Chowell, F. Sanchez, J. Snyder and C. Castillo-Chavez, The role of vaccination in the control of SARS,, Math. Biosci. Eng., 2 (2005), 753.  doi: 10.3934/mbe.2005.2.753.  Google Scholar [11] S. Goldstein, F. Zhou, S. C. Hadler, B. P. Bell, E. E. Mast and H. S. Margolis, A mathematical model to estimate global hepatitis B disease burden and vaccination impact,, Int. J. Epidemiol., 34 (2005), 1329.  doi: 10.1093/ije/dyi206.  Google Scholar [12] B. Gomero, Latin Hypercube Sampling and Partial Rank Correlation Coefficient Analysis Applied to an Optimal Control Problem,, Master Thesis, (2012).   Google Scholar [13] A. B. Gumel and S. M. Moghadas, A qualitative study of a vaccination model with non-linear incidence,, Appl. Math. Comp, 143 (2003), 409.  doi: 10.1016/S0096-3003(02)00372-7.  Google Scholar [14] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.  doi: 10.3934/mbe.2006.3.513.  Google Scholar [15] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89.  doi: 10.3934/mbe.2006.3.89.  Google Scholar [16] D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on the transmission dynamics of infectious diseases,, Math. Biosci. Eng., 8 (2011), 711.  doi: 10.3934/mbe.2011.8.711.  Google Scholar [17] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707.  doi: 10.1007/BF02460601.  Google Scholar [18] A. Korobeinikov, Global properties of sir and seir epidemic models with multiple parallel infectious stages,, Bull. Math. Bio., 71 (2009), 75.  doi: 10.1007/s11538-008-9352-z.  Google Scholar [19] J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar [20] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol, 254 (2008), 178.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar [21] G. F. Medley, N. A. Lindop, W. J. Edmunds and D. J. Nokes, Hepatitis-B virus edemicity: Heterogeneity, catastrophic dynamics and control,, Nat. Med., 7 (2001), 617.   Google Scholar [22] R. Naresh, S. Pandey and A. K. Misra, Analysis of a vaccination model for carrier dependent infectious diseases with environmental effects,, Nonlinear Analysis: Modelling and Control, 13 (2008), 331.   Google Scholar [23] M. M. Riggs, A. K. Sethi, T. F. Zabarsky, E. C. Eckstein, R. L. Jump and C. J. Donskey, Asymptomatic carriers are a potential source for transmission of epidemic and nonepidemic Clostridium diffcile strains among long-term care facility residents,, Clin. Infect. Dis., 45 (2007), 992.   Google Scholar [24] P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301.  doi: 10.1126/science.1134933.  Google Scholar [25] C. L. Trotter, N. J. Gay and W. J. Edmunds, Dynamic models of meningococcal carriage, disease, and the impact of serogroup C conjugate vaccination,, Am. J. Epidemiol., 162 (2005), 89.  doi: 10.1093/aje/kwi160.  Google Scholar [26] S. Zhao, Z. Xu and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China,, Int. J. Epidemiol., 29 (2000), 744.  doi: 10.1093/ije/29.4.744.  Google Scholar [27] L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China,, J. Theor. Biol., 262 (2010), 330.  doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar [28] "The ABCs of Hepatitis", Center for Disease Control and Prevention (CDC), 2015., Available from: , ().   Google Scholar [29] WHO, "Fact Sheet N$^o$ 204 on Hepatitis B",, July 2015. Available from: , (2015).   Google Scholar

show all references

##### References:
 [1] H. Abboubakar, J. C. Kamgang, L. N. Nkamba, D. Tieudjo and L. Emini, Modeling the dynamics of arboviral diseases with vaccination perspective,, Biomath, 4 (2015).  doi: 10.11145/j.biomath.2015.07.241.  Google Scholar [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991).   Google Scholar [3] 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.  doi: 10.1137/S0036139902413829.  Google Scholar [4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Springer, (2001).  doi: 10.1007/978-1-4757-3516-1.  Google Scholar [5] C. Castillo-Chavez and B. Song, Dynamical model of tuberclosis and their applications,, Math.Biosci.Eng, 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar [6] 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 [7] C. P. Farrington, On vaccine efficacy and reproduction numbers,, Math. Biosci., 185 (2003), 89.  doi: 10.1016/S0025-5564(03)00061-0.  Google Scholar [8] G. Francois, M. Kew, P. Van Damme, M. J. Mphahlele and A. Meheus, Mutant hepatitis B viruses: A matter of academic interest only or a problem with far-reaching implications,, Vaccine, 19 (2001), 3799.  doi: 10.1016/S0264-410X(01)00108-6.  Google Scholar [9] M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modelling the spread of carrier-dependent infectious diseases with environmental effect,, Appl. Math. Comput., 152 (2004), 385.  doi: 10.1016/S0096-3003(03)00564-2.  Google Scholar [10] J. Gjorgjieva, K. Smith, G. Chowell, F. Sanchez, J. Snyder and C. Castillo-Chavez, The role of vaccination in the control of SARS,, Math. Biosci. Eng., 2 (2005), 753.  doi: 10.3934/mbe.2005.2.753.  Google Scholar [11] S. Goldstein, F. Zhou, S. C. Hadler, B. P. Bell, E. E. Mast and H. S. Margolis, A mathematical model to estimate global hepatitis B disease burden and vaccination impact,, Int. J. Epidemiol., 34 (2005), 1329.  doi: 10.1093/ije/dyi206.  Google Scholar [12] B. Gomero, Latin Hypercube Sampling and Partial Rank Correlation Coefficient Analysis Applied to an Optimal Control Problem,, Master Thesis, (2012).   Google Scholar [13] A. B. Gumel and S. M. Moghadas, A qualitative study of a vaccination model with non-linear incidence,, Appl. Math. Comp, 143 (2003), 409.  doi: 10.1016/S0096-3003(02)00372-7.  Google Scholar [14] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.  doi: 10.3934/mbe.2006.3.513.  Google Scholar [15] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89.  doi: 10.3934/mbe.2006.3.89.  Google Scholar [16] D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on the transmission dynamics of infectious diseases,, Math. Biosci. Eng., 8 (2011), 711.  doi: 10.3934/mbe.2011.8.711.  Google Scholar [17] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707.  doi: 10.1007/BF02460601.  Google Scholar [18] A. Korobeinikov, Global properties of sir and seir epidemic models with multiple parallel infectious stages,, Bull. Math. Bio., 71 (2009), 75.  doi: 10.1007/s11538-008-9352-z.  Google Scholar [19] J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar [20] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol, 254 (2008), 178.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar [21] G. F. Medley, N. A. Lindop, W. J. Edmunds and D. J. Nokes, Hepatitis-B virus edemicity: Heterogeneity, catastrophic dynamics and control,, Nat. Med., 7 (2001), 617.   Google Scholar [22] R. Naresh, S. Pandey and A. K. Misra, Analysis of a vaccination model for carrier dependent infectious diseases with environmental effects,, Nonlinear Analysis: Modelling and Control, 13 (2008), 331.   Google Scholar [23] M. M. Riggs, A. K. Sethi, T. F. Zabarsky, E. C. Eckstein, R. L. Jump and C. J. Donskey, Asymptomatic carriers are a potential source for transmission of epidemic and nonepidemic Clostridium diffcile strains among long-term care facility residents,, Clin. Infect. Dis., 45 (2007), 992.   Google Scholar [24] P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301.  doi: 10.1126/science.1134933.  Google Scholar [25] C. L. Trotter, N. J. Gay and W. J. Edmunds, Dynamic models of meningococcal carriage, disease, and the impact of serogroup C conjugate vaccination,, Am. J. Epidemiol., 162 (2005), 89.  doi: 10.1093/aje/kwi160.  Google Scholar [26] S. Zhao, Z. Xu and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China,, Int. J. Epidemiol., 29 (2000), 744.  doi: 10.1093/ije/29.4.744.  Google Scholar [27] L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China,, J. Theor. Biol., 262 (2010), 330.  doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar [28] "The ABCs of Hepatitis", Center for Disease Control and Prevention (CDC), 2015., Available from: , ().   Google Scholar [29] WHO, "Fact Sheet N$^o$ 204 on Hepatitis B",, July 2015. Available from: , (2015).   Google Scholar
 [1] Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134 [2] Dennis L. Chao, Dobromir T. Dimitrov. Seasonality and the effectiveness of mass vaccination. Mathematical Biosciences & Engineering, 2016, 13 (2) : 249-259. doi: 10.3934/mbe.2015001 [3] Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073 [4] 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 [5] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [6] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [7] Jane M. Heffernan, Yijun Lou, Marc Steben, Jianhong Wu. Cost-effectiveness evaluation of gender-based vaccination programs against sexually transmitted infections. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 447-466. doi: 10.3934/dcdsb.2014.19.447 [8] Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination. Mathematical Biosciences & Engineering, 2017, 14 (4) : 975-999. doi: 10.3934/mbe.2017051 [9] Muntaser Safan, Klaus Dietz. On the eradicability of infections with partially protective vaccination in models with backward bifurcation. Mathematical Biosciences & Engineering, 2009, 6 (2) : 395-407. doi: 10.3934/mbe.2009.6.395 [10] Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 [11] Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631 [12] Hui Cao, Yicang Zhou, Zhien Ma. Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1399-1417. doi: 10.3934/mbe.2013.10.1399 [13] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [14] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [15] Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449 [16] Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635 [17] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [18] Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009 [19] Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087 [20] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

2018 Impact Factor: 1.313