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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), 1507241, 30pp. doi: 10.11145/j.biomath.2015.07.241. [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991. [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), 260-276. doi: 10.1137/S0036139902413829. [4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [5] C. Castillo-Chavez and B. Song, Dynamical model of tuberclosis and their applications, Math.Biosci.Eng, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [7] C. P. Farrington, On vaccine efficacy and reproduction numbers, Math. Biosci., 185 (2003), 89-109. doi: 10.1016/S0025-5564(03)00061-0. [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), 3799-3815. doi: 10.1016/S0264-410X(01)00108-6. [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), 385-402. doi: 10.1016/S0096-3003(03)00564-2. [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), 753-769. doi: 10.3934/mbe.2005.2.753. [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), 1329-1339. doi: 10.1093/ije/dyi206. [12] B. Gomero, Latin Hypercube Sampling and Partial Rank Correlation Coefficient Analysis Applied to an Optimal Control Problem, Master Thesis, University of Tennessee, Knoxville, 2012. [13] A. B. Gumel and S. M. Moghadas, A qualitative study of a vaccination model with non-linear incidence, Appl. Math. Comp, 143 (2003), 409-419. doi: 10.1016/S0096-3003(02)00372-7. [14] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. [15] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models, Math. Biosci. Eng., 3 (2006), 89-100. doi: 10.3934/mbe.2006.3.89. [16] D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on the transmission dynamics of infectious diseases, Math. Biosci. Eng., 8 (2011), 711-722. doi: 10.3934/mbe.2011.8.711. [17] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model, Bull. Math. Bio., 40 (1978), 707-718. doi: 10.1007/BF02460601. [18] A. Korobeinikov, Global properties of sir and seir epidemic models with multiple parallel infectious stages, Bull. Math. Bio., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z. [19] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [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), 178-196. doi: 10.1016/j.jtbi.2008.04.011. [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), 617-624. [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), 331-350. [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), 992-998. [24] P. Roumagnac, et al., Evolutionary history of Salmonella typhi, Science, 314 (2006), 1301-1304. doi: 10.1126/science.1134933. [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), 89-100. doi: 10.1093/aje/kwi160. [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), 744-752. doi: 10.1093/ije/29.4.744. [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), 330-338. doi: 10.1016/j.jtbi.2009.09.035. [28] "The ABCs of Hepatitis", Center for Disease Control and Prevention (CDC), 2015. Available from: http://www.cdc.gov/hepatitis/Resources/Professionals/PDFs/ABCTable.pdf [29] WHO, "Fact Sheet N$^o$ 204 on Hepatitis B", July 2015. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/

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), 1507241, 30pp. doi: 10.11145/j.biomath.2015.07.241. [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991. [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), 260-276. doi: 10.1137/S0036139902413829. [4] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [5] C. Castillo-Chavez and B. Song, Dynamical model of tuberclosis and their applications, Math.Biosci.Eng, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [7] C. P. Farrington, On vaccine efficacy and reproduction numbers, Math. Biosci., 185 (2003), 89-109. doi: 10.1016/S0025-5564(03)00061-0. [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), 3799-3815. doi: 10.1016/S0264-410X(01)00108-6. [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), 385-402. doi: 10.1016/S0096-3003(03)00564-2. [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), 753-769. doi: 10.3934/mbe.2005.2.753. [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), 1329-1339. doi: 10.1093/ije/dyi206. [12] B. Gomero, Latin Hypercube Sampling and Partial Rank Correlation Coefficient Analysis Applied to an Optimal Control Problem, Master Thesis, University of Tennessee, Knoxville, 2012. [13] A. B. Gumel and S. M. Moghadas, A qualitative study of a vaccination model with non-linear incidence, Appl. Math. Comp, 143 (2003), 409-419. doi: 10.1016/S0096-3003(02)00372-7. [14] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. [15] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models, Math. Biosci. Eng., 3 (2006), 89-100. doi: 10.3934/mbe.2006.3.89. [16] D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on the transmission dynamics of infectious diseases, Math. Biosci. Eng., 8 (2011), 711-722. doi: 10.3934/mbe.2011.8.711. [17] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model, Bull. Math. Bio., 40 (1978), 707-718. doi: 10.1007/BF02460601. [18] A. Korobeinikov, Global properties of sir and seir epidemic models with multiple parallel infectious stages, Bull. Math. Bio., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z. [19] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [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), 178-196. doi: 10.1016/j.jtbi.2008.04.011. [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), 617-624. [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), 331-350. [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), 992-998. [24] P. Roumagnac, et al., Evolutionary history of Salmonella typhi, Science, 314 (2006), 1301-1304. doi: 10.1126/science.1134933. [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), 89-100. doi: 10.1093/aje/kwi160. [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), 744-752. doi: 10.1093/ije/29.4.744. [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), 330-338. doi: 10.1016/j.jtbi.2009.09.035. [28] "The ABCs of Hepatitis", Center for Disease Control and Prevention (CDC), 2015. Available from: http://www.cdc.gov/hepatitis/Resources/Professionals/PDFs/ABCTable.pdf [29] WHO, "Fact Sheet N$^o$ 204 on Hepatitis B", July 2015. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/
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