# American Institute of Mathematical Sciences

2011, 8(3): 711-722. doi: 10.3934/mbe.2011.8.711

## Modeling the effects of carriers on transmission dynamics of infectious diseases

 1 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received  September 2010 Revised  March 2011 Published  June 2011

An $S$-$I_c$-$I$-$R$ epidemic model is investigated for infectious diseases that can be transmitted through carriers, infected individuals who are contagious but do not show any disease symptoms. Mathematical analysis is carried out that completely determines the global dynamics of the model. The impacts of disease carriers on the transmission dynamics are discussed through the basic reproduction number and through numerical simulations.
Citation: Darja Kalajdzievska, Michael Yi Li. Modeling the effects of carriers on transmission dynamics of infectious diseases. Mathematical Biosciences & Engineering, 2011, 8 (3) : 711-722. doi: 10.3934/mbe.2011.8.711
##### References:
 [1] 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. [2] S. Goldstein, F. Zhou, S. C. Hadler, B. P. Bell, E. E. Mast and H. S. Margolis, A mathematical model to estimate global hepatits B disease burden and vaccination impact,, Int. J. Epidemiol., 34 (2005), 1329. doi: 10.1093/ije/dyi206. [3] H. Guo, Global dynamics of a mathematical model of tuberculosis,, Canadian Appl. Math. Quart., 13 (2005), 313. [4] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513. [5] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89. [6] D. Kalajdzievska, "Modeling the Effects of Carriers on the Transmission Dynamics of Infectious Diseases,", M.Sc. thesis, (2006). [7] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707. [8] A. Korobeinikov and P. K. Maini, A Lyaponov function and global properties for SIR and SEIR epedimiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57. [9] J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). [10] 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. doi: 10.1038/87953. [11] 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. [12] 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 difficile strains among long-term care facility residents,, Clin. Infect. Dis., 45 (2007), 992. doi: 10.1086/521854. [13] P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301. doi: 10.1126/science.1134933. [14] 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. [15] 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. [16] "The ABCs of Hepatitis,", Center for Disease Control and Prevention (CDC), 2009., Available from: \url{http://www.cdc.gov/hepatitis/Resources/Professionals/PDFs/ABCTable_BW.pdf}., 2009 (). [17] "Viral Hepatitis and Emerging Bloodborne Pathogens in Canada,", CCDR, 27S3,, Public Health Agency of Canada (PHAC), (2001). [18] WHO, "Fact Sheet on Hepatitis B," 2008., Available from: \url{http://www.who.int/mediacentre/factsheets/fs204/en/index.html}., 2008 ().

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##### References:
 [1] 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. [2] S. Goldstein, F. Zhou, S. C. Hadler, B. P. Bell, E. E. Mast and H. S. Margolis, A mathematical model to estimate global hepatits B disease burden and vaccination impact,, Int. J. Epidemiol., 34 (2005), 1329. doi: 10.1093/ije/dyi206. [3] H. Guo, Global dynamics of a mathematical model of tuberculosis,, Canadian Appl. Math. Quart., 13 (2005), 313. [4] H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513. [5] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89. [6] D. Kalajdzievska, "Modeling the Effects of Carriers on the Transmission Dynamics of Infectious Diseases,", M.Sc. thesis, (2006). [7] J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707. [8] A. Korobeinikov and P. K. Maini, A Lyaponov function and global properties for SIR and SEIR epedimiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57. [9] J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). [10] 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. doi: 10.1038/87953. [11] 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. [12] 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 difficile strains among long-term care facility residents,, Clin. Infect. Dis., 45 (2007), 992. doi: 10.1086/521854. [13] P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301. doi: 10.1126/science.1134933. [14] 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. [15] 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. [16] "The ABCs of Hepatitis,", Center for Disease Control and Prevention (CDC), 2009., Available from: \url{http://www.cdc.gov/hepatitis/Resources/Professionals/PDFs/ABCTable_BW.pdf}., 2009 (). [17] "Viral Hepatitis and Emerging Bloodborne Pathogens in Canada,", CCDR, 27S3,, Public Health Agency of Canada (PHAC), (2001). [18] WHO, "Fact Sheet on Hepatitis B," 2008., Available from: \url{http://www.who.int/mediacentre/factsheets/fs204/en/index.html}., 2008 ().
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