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. Google Scholar

[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. Google Scholar

[3]

H. Guo, Global dynamics of a mathematical model of tuberculosis,, Canadian Appl. Math. Quart., 13 (2005), 313. Google Scholar

[4]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513. Google Scholar

[5]

J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89. Google Scholar

[6]

D. Kalajdzievska, "Modeling the Effects of Carriers on the Transmission Dynamics of Infectious Diseases,", M.Sc. thesis, (2006). Google Scholar

[7]

J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707. Google Scholar

[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. Google Scholar

[9]

J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301. doi: 10.1126/science.1134933. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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 (). Google Scholar

[17]

"Viral Hepatitis and Emerging Bloodborne Pathogens in Canada,", CCDR, 27S3,, Public Health Agency of Canada (PHAC), (2001). Google Scholar

[18]

WHO, "Fact Sheet on Hepatitis B," 2008., Available from: \url{http://www.who.int/mediacentre/factsheets/fs204/en/index.html}., 2008 (). Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

H. Guo, Global dynamics of a mathematical model of tuberculosis,, Canadian Appl. Math. Quart., 13 (2005), 313. Google Scholar

[4]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513. Google Scholar

[5]

J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Eng., 3 (2006), 89. Google Scholar

[6]

D. Kalajdzievska, "Modeling the Effects of Carriers on the Transmission Dynamics of Infectious Diseases,", M.Sc. thesis, (2006). Google Scholar

[7]

J. T. Kemper, The effects of asymptotic attacks on the spread of infectious disease: A deterministic model,, Bull. Math. Bio., 40 (1978), 707. Google Scholar

[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. Google Scholar

[9]

J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

P. Roumagnac, et al., Evolutionary history of Salmonella typhi,, Science, 314 (2006), 1301. doi: 10.1126/science.1134933. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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 (). Google Scholar

[17]

"Viral Hepatitis and Emerging Bloodborne Pathogens in Canada,", CCDR, 27S3,, Public Health Agency of Canada (PHAC), (2001). Google Scholar

[18]

WHO, "Fact Sheet on Hepatitis B," 2008., Available from: \url{http://www.who.int/mediacentre/factsheets/fs204/en/index.html}., 2008 (). Google Scholar

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