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

[1]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[2]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[3]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[4]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[5]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117

[6]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[7]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[8]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[9]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[10]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[11]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[13]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[14]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[15]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[16]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[17]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[18]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[19]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[20]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]