American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Evolutionary branching patterns in predator-prey structured populations
  • DCDS-B Home
  • This Issue
  • Next Article
    A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver
November  2013, 18(9): 2239-2265. doi: 10.3934/dcdsb.2013.18.2239

Epidemic models with age of infection, indirect transmission and incomplete treatment

Liming Cai 1, , Maia Martcheva 2, and  Xue-Zhi Li 1,

1. 

Department of Mathematics, Xinyang Normal University, Xinyang 464000, China, China
2. 

Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, United States

Received  July 2012 Revised  July 2013 Published  September 2013

An infection-age-structured epidemic model with environmental bacterial infection is investigated in this paper. It is assumed that the infective population is structured according to age of infection, and the infectivity of the treated individuals is reduced but varies with the infection-age. An explicit formula for the reproductive number $ \Re_0$ of the model is obtained. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$. It is also shown that if the reproduction number $\Re_0>1$, then the system has a unique endemic equilibrium which is locally asymptotically stable. Furthermore, if the reproduction number $\Re_0>1$, the system is permanent. When the treatment rate and the transmission rate are both independent of infection age, the system of partial differential equations (PDEs) reduces to a system of ordinary differential equations (ODEs). In this special case, it is shown that the global dynamics of the system can be determined by the basic reproductive number.
Keywords: Lyapunov function, infection-age-structured., global stability, Epidemic model.
Mathematics Subject Classification: Primary: 92D30; Secondary: 92D2.
Citation: Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239
References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1991).   Google Scholar

[2]

J. A. Crump, S. P. Luby and E. D. Mintz, The global burden of typhoid fever,, Bull. World Health Organ., 82 (2004), 346.   Google Scholar

[3]

O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, Journal of Mathematical Biology, 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[4]

J. Z. Farkas and T. C. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback,, Communications on Pure and Applied Analysis (CPAA), 8 (2009), 1825.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[5]

M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population,, Nonlinear Analysis: Real World Applications, 7 (2006), 341.  doi: 10.1016/j.nonrwa.2005.03.005.  Google Scholar

[6]

J. Gonzlez-Guzmn, An epidemiological model for direct and indirect transmission of Typhoid fever,, Mathematical Biosciences, 96 (1989), 33.  doi: 10.1016/0025-5564(89)90081-3.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[9]

D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, Plos Med., 3 (2006), 63.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[10]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Math., 9 (2005), 151.   Google Scholar

[11]

, , ().   Google Scholar

[12]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection Model,, SIAM Journal on Applied Mathematics, 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[13]

J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment,, Journal of Biological Dynamics, 1 (2007), 109.  doi: 10.1080/17513750601040383.  Google Scholar

[14]

M. Iannelli, M. Martcheva and X. Z. Li, Strain replacement in an epidemic model with super-infection and perfect vaccination,, Mathematical Biosciences, 195 (2005), 23.  doi: 10.1016/j.mbs.2005.01.004.  Google Scholar

[15]

R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold,, Bulletin of Mathematical Biology, 71 (2009), 845.  doi: 10.1007/s11538-008-9384-4.  Google Scholar

[16]

J. Li, L. Wang, H. Zhao and Z. Ma, Dynamical behavior of an epidemic model with coinfection of two diseases,, Rocky Mountain Journal of Mathematics, 38 (2008), 1457.  doi: 10.1216/RMJ-2008-38-5-1457.  Google Scholar

[17]

J. Li, Y. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination,, Nonlinear Analysis: Real World Applications, 12 (2011), 2163.  doi: 10.1016/j.nonrwa.2010.12.030.  Google Scholar

[18]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Models and Dynamics of Infectious Diseases,", China Sciences Press, (2004).   Google Scholar

[19]

M. Martcheva and S. Pilyugin, The role of coinfection in multidisease dynamics,, SIAM Journal on Applied Mathematics, 66 (2006), 843.  doi: 10.1137/040619272.  Google Scholar

[20]

M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection,, Journal of Mathematical Biology, 46 (2003), 385.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

[21]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf Bifurcation in age structured models,, Memoirs of the American Mathematical Society, 202 (2009).  doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[22]

Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment,, Journal of Dynamics and Differential Equations, 22 (2010), 823.  doi: 10.1007/s10884-010-9178-x.  Google Scholar

[23]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731.  doi: 10.1137/060663945.  Google Scholar

[24]

R. P. Sanches, C. P. Ferreira and R. A. Kraenkel, The role of immunity and seasonality in cholera epidemics,, Bulletin of Mathematical Biology, 73 (2011), 2916.  doi: 10.1007/s11538-011-9652-6.  Google Scholar

[25]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Mathematical Biosciences, 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).   Google Scholar

[28]

K. Yosida, "Functional Analysis,", second edition, (1968).   Google Scholar

[29]

L. Zou, S. Ruan and W. Zhang, An age-structured model for the transmission dynamics of Hepatitis B,, SIAM Journal on Applied Mathematics, 70 (2010), 3121.  doi: 10.1137/090777645.  Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1991).   Google Scholar

[2]

J. A. Crump, S. P. Luby and E. D. Mintz, The global burden of typhoid fever,, Bull. World Health Organ., 82 (2004), 346.   Google Scholar

[3]

O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, Journal of Mathematical Biology, 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[4]

J. Z. Farkas and T. C. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback,, Communications on Pure and Applied Analysis (CPAA), 8 (2009), 1825.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[5]

M. Ghosh, P. Chandra, P. Sinha and J. B. Shukla, Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population,, Nonlinear Analysis: Real World Applications, 7 (2006), 341.  doi: 10.1016/j.nonrwa.2005.03.005.  Google Scholar

[6]

J. Gonzlez-Guzmn, An epidemiological model for direct and indirect transmission of Typhoid fever,, Mathematical Biosciences, 96 (1989), 33.  doi: 10.1016/0025-5564(89)90081-3.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).   Google Scholar

[8]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Springer-Verlag, (1993).   Google Scholar

[9]

D. M. Hartley, J. G. Morris and D. L. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, Plos Med., 3 (2006), 63.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[10]

S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese Journal of Math., 9 (2005), 151.   Google Scholar

[11]

, , ().   Google Scholar

[12]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection Model,, SIAM Journal on Applied Mathematics, 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[13]

J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment,, Journal of Biological Dynamics, 1 (2007), 109.  doi: 10.1080/17513750601040383.  Google Scholar

[14]

M. Iannelli, M. Martcheva and X. Z. Li, Strain replacement in an epidemic model with super-infection and perfect vaccination,, Mathematical Biosciences, 195 (2005), 23.  doi: 10.1016/j.mbs.2005.01.004.  Google Scholar

[15]

R. I. Joh, H. Wang, H. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold,, Bulletin of Mathematical Biology, 71 (2009), 845.  doi: 10.1007/s11538-008-9384-4.  Google Scholar

[16]

J. Li, L. Wang, H. Zhao and Z. Ma, Dynamical behavior of an epidemic model with coinfection of two diseases,, Rocky Mountain Journal of Mathematics, 38 (2008), 1457.  doi: 10.1216/RMJ-2008-38-5-1457.  Google Scholar

[17]

J. Li, Y. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination,, Nonlinear Analysis: Real World Applications, 12 (2011), 2163.  doi: 10.1016/j.nonrwa.2010.12.030.  Google Scholar

[18]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Models and Dynamics of Infectious Diseases,", China Sciences Press, (2004).   Google Scholar

[19]

M. Martcheva and S. Pilyugin, The role of coinfection in multidisease dynamics,, SIAM Journal on Applied Mathematics, 66 (2006), 843.  doi: 10.1137/040619272.  Google Scholar

[20]

M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection,, Journal of Mathematical Biology, 46 (2003), 385.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

[21]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf Bifurcation in age structured models,, Memoirs of the American Mathematical Society, 202 (2009).  doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[22]

Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment,, Journal of Dynamics and Differential Equations, 22 (2010), 823.  doi: 10.1007/s10884-010-9178-x.  Google Scholar

[23]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731.  doi: 10.1137/060663945.  Google Scholar

[24]

R. P. Sanches, C. P. Ferreira and R. A. Kraenkel, The role of immunity and seasonality in cholera epidemics,, Bulletin of Mathematical Biology, 73 (2011), 2916.  doi: 10.1007/s11538-011-9652-6.  Google Scholar

[25]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Mathematical Biosciences, 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,", Marcel Dekker, (1985).   Google Scholar

[28]

K. Yosida, "Functional Analysis,", second edition, (1968).   Google Scholar

[29]

L. Zou, S. Ruan and W. Zhang, An age-structured model for the transmission dynamics of Hepatitis B,, SIAM Journal on Applied Mathematics, 70 (2010), 3121.  doi: 10.1137/090777645.  Google Scholar

[1]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[2]

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
[3]

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
[4]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[5]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060
[6]

Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva. An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences & Engineering, 2010, 7 (1) : 123-147. doi: 10.3934/mbe.2010.7.123
[7]

Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048
[8]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
[9]

Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186
[10]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[11]

Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022
[12]

Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23
[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]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019226
[15]

Mamadou L. Diagne, Ousmane Seydi, Aissata A. B. Sy. A two-group age of infection epidemic model with periodic behavioral changes. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019202
[16]

Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002
[17]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69
[18]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[19]

Jean-Baptiste Burie, Arnaud Ducrot, Abdoul Aziz Mbengue. Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2879-2905. doi: 10.3934/dcdsb.2017155
[20]

Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences