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Epidemic models with age of infection, indirect transmission and incomplete treatment
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 |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, London, 1991. |
[2] |
J. A. Crump, S. P. Luby and E. D. Mintz, The global burden of typhoid fever, Bull. World Health Organ., 82 (2004), 346-353. |
[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), 365-382.
doi: 10.1007/BF00178324. |
[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), 1825-1839.
doi: 10.3934/cpaa.2009.8.1825. |
[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), 341-363.
doi: 10.1016/j.nonrwa.2005.03.005. |
[6] |
J. Gonzlez-Guzmn, An epidemiological model for direct and indirect transmission of Typhoid fever, Mathematical Biosciences, 96 (1989), 33-46.
doi: 10.1016/0025-5564(89)90081-3. |
[7] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, 1988. |
[8] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Springer-Verlag, Berlin, 1993. |
[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), 63-69.
doi: 10.1371/journal.pmed.0030007. |
[10] |
S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese Journal of Math., 9 (2005), 151-173. |
[11] |
, , ().
|
[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), 25-38.
doi: 10.1137/110826588. |
[13] |
J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment, Journal of Biological Dynamics, 1 (2007), 109-131.
doi: 10.1080/17513750601040383. |
[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), 23-46.
doi: 10.1016/j.mbs.2005.01.004. |
[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), 845-862.
doi: 10.1007/s11538-008-9384-4. |
[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), 1457-1479.
doi: 10.1216/RMJ-2008-38-5-1457. |
[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), 2163-2173.
doi: 10.1016/j.nonrwa.2010.12.030. |
[18] |
Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Models and Dynamics of Infectious Diseases," China Sciences Press, Beijing, 2004. |
[19] |
M. Martcheva and S. Pilyugin, The role of coinfection in multidisease dynamics, SIAM Journal on Applied Mathematics, 66 (2006), 843-872.
doi: 10.1137/040619272. |
[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), 385-424.
doi: 10.1007/s00285-002-0181-7. |
[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), 951.
doi: 10.1090/S0065-9266-09-00568-7. |
[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), 823-851.
doi: 10.1007/s10884-010-9178-x. |
[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), 731-756.
doi: 10.1137/060663945. |
[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), 2916-2931.
doi: 10.1007/s11538-011-9652-6. |
[25] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Mathematical Biosciences, 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[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), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[27] |
G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. |
[28] |
K. Yosida, "Functional Analysis," second edition, Berlin-Heidelberg, New York, Springer-Verlag, 1968. |
[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), 3121-3139.
doi: 10.1137/090777645. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, London, 1991. |
[2] |
J. A. Crump, S. P. Luby and E. D. Mintz, The global burden of typhoid fever, Bull. World Health Organ., 82 (2004), 346-353. |
[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), 365-382.
doi: 10.1007/BF00178324. |
[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), 1825-1839.
doi: 10.3934/cpaa.2009.8.1825. |
[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), 341-363.
doi: 10.1016/j.nonrwa.2005.03.005. |
[6] |
J. Gonzlez-Guzmn, An epidemiological model for direct and indirect transmission of Typhoid fever, Mathematical Biosciences, 96 (1989), 33-46.
doi: 10.1016/0025-5564(89)90081-3. |
[7] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," AMS, Providence, 1988. |
[8] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Springer-Verlag, Berlin, 1993. |
[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), 63-69.
doi: 10.1371/journal.pmed.0030007. |
[10] |
S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese Journal of Math., 9 (2005), 151-173. |
[11] |
, , ().
|
[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), 25-38.
doi: 10.1137/110826588. |
[13] |
J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment, Journal of Biological Dynamics, 1 (2007), 109-131.
doi: 10.1080/17513750601040383. |
[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), 23-46.
doi: 10.1016/j.mbs.2005.01.004. |
[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), 845-862.
doi: 10.1007/s11538-008-9384-4. |
[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), 1457-1479.
doi: 10.1216/RMJ-2008-38-5-1457. |
[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), 2163-2173.
doi: 10.1016/j.nonrwa.2010.12.030. |
[18] |
Z. Ma, Y. Zhou, W. Wang and Z. Jin, "Mathematical Models and Dynamics of Infectious Diseases," China Sciences Press, Beijing, 2004. |
[19] |
M. Martcheva and S. Pilyugin, The role of coinfection in multidisease dynamics, SIAM Journal on Applied Mathematics, 66 (2006), 843-872.
doi: 10.1137/040619272. |
[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), 385-424.
doi: 10.1007/s00285-002-0181-7. |
[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), 951.
doi: 10.1090/S0065-9266-09-00568-7. |
[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), 823-851.
doi: 10.1007/s10884-010-9178-x. |
[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), 731-756.
doi: 10.1137/060663945. |
[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), 2916-2931.
doi: 10.1007/s11538-011-9652-6. |
[25] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Mathematical Biosciences, 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[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), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[27] |
G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. |
[28] |
K. Yosida, "Functional Analysis," second edition, Berlin-Heidelberg, New York, Springer-Verlag, 1968. |
[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), 3121-3139.
doi: 10.1137/090777645. |
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