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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

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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

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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

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