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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

 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.
Citation: Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete and 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, 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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