# American Institute of Mathematical Sciences

2014, 11(3): 449-469. doi: 10.3934/mbe.2014.11.449

## The global stability of an SIRS model with infection age

 1 Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi, China, China, China

Received  November 2012 Revised  October 2013 Published  January 2014

Infection age is an important factor affecting the transmission of infectious diseases. In this paper, we consider an SIRS model with infection age, which is described by a mixed system of ordinary differential equations and partial differential equations. The expression of the basic reproduction number $\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then the model only has the disease-free equilibrium, while if $\mathscr{R}_0>1$ then besides the disease-free equilibrium the model also has an endemic equilibrium. Moreover, if $\mathscr{R}_0<1$ then the disease-free equilibrium is globally asymptotically stable otherwise it is unstable; if $\mathscr{R}_0>1$ then the endemic equilibrium is globally asymptotically stable under additional conditions. The local stability is established through linearization. The global stability of the disease-free equilibrium is shown by applying the fluctuation lemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
Citation: 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
##### References:
 [1] À. Calsina, J. M. Palmada and J. Ripoll, Optimal latent period in a bacteriophage population model structured by infection-age,, Math. Models Methods Appl. Sci., 21 (2011), 693. doi: 10.1142/S0218202511005180. [2] C. Castillo-Chavez et al., Epidemiological models with age structure, proportionate mixing, and cross-immunity,, J. Math. Bioi., 27 (1989), 233. doi: 10.1007/BF00275810. [3] B. Buonomo and S. Rionero, On the Lyapunov stability for SIRS epidemic models with generalized nonlinear incidence rate,, Appl. Math. Comput., 217 (2010), 4010. doi: 10.1016/j.amc.2010.10.007. [4] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, Nonlineaity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. [5] Z. Feng, M. Iannelli and F. A. Milner, A two-strain tuberculosis model with age of infection,, SIAM. J. Appl. Math., 62 (2002), 1634. doi: 10.1137/S003613990038205X. [6] D. F. Francis et al., Infection of chimpanzees with lymphadenopathy-associated virus,, Lancet, 2 (1984), 1276. [7] H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control,, Springer-Verlag, (1984). [8] W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733. doi: 10.1002/cpa.3160380607. [9] J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment,, J. Biol. Dyn., 1 (2007), 109. doi: 10.1080/17513750601040383. [10] H. Inaba and H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity,, Math. Biosci., 190 (2004), 39. doi: 10.1016/j.mbs.2004.02.004. [11] A. Lahrouz et al., Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination,, Appl. Math. Comput., 218 (2012), 6519. doi: 10.1016/j.amc.2011.12.024. [12] J. M. A. Lange et al., Persistent HIV antigenaemia and decline of HIV core antibodies associated with transition to AIDS,, British Medical J., 293 (1986), 1459. [13] J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with trasport-related infection,, Chaos Solitons Fractals, 40 (2009), 145. doi: 10.1016/j.chaos.2007.07.047. [14] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Z. Angew. Math. Phys., 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. [15] P. Magal, Compact attrators for time-periodic age-structured population models,, Electron. J. Differntial Equations, 2001 (2001). [16] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov fucntional and global asymptoticalc stability for an infection-age model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. [17] P. Magal and X.-Q. Zhao, Global attractors in uniformly persistent dynamical systems,, SIAM J. Mah. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. [18] M. Martcheva and S. S. Pilyugin, The role of coinfection in multidisease dynamics,, SIAM J. Appl. Math., 66 (2006), 843. doi: 10.1137/040619272. [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [20] C. Pedersen et al., Temporal relation of antigenaemia and loss of antibodies to core core antigens to development of clinical disease in HIV infection,, British Medical J., 295 (1987), 567. [21] S. Z. Salahuddin et al., HLTV-III in symptom-free seronegative persons,, Lancet, 2 (1984), 1418. [22] H. R. Thieme, Semiflows generated by Lipschitz pertrubations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. [23] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in Advances in Mathematical Population Dynamics-Molecules, (1997), 691. [24] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. [25] J.-Y. Yang, X.-Z. Li and M. Martcheva, Global stability of a DS-DI epidemic model with age of infection,, J. Math. Anal. Appl., 385 (2012), 655. doi: 10.1016/j.jmaa.2011.06.087. [26] J.-Y. Yang et al., Intrinsic transmission global dynamics of tuberculosis with age structure,, Int. J. Biomath., 4 (2011), 329. doi: 10.1142/S1793524511001222. [27] Z. Zhang and J. Peng, A SIRS epiemic model with infection-age dependence,, J. Math. Anal. Appl., 331 (2007), 1396. doi: 10.1016/j.jmaa.2006.09.061. [28] Y. Zhou et al., Modeling and prediction of HIV in China: Transmission rates structured by infection ages,, Math. Biosci. Eng., 5 (2008), 403. doi: 10.3934/mbe.2008.5.403.

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##### References:
 [1] À. Calsina, J. M. Palmada and J. Ripoll, Optimal latent period in a bacteriophage population model structured by infection-age,, Math. Models Methods Appl. Sci., 21 (2011), 693. doi: 10.1142/S0218202511005180. [2] C. Castillo-Chavez et al., Epidemiological models with age structure, proportionate mixing, and cross-immunity,, J. Math. Bioi., 27 (1989), 233. doi: 10.1007/BF00275810. [3] B. Buonomo and S. Rionero, On the Lyapunov stability for SIRS epidemic models with generalized nonlinear incidence rate,, Appl. Math. Comput., 217 (2010), 4010. doi: 10.1016/j.amc.2010.10.007. [4] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, Nonlineaity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. [5] Z. Feng, M. Iannelli and F. A. Milner, A two-strain tuberculosis model with age of infection,, SIAM. J. Appl. Math., 62 (2002), 1634. doi: 10.1137/S003613990038205X. [6] D. F. Francis et al., Infection of chimpanzees with lymphadenopathy-associated virus,, Lancet, 2 (1984), 1276. [7] H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control,, Springer-Verlag, (1984). [8] W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733. doi: 10.1002/cpa.3160380607. [9] J. M. Hyman and J. Li, Infection-age structured epidemic models with behavior change or treatment,, J. Biol. Dyn., 1 (2007), 109. doi: 10.1080/17513750601040383. [10] H. Inaba and H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity,, Math. Biosci., 190 (2004), 39. doi: 10.1016/j.mbs.2004.02.004. [11] A. Lahrouz et al., Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination,, Appl. Math. Comput., 218 (2012), 6519. doi: 10.1016/j.amc.2011.12.024. [12] J. M. A. Lange et al., Persistent HIV antigenaemia and decline of HIV core antibodies associated with transition to AIDS,, British Medical J., 293 (1986), 1459. [13] J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with trasport-related infection,, Chaos Solitons Fractals, 40 (2009), 145. doi: 10.1016/j.chaos.2007.07.047. [14] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Z. Angew. Math. Phys., 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. [15] P. Magal, Compact attrators for time-periodic age-structured population models,, Electron. J. Differntial Equations, 2001 (2001). [16] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov fucntional and global asymptoticalc stability for an infection-age model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. [17] P. Magal and X.-Q. Zhao, Global attractors in uniformly persistent dynamical systems,, SIAM J. Mah. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. [18] M. Martcheva and S. S. Pilyugin, The role of coinfection in multidisease dynamics,, SIAM J. Appl. Math., 66 (2006), 843. doi: 10.1137/040619272. [19] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [20] C. Pedersen et al., Temporal relation of antigenaemia and loss of antibodies to core core antigens to development of clinical disease in HIV infection,, British Medical J., 295 (1987), 567. [21] S. Z. Salahuddin et al., HLTV-III in symptom-free seronegative persons,, Lancet, 2 (1984), 1418. [22] H. R. Thieme, Semiflows generated by Lipschitz pertrubations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. [23] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in Advances in Mathematical Population Dynamics-Molecules, (1997), 691. [24] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. [25] J.-Y. Yang, X.-Z. Li and M. Martcheva, Global stability of a DS-DI epidemic model with age of infection,, J. Math. Anal. Appl., 385 (2012), 655. doi: 10.1016/j.jmaa.2011.06.087. [26] J.-Y. Yang et al., Intrinsic transmission global dynamics of tuberculosis with age structure,, Int. J. Biomath., 4 (2011), 329. doi: 10.1142/S1793524511001222. [27] Z. Zhang and J. Peng, A SIRS epiemic model with infection-age dependence,, J. Math. Anal. Appl., 331 (2007), 1396. doi: 10.1016/j.jmaa.2006.09.061. [28] Y. Zhou et al., Modeling and prediction of HIV in China: Transmission rates structured by infection ages,, Math. Biosci. Eng., 5 (2008), 403. doi: 10.3934/mbe.2008.5.403.
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