The global stability of an SIRS model with infection age

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

The global stability of an SIRS model with infection age

Yuming Chen ^1, , Junyuan Yang ^1, and Fengqin Zhang ^1,

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

Received November 2012 Revised October 2013 Published January 2014

Abstract
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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.

Keywords: global stability, persistence., SIRS model, infection age.

Mathematics Subject Classification: Primary: 34D23; Secondary: 34G20, 35B35, 92D3.

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.

show all references

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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