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Bulletin of Mathematical Biology

, Volume 80, Issue 8, pp 2049–2087 | Cite as

Transmission Dynamics of an SIS Model with Age Structure on Heterogeneous Networks

  • Shanshan Chen
  • Michael Small
  • Yizhou Tao
  • Xinchu Fu
Original Article

Abstract

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).

Keywords

Basic reproduction number Infection age Scale-free network Global stability 

Notes

Acknowledgements

This work was jointly supported by the NSFC Grants 11572181 and 11331009. SC was also supported with funding from ARC Linkage Grant LP130101055. Part of this work was done while SC, YT and XF visited the Center for Mathematical Sciences at Huazhong University of Science and Technology, Wuhan, China.

References

  1. Adams RA, Fournier JJF (2003) Sobolev spaces, 2nd edn. Elsevier, AmsterdamzbMATHGoogle Scholar
  2. Anderson RM, May RM (1991) Infectious diseases of humans. Oxford University Press, OxfordGoogle Scholar
  3. Bailey NTJ (1975) The mathematical theory of infectious diseases and its applications, 2nd edn. Hafner, New YorkzbMATHGoogle Scholar
  4. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512MathSciNetCrossRefzbMATHGoogle Scholar
  5. Browne CJ, Pilyugin SS (2013) Global analysis of age-structured within-host virus model. Discrete Contin Dyn Syst: Ser B 18(8):1999–2017MathSciNetCrossRefzbMATHGoogle Scholar
  6. Colizzaa V, Barrat A, Barthélemy M, Vespignani A (2006) The modeling of global epidemics: stochastic dynamics and predictability. Bull Math Biol 68:1893–1921MathSciNetCrossRefzbMATHGoogle Scholar
  7. Durrett R, Jung P (2007) Two phase transitions for the contact process on small worlds. Stoch Proc Their Appl 117:1910–1927MathSciNetCrossRefzbMATHGoogle Scholar
  8. Elveback L et al (1971) Stochastic two-agent epidemic simulation models for a 379 community of families. Am J Epidemiol 93:267–280CrossRefGoogle Scholar
  9. Fu XC, Small M, Walker DM, Zhang HF (2008) Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization. Phys Rev E 77:036113MathSciNetCrossRefGoogle Scholar
  10. Hale JK (1971) Functional differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  11. Hale JK (1988) Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol 25. American Mathematical Society, ProvidenceGoogle Scholar
  12. Hale JK, Verduyn LSM (1993) Introduction to functional differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  13. Hale JK, Waltman P (1989) Persistence in infinite-dimensional systems. SIAM J Math Anal 20(2):388–395MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hoppensteadt F (1974) An age-dependent epidemic model. J Franklin Inst 297:325–338CrossRefzbMATHGoogle Scholar
  15. Hoppensteadt F (1975) Mathematical theories of populations: demographics, genetics and epidemics. SIAM Publications, PhiladelphiaCrossRefzbMATHGoogle Scholar
  16. House T, Davies G, Danon L, Keeling MJ (2009) A motif-based approach to network epidemics. Bull Math Biol 71:1693–1706MathSciNetCrossRefzbMATHGoogle Scholar
  17. Keeling MJ, Eames KTD (2005) Networks and epidemic models. J R Soc Interface 2:295–307CrossRefGoogle Scholar
  18. Kermack WO, McKendrick AG (1927) Contributions to the mathematical theory of epidemics. Proc R Soc A 115:700–721CrossRefzbMATHGoogle Scholar
  19. Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics. Proc R Soc A 138:55–83CrossRefzbMATHGoogle Scholar
  20. Levin SA, Durrett R (1996) From individuals to epidemics. Philos Trans R Soc Lond B 351:1615–1621CrossRefGoogle Scholar
  21. Li MY, Shuai ZS (2010) Global-stability problem for coupled systems of differential equations on networks. J Differ Equ 248:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  22. Liu J (2011) Threshold dynamics for a HFMD epidemic model with periodic transmission rate. Nonlinear Dyn 64(1):89–95MathSciNetCrossRefzbMATHGoogle Scholar
  23. Liu J, Tang Y, Yang Z (2004) The spread of disease with birth and death on networks. J Stat Mech 2004(08):P08008CrossRefzbMATHGoogle Scholar
  24. Liu LL, Wang JL, Liu XN (2015) Global stability of an SEIR epidemic model with age-dependent latency and relapse. Nonlinear Anal RWA 24:18–35MathSciNetCrossRefzbMATHGoogle Scholar
  25. Magal P, McCluskey C (2013) Two group infection age model including an application to noscomial infection. SIAM J Appl Math 73(2):1058–1095MathSciNetCrossRefzbMATHGoogle Scholar
  26. Magal P, Zhao XQ (2005) Global attractors and steady states for uniformly persistent dynamical systems. SIAM J Math Anal 37(1):251–275MathSciNetCrossRefzbMATHGoogle Scholar
  27. Magal P, McCluskey CC, Webb GF (2010) Lyapunov functional and global asymptotic stability for an infection-age model. Appl Anal 89:1109–1140MathSciNetCrossRefzbMATHGoogle Scholar
  28. McCluskey CC (2012) Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Math Biosci Eng 9(4):819–841MathSciNetCrossRefzbMATHGoogle Scholar
  29. Olinky R, Stone L (2004) Unexpected epidemic threshold in heterogeneous networks: the role of disease transmission. Phys Rev E 70:030902CrossRefGoogle Scholar
  30. Pastor-Satorras R, Vespignani A (2001a) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200–3203CrossRefGoogle Scholar
  31. Pastor-Satorras R, Vespignani A (2001b) Epidemic dynamics and endemic states in complex networks. Phys Rev E 63:066117CrossRefGoogle Scholar
  32. Pastor-Satorras R, Vespignani A (2002) Epidemic dynamics in finite size scale-free networks. Phys Rev E 65:035108CrossRefGoogle Scholar
  33. Read JM, Keeling MJ (2003) Disease evolution on networks: the role of contact structure. Proc R Soc B 270:699–708CrossRefGoogle Scholar
  34. Samanta GP (2014) Analysis of a delayed hand-foot-mouth disease epidemic model with pulse vaccination. Syst Sci Control Eng 2(1):61–73CrossRefGoogle Scholar
  35. Shuai Z, Driessche PVD (2013) Global stability of infectious disease models using Lyapunov functions. SIAM J Appl Math 73(4):1513–1532MathSciNetCrossRefzbMATHGoogle Scholar
  36. Smith HL, Thieme HR (2011) Dynamical systems and population persistence. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  37. Takahashi S, Liao Q, Van Boeckel TP et al (2016) Hand, foot, and mouth disease in China: modeling epidemic dynamics of enterovirus serotypes and implications for vaccination [J]. PLoS Med 13(2):e1001958CrossRefGoogle Scholar
  38. Tarkhanov N (2008) Lyapunov stability for an age-structured population model. Ecol Model 216:232–239CrossRefGoogle Scholar
  39. Thieme HR (1990) Semiflows generated by Lipschitz perturbations of non-densely defined operators. Differ Integral Equ 3:1035–1066MathSciNetzbMATHGoogle Scholar
  40. Wang L, Dai G (2008) Global stability of virus spreading in complex heterogeneous networks. SIAM J Appl Math 68:1495–1502MathSciNetCrossRefzbMATHGoogle Scholar
  41. Wang Y, Feng ZJ, Yang Y, Self S, Gao YJ, Longini Ira M, Wakefield J, Zhang Z, Wang LP, Chen X, Yao LN, Stanaway JD, Wang ZJ, Yang WZ (2011) Hand, foot and mouth disease in China: patterns of spread and transmissibility during 2008–2009. Epidemiology 22(6):781–792.  https://doi.org/10.1097/EDE.0b013e318231d67a CrossRefGoogle Scholar
  42. Wang J, Zhang R, Kuniya T (2015) Global dynamics for a class of age-infection hiv models with nonlinear infection rate. J Math Anal Appl 432(1):289–313MathSciNetCrossRefzbMATHGoogle Scholar
  43. Wang LW, Liu ZJ, Zhang XG (2016) Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination. Nonlinear Anal: Real World Appl 32:136–158MathSciNetCrossRefzbMATHGoogle Scholar
  44. Webb GF (1985) Theory of nonlinear age-dependent population dynamics. Marcel Dekker, New YorkzbMATHGoogle Scholar
  45. Yang J, Qiu Z, Li XZ (2014) Global stability of an age-structured cholera model. Math Biosci Eng 11(3):641MathSciNetCrossRefzbMATHGoogle Scholar
  46. Yang JY, Chen YM, Xu F (2016) Effect of infection age on an SIS epidemic model on complex networks. J Math Biol 73:1227–1249MathSciNetCrossRefzbMATHGoogle Scholar
  47. Zhang HF, Fu XC (2009) Spreading of epidemics on scale-free networks with nonlinear infectivity. Nonlinear Anal TMA 70:3273–3278MathSciNetCrossRefzbMATHGoogle Scholar
  48. Zhou T, Liu J, Bai W, Chen GR, Wang B (2006) Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Phys Rev E 74:056109CrossRefGoogle Scholar
  49. Zhu GH, Fu XC, Chen GR (2012) Global attractivity of a network-based epidemic SIS model with nonlinear infectivity. Commun Nonlinear Sci Numer Simul 17:2588–2594MathSciNetCrossRefzbMATHGoogle Scholar
  50. Zhu GH, Chen GR, Xu XJ, Fu XC (2013) Epidemic spreading on contact networks with adaptive weights. J Theor Biol 317:133–139MathSciNetCrossRefzbMATHGoogle Scholar
  51. Zhu YT, Xu BY, Lian XZ, Lin W, Zhou ZM, Wang WM (2014) A hand-foot-and-mouth disease model with periodic transmission rate in Wenzhou, China. Hindawi Publishing corporation abstract and applied analysis. Article ID 234509, vol 2014, p 11.  https://doi.org/10.1155/2014/234509

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.School of Mathematics and StatisticsUniversity of Western AustraliaCrawleyAustralia
  3. 3.Mineral ResourcesCSIROKensingtonAustralia

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