### Mathematical Biosciences and Engineering

2019, Issue 5: 6071-6102. doi: 10.3934/mbe.2019304
Research article Special Issues

# Mathematical analysis for an age-structured SIRS epidemic model

• Received: 27 March 2019 Accepted: 19 June 2019 Published: 01 July 2019
• In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number $R_0$ to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on $R_0$ and the critical coverage of immunization based on the reinfection threshold.

Citation: Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya. Mathematical analysis for an age-structured SIRS epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 6071-6102. doi: 10.3934/mbe.2019304

### Related Papers:

• In this paper, we investigate an SIRS epidemic model with chronological age structure in a demographic steady state. Although the age-structured SIRS model is a simple extension of the well-known age-structured SIR epidemic model, we have to develop new technique to deal with problems due to the reversion of susceptibility for recovered individuals. First we give a standard proof for the well-posedness of the normalized age-structured SIRS model. Next we examine existence of endemic steady states by fixed point arguments and bifurcation method, where we introduce the next generation operator and the basic reproduction number $R_0$ to formulate endemic threshold results. Thirdly we investigate stability of steady states by the bifurcation calculation and the comparison method, and we show existence of a compact attractor and discuss the global behavior based on the population persistence theory. Finally we give some numerical examples and discuss the effect of mass-vaccination on $R_0$ and the critical coverage of immunization based on the reinfection threshold.

 [1] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics II. The problem of endemicity, Proc. Roy. Soc., 138A(1932), 55–83; reprinted in Bull. Math. Biol. 53(1991), 57–87. [2] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics III. Further studies of the problem of endemicity, Proc. Roy. Soc., 141A, 94–122; reprinted in Bull. Math. Biol., 53(1991), 89–118. [3] H. Inaba, Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases, Jap. J. Indust. Appl. Math., 18(2001), 273–292. [4] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Castillo-Chaves, C. et al. (eds.), The IMA Volumes in Mathematics and its Applications 126, Springer, (2002), 337–359. [5] H. R. Thieme and J. Yang, An endemic model with variable re–infection rate and applications to influenza, Math. Biosci., 180(2002), 207–235. [6] H. Inaba, Endemic threshold analysis for the Kermack–McKendrick reinfection model, Josai Math. Monograph., 9(2016), 105–133. [7] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. [8] H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28(1976), 335–356. [9] J. L. Aron, Dynamics of acquired immunity boosted by exposure to infection, Math. Biosci.,64(1983), 249–259. [10] J. L. Aron, Acquired immunity dependent upon exposure in an SIRS epidemic model, Math. Biosci., 88(1988a), 37–47. [11] J. L. Aron, Mathematical modelling of immunity of malaria, Math. Biosci., 90(1988b), 385–396. [12] P. K. Tapaswi and J. Chattopadhyay, Global stability results of a "susceptible-infective-immune-susceptible" (SIRS) epidemic model, Ecol. Modell., 87(1996), 223–226. [13] S. Busenberg and K. P. Hadeler, Demography and epidemics, Math. Biosci.. 101(1990), 63–74. [14] S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28, 257–270. [15] Y. Nakata, Y. Enatsu, H. Inaba, et al., Stability of epidemic models with waning immunity, SUT J. Math., 50(2014), 205–245. [16] D. Breda, O. Diekmann, W. F. de Graaf, et al., On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6(2012), S103–S117. [17] D. W. Tudor, An age-dependent epidemic model with application to measles, Math. Biosci., 73(1985), 131–147. [18] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28(1990), 411–434. [19] M. C. M. De Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size ?, Epidemic Models: Their Structure and Relation to Data, D. Mollison (ed.), Cambridge U. P., Cambridge, (1995), 84–94. [20] M. van Boven, H. E. de Melker, J. F. P. Schellekens, et al., Waning immunity and sub-clinical infection in an epidemic model: implications for pertussis in the Netherlands, Math. Biosci., 164(2000), 161–182. [21] S. Tsutsui, Mathematical analysis for an age-structured epidemic model with waning immunity and subclinical infection, Master Thesis, Graduate School of Mathematical Sciences, The Univer-sity of Tokyo, 2010. [22] H. L. Smith and H. R. Thieme Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118, Amer. Math. Soc. Providence, Rhode Island, 2011. [23] S. Busenberg, M. Iannelli, and H. Thieme, Global behaviour of an age-structured S-I-S epidemic model, SIAM J. Math. Anal., 22(1991), 1065–1080. [24] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28(1990), 365–382. [25] O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infec-tious Disease Dynamics, Princeton University Press, Princeton and Oxford, 2013. [26] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65(2012), 309–348. [27] C. Barril, À. Calsina and J. Ripoll, A practical approach to R0 in continuous-time ecological models, Math. Meth. Appl. Sci., 41(2017), 8432–8445. [28] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [29] N. Dunford and J. T. Schwartz Linear Operators Part I: General Theory, New York: Interscience publishers, 1958. [30] I. Sawashima, On spectral properties of some positive operators, Nat. Sci. Rep. Ochanomizu Univ., 15(1964), 53–64. [31] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19(1970), 607–628. [32] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmis-sion, Disc. Conti. Dyn. Sys., Series B, 6(2006), 69–96. [33] H. J. Heijmans, The dynamical behaviour of the age-size-distribution of a cell population, The Dynamics of Physiologically Structured Populations, Springer, Berlin, Heidelberg, (1986), 185–202. [34] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer Science & Business Media, 1999. [35] T. Kato, Perturbation Theory for Linear Operators, 2nd Edition, Springer, Berlin, 1984. [36] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995. [37] M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics, Springer, The Netherlands, 2017. [38] H. R. Thieme, Disease extinction and disease persistence in age structured epidemic models, Nonl. Anal., 47(2001), 6181–6194. [39] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton and Oxford, 2003. [40] M. G. Gomes, L. J. White and G. F. Medley, Infection, reinfection, and vaccination under subop-timal immune protection: epidemiological perspectives, J. Theor. Biol., 228(2004), 539–549. [41] M. G. Gomes, L. J. White and G. F. Medley, The reinfection threshold, J. Theor. Biol., 236(2005), 111-113.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3

Article outline

Figures(4)

• On This Site