Mathematical analysis for an age-structured SIRS epidemic model

Kento Okuwa; Hisashi Inaba; Toshikazu Kuniya; Kento Okuwa; Hisashi Inaba; Toshikazu Kuniya

doi:10.3934/mbe.2019304

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 6071-6102. doi: 10.3934/mbe.2019304

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Mathematical analysis for an age-structured SIRS epidemic model

1.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914 Japan
2.
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501 Japan

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.
- SIRS epidemic,
- basic reproduction number,
- age structure,
- forward bifurcation,
- persistence,
- compact attractor
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

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Abstract

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.

References

[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.

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