
Mathematical Biosciences and Engineering, 2019, 16(5): 60716102. doi: 10.3934/mbe.2019304.
Research article Special Issues
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Mathematical analysis for an agestructured SIRS epidemic model
1 Graduate School of Mathematical Sciences, The University of Tokyo, 381 Komaba Meguroku Tokyo 1538914 Japan
2 Graduate School of System Informatics, Kobe University, 11 Rokkodaicho, Nadaku, Kobe 6578501 Japan
Received: , Accepted: , Published:
Special Issues: Mathematical Modeling to Solve the Problems in Life Sciences
Keywords: SIRS epidemic; basic reproduction number; age structure; forward bifurcation; persistence; compact attractor
Citation: Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya. Mathematical analysis for an agestructured SIRS epidemic model. Mathematical Biosciences and Engineering, 2019, 16(5): 60716102. doi: 10.3934/mbe.2019304
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, CastilloChaves, 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, AgeStructured 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 "susceptibleinfectiveimmunesusceptible" (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 agedependent epidemic model with application to measles, Math. Biosci., 73(1985), 131–147.
 18. H. Inaba, Threshold and stability results for an agestructured 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 subclinical infection in an epidemic model: implications for pertussis in the Netherlands, Math. Biosci., 164(2000), 161–182.
 21. S. Tsutsui, Mathematical analysis for an agestructured epidemic model with waning immunity and subclinical infection, Master Thesis, Graduate School of Mathematical Sciences, The University 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 agestructured SIS 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 R_{0} 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 Infectious 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 R_{0} in continuoustime 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 agestructured SIR epidemic model with vertical transmission, Disc. Conti. Dyn. Sys., Series B, 6(2006), 69–96.
 33. H. J. Heijmans, The dynamical behaviour of the agesizedistribution of a cell population, The Dynamics of Physiologically Structured Populations, Springer, Berlin, Heidelberg, (1986), 185–202.
 34. K. J. Engel and R. Nagel, OneParameter 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 AgeStructured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995.
 37. M. Iannelli and F. Milner, The Basic Approach to AgeStructured 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 suboptimal 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), 111113.
Reader Comments
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *