Abstract
Age-structured epidemic models are based on age-structured population models. Typically the linear age-structured population model, called the Lotka-McKendrick model, is used as a baseline population model in epidemic systems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.M. Anderson, R.M. May, Population biology of infectious disease: part I. Nature 280, 361–367 (1979)
R. Anderson, R. May, Age-related changes in the rate of disease transmission: implication for the design of vaccination programmes. Camb. J. Hyg. 94, 365–436 (1985)
N. Bacaër, A Short History of Mathematica Population Dynamics (2011). https://doi.org/10.1007/978-0-85729-115-8
S. Busenberg, M. Iannelli, H. Thieme, Global behavior of an age-structured epidemic model. SIAM J. Math. Anal. 22, 1065–1080 (1991)
S. Busenberg, M. Iannelli, H. Thieme, Dynamics of an age structured epidemic model, in Dynamical systems (Tianjin, 1990/1991). Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 4 (World Scientific Publishing, River Edge, NJ, 1993), pp. 1–19
C. Castillo-Chavez, H.W. Hethcote, V. Andreasen, S.A. Levin, W.M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27, 233–258 (1989)
Y. Cha, M. Iannelli, F.A. Milner, Are multiple endemic equilibria possible? in Advances in Mathematical Population Dynamics—Molecules, Cells and Man (Houston, TX, 1995). Series in Mathematical Biology and Medicine, vol. 6 (World Scientific Publishing, River Edge, NJ, 1997), pp. 779–788
Y. Cha, M. Iannelli, F.A. Milner, Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model. Math. Biosci. 150, 177–190 (1998)
M.S. Cohen, Preventing sexual transmission of HIV. Clin. Infect. Dis. 45, S287–S292 (2007)
J.M. Cushing, An Introduction to Structured Population Dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 71 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998)
W. Feller, On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267 (1941)
A. Franceschetti, A. Pugliese, D. Breda, Multiple endemic states in age-structured sir epidemic models. Math. Biosci. Eng. 9, 577–599 (2012)
D. Greenhalgh, Analytical threshold and stability results on age-structured epidemic models with vaccination. Theor. Popul. Biol. 33, 266–290 (1988)
M.E. Gurtin, R.C. MacCamy, Non-linear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281–300 (1974)
M.E. Gurtin, R.C. MacCamy, Some simple models for nonlinear age-dependent population dynamics. Math. Biosci. 43, 199–211 (1979)
M.E. Gurtin, R.C. MacCamy, Product solutions and asymptotic behavior for age-dependent, dispersing populations. Math. Biosci. 62, 157–167 (1982)
M. Gyllenberg, Mathematical aspects of physiologically structured populations: the contributions of J. A. J. Metz. J. Biol. Dyn. 1, 3–44 (2007)
M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics (Giardini, Pisa, 1995)
M. Iannelli, M. Martcheva, Homogeneous dynamical systems and the age-structured SIR model with proportionate mixing incidence, in Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics (Levico Terme, 2000). Progress in Nonlinear Differential Equations, vol. 55 (Birkhäuser, Basel, 2003), pp. 227–251
M. Iannelli, F. Milner, The Basic Approach to Age-Structured Population Dynamics (Springer, New York, 2017)
H. Inaba, Threshold and stability results for an age-structured epidemic model. J. Math. Biol. 28, 411–434 (1990)
H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model. J. Math. Biol. 54, 101–146 (2007)
H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology (Springer, Singapore, 2017)
W. Kermack, A. McKendrick, A contribution to mathematical theory of epidemics-iii–further studies of the problem of endemicity. Proc. R. Soc. Lond. A 141, 94–122 (1933)
A. Lotka, On an integral equation in population analysis. Ann. Math. Stat. 10, 144–161 (1939)
R.M. May, R.M. Anderson, Population biology of infectious diseases: part II. Nature 280, 455–461 (1979)
A. McKendrick, Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1926)
J.A.J. Metz, O. Diekmann, Age dependence, in The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, vol. 68 (Springer, Berlin, 1986), pp. 136–184
J.A.J. Metz, O. Diekmann, A gentle introduction to structured population models: three worked examples, in The Dynamics of Physiologically Structured Populations (Amsterdam, 1983). Lecture Notes in Biomathematics, vol. 68 (Springer, Berlin, 1986), pp. 3–45
F. Sharpe, A.J. Lotka, A problem in age distributions. Phil. Mag. 21, 435–438 (1911)
G.F. Webb, A semigroup proof of the Sharpe-Lotka theorem, in Infinite-Dimensional Systems (Retzhof, 1983). Lecture Notes in Mathematics, vol. 1076 (Springer, Berlin, 1984), pp. 254–268
G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and Textbooks in Pure and Applied Mathematics, vol. 89 (Marcel Dekker, New York, 1985)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Li, XZ., Yang, J., Martcheva, M. (2020). Linear Age-Structured Population Models as a Base of Age-Structured Epidemic Models. In: Age Structured Epidemic Modeling. Interdisciplinary Applied Mathematics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-42496-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-42496-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-42495-4
Online ISBN: 978-3-030-42496-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)