Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction: A Prelude to Mathematical Epidemiology Read chapter Read first chapter

2019 | OriginalPaper | Chapter

5. Models with Heterogeneous Mixing

Authors : Fred Brauer, Carlos Castillo-Chavez, Zhilan Feng

Published in: Mathematical Models in Epidemiology

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

To cope with annual seasonal influenza epidemics there is a program of vaccination before the “flu” season begins. Each year a vaccine is produced aimed at protecting against the three influenza strains considered most dangerous for the coming season. We formulate a model to add vaccination to the simple SIR model under the assumption that vaccination reduces susceptibility (the probability of infection if a contact with an infected member of the population is made).

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Epidemic Models
next chapter Models for Diseases Transmitted by Vectors
Literature
1.
Adler, F.R. (1992) The effects of averaging on the basic reproduction ratio, Math Biosci. 111(1): 89–98.MathSciNetMATH
2.
Allen,L.J. and P. van den Driessche (2008) The basic reproduction number in some discrete-time epidemic models, J. Diff. Equ. App. 14: 1127–1147.MathSciNetMATH
3.
Andersson, H. & Britton, T. (1998) Heterogeneity in epidemic models and its effect on the spread of infection, J. Appl. Prob. 35: 651–661.MathSciNetMATH
4.
Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2006) Simple models for containment of a pandemic, J. Roy. Soc. Interface, 3: 453–457.
5.
Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2007) A final size relation for epidemic models, Math. Biosc. & Eng. 4: 159–176.MathSciNetMATH
6.
Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2008) A model for influenza with vaccination and antiviral treatment, Theor. Pop. Biol. 253: 118–130.MathSciNetMATH
7.
Berman, A. & R.J. Plemmons (1994) Nonnegative Matrices in the Mathematical Sciences, SIAM, Vol. 9, 1994.MATH
8.
Blythe, S.P., S. Busenberg & C. Castillo-Chavez (1995) Affinity and paired-event probability, Math. Biosc. 128: 265–84 .MathSciNetMATH
9.
Blythe, S.P. , C. Castillo-Chavez, J. Palmer & M. Cheng (1991) Towards a unified theory of mixing and pair formation, Math. Biosc. 107: 379–405.MATH
10.
11.
Brauer, F. (2008) Epidemic models with treatment and heterogeneous mixing, Bull. Math. Biol. 70: 1869–1885.MathSciNetMATH
12.
Brauer, F. & J. Watmough (2009) Age of infection epidemic models with heterogeneous mixing, J. Biol. Dynamics 3: 324–330.MathSciNetMATH
13.
Busenberg, S. & C. Castillo-Chavez (1989) Interaction, pair formation and force of infection terms in sexually transmitted diseases, In Mathematical and Statistical Approaches to AIDS Epidemiology, Lect. Notes Biomath. 83, C. Castillo-Chavez (ed.), Springer-Verlag, Berlin-Heidelberg-New York, 289–300.
14.
Busenberg, S. & C. Castillo-Chavez (1991) A general solution of the problem of mixing of sub-populations and its application to risk- and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol., 8: 1–29.MathSciNetMATH
15.
Chow, L., M. Fan, and Z. Feng (2011) Dynamics of a multi-group epidemiological model with group-targeted vaccination strategies, J. Theor. Biol. 29: 56–64.MATH
16.
De Jong, M., O. Diekmann and J.A.P. Heesterbeek (1994) The computation of R0 for discrete-time epidemic models with dynamic heterogeneity, Math. Biosci. 119(1): 97–114.MATH
17.
Del Valle, S. Y., J.M. Hyman, H.W. Hethcote & S.G. Eubank (2007) Mixing patterns between age groups in social networks, Social Networks, 29(4): 539–554.
18.
Diekmann, O. & J.A.P. Heesterbeek (2000) Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester (2000)
19.
Diekmann, O., J.A.P. Heesterbeek & J.A.J. Metz (1990) On the definition and the computation of the basic reproductive ratio \(\mathcal {R}_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28: 365–382.
20.
Diekmann, O., J.A.P. Heesterbeek, & T. Britton (2012) Mathematical tools for understanding infectious disease dynamics, 2012, Princeton University Press.MATH
21.
Feng, Z., A.N. Hill, P.J. Smith, J.W. Glasser (2015) An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing, J. Theor. Biol. 386: 177–187.MathSciNetMATH
22.
Feng, Z., A.N. Hill, A.T. Curns, J.W. Glasser (2007) Evaluating targeted interventions via meta-population models with multi-level mixing, Math. Biosci. 287: 93–104.MathSciNetMATH
23.
Ferguson, N.M., D.A.T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, & D.S. Burke (2005) Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437: 209–214.
24.
Gani, R., H. Hughes, T. Griffin, J. Medlock, & S. Leach (2005) Potential impact of antiviral use on hospitalizations during influenza pandemic, Emerg. Infect. Dis. 11: 1355–1362.
25.
Glasser, J., Z. Feng, A. Moylan, S. Del Valle & C. Castillo-Chavez (2012) Mixing in age-structured population models of infectious diseases, Math. Biosci., 235(1): 1–7.MathSciNetMATH
26.
Glasser, J.W., Z. Feng, S.B. Omer, P.J. Smith, L.E. Rodewald (2016) The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study, 16(5): 599–605.
27.
Hethcote, H.W. & J.A. Yorke (1984) Gonorrhea Transmission Dynamics and Control, Lect. Notes in Biomath. 56, Springer-Verlag, Berlin-Heidelberg-New York (1984).
28.
Hethcote, H.W. & J.W. van Ark (1987) Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs. Math. Biosci., 84(1): 85–118.MathSciNetMATH
29.
Hirsch, M.W. & S. Smale, Differential Equations (1974) Dynamical Systems, and Linear Algebra, Academic Press, Orlando, FL (1974).MATH
30.
Jacquez, J.A., C.P. Simon, J. Koopman, L. Sattenspiel, & T. Perry (1988) Modeling and analyzing HIV transmission: the effect of contact patterns, Math. Biosci., 92: 119–199.MathSciNetMATH
31.
Lewis, M.A., J. Renclawowicz, P. van Den Driessche, and M. Wonham (2006) A comparison of continuous and discrete-time West Nile Virus models, Bull. Math. Biol. 68(3): 491–509.MathSciNetMATH
32.
Longini, I.M., M.E. Halloran, A. Nizam, & Y. Yang (2004) Containing pandemic influenza with antiviral agents, Am. J. Epidem. 159: 623–633.
33.
Longini, I.M., A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D.A.T. Cummings, & M.E. Halloran (2005) Containing pandemic influenza at the source, Science 309, 1083–1087.
34.
Longini, I. M. & M. E. Halloran (2005) Strategy for distribution of influenza vaccine to high - risk groups and children, Am. J. Epidem. 161: 303–306.
35.
Ma, J. & D.J.D. Earn (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol. 68: 679–702.MathSciNetMATH
36.
May, R.M. & R.M. Anderson (1984) Spatial heterogeneity and the design of immunization programs. Math Biosci. 72(1): 83–111.MathSciNetMATH
37.
May, R.M. & R.M. Anderson (1984) Spatial, temporal and genetic heterogeneity in host populations and the design of immunization programmes. IMA J Math Appl Math Biol. 1(3): 233–66.MathSciNetMATH
38.
Meyers, L.A. (2007) Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc. 44, 63–86.MathSciNetMATH
39.
Meyers, L.A. , M.E.J. Newman & B. Pourbohloul (2006) Predicting epidemics on directed contact networks, J. Theor. Biol. 240, 400–418.MathSciNet
40.
Meyers, L.A., B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, & R.C. Brunham (2005) Network theory and SARS: predicting outbreak diversity. J. Theor. Biol., 232: 71–81.MathSciNet
41.
Mossong, J., N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikolajczyk, …& W. J. Edmunds (2008) Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Medicine, 5(3): e74.
42.
43.
Poghotanyan, G., Z. Feng, J.W. Glasser, A.N. Hill (2018) Constrained minimization problems for the reproduction number in meta-population models, J. Math. Biol. https:/doi.org/10.1007/s00285-018-1216-z.
44.
45.
van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc. 180: 29–48.MathSciNetMATH
46.
van den Driessche, P. & J. Watmough (2002) Further notes on the basic reproduction number, in Mathematical Epidemiology, F. Brauer, P. van den Driessche, & J. Wu (eds.) Springer Lecture Notes, Vol. 1945.
47.
Wallinga, J., P. Teunis, & M. Kretzschmar (2006) Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents, American Journal of Epi, 164(10): 936–944.
48.
Wesley, C. L., L.J. Allen, C.B. Jonsson, Y.-K. Chu, R.D. Owen (2009) A discrete-time rodent-hantavirus model structured by infection and developmental stages, Adv. Stu. Pure Math. 53: 387–398.MathSciNetMATH
49.
Zagheni, E., F.C. Billari, P. Manfredi, A. Melegaro, J. Mossong & W.J. Edmunds (2008) Using time-use data to parameterize models for the spread of close-contact infectious diseases, Am. J. Epi., 168(9), 1082–1090.
Metadata
Title
Models with Heterogeneous Mixing
Authors
Fred Brauer
Carlos Castillo-Chavez
Zhilan Feng
Copyright Year
2019
Publisher
Springer New York
Book
Mathematical Models in Epidemiology

Print ISBN: 978-1-4939-9826-5

Electronic ISBN: 978-1-4939-9828-9

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-1-4939-9828-9_5

Premium Partner

    Image Credits