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Coupled Systems of Renewal Equations for Forces of Infection through a Contact Network

Part of: Mathematical biology in general

Published online by Cambridge University Press:  04 December 2019

Mahnaz Alavinejad  and
Jianhong Wu
Mahnaz Alavinejad
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada Email: mahnazal@yorku.cawujh@mathstat.yorku.ca
Jianhong Wu
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada Email: mahnazal@yorku.cawujh@mathstat.yorku.ca
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Abstract

We formulate a coupled system of renewal equations for the forces of infections in interacting subgroups through a contact network. We use the theory of order-preserving and sub-homogeneous discrete dynamical systems to show the existence and uniqueness of the disease outbreak final sizes in the sub-populations. We illustrate the general theory through a simple SIR model with exponentially and non-exponentially distributed infectious period.

Keywords

force of infectionrenewal equationepidemic final size

MSC classification

Primary: 92B05: General biology and biomathematics
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 624 - 632
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

This research has been partially supported by NSERC, the Canada Research Chair Program, and the NSERC-Sanofi Industrial Research Chair Program in Vaccine Mathematics, Modeling, and Manufacturing.

References

Brauer, F., Age-of-infection and the final size relation. Math. Biosci. Eng. 5(2008), 681–690. https://doi.org/10.3934/mbe.2008.5.681Google ScholarPubMed
Brauer, F., Heterogeneous mixing in epidemic models. Can. Appl. Math. Q. 20(2012), 1–13.Google Scholar
Breda, D., Diekmann, O., de Graaf, W. F., Pugliese, A., and Vermiglio, R., On the formulation of epidemic models (an appraisal of Kermack and Mckendrick). J. Biol. Dyn. 7(2012), 103–117. https://doi.org/10.1080/17513758.2012.716454CrossRefGoogle Scholar
Cui, J., Zhang, Y., and Feng, Z., Influence of non-homogeneous mixing on final epidemic size in a meta-population model. J. Biol. Dyn. 13(2019), 31–46. https://doi.org/10.1080/17513758.2018.1484186CrossRefGoogle Scholar
Diekmann, O., Limiting behaviour in an epidemic model. Nonlinear Anal. 1(1976/77), 459–470. https://doi.org/10.1016/0362-546X(77)90011-6CrossRefGoogle Scholar
Gripenberg, G., Londen, S. O., and Staffans, O., Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511662805CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R., Matrix analysis. Second ed., Cambridge University Press, Cambridge, 2013.Google Scholar
Kermack, W. O. and McKendrick, A. G., A contribution to the mathematical theory of epidemics. Proc. R. Soc. 105A(700)(1927).Google Scholar
Krylova, O. and Earn, D. J. D., Effects of the infectious period distribution on predicted transitions in childhood disease dynamics. J. R. Soc. Interface 10(84)(2013).Google Scholar
Lloyd, A. L., Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretical Population Biology 60(2001), 59–71.10.1006/tpbi.2001.1525CrossRefGoogle ScholarPubMed
Zhao, X.-Q., Dynamical systems in population biology. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, Berlin, 2003. https://doi.org/10.1007/978-0-387-21761-1CrossRefGoogle Scholar