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An Extension of Discrete Tribocharging Models to Continuous Size Distributions

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Continuous and discrete SIR-models with spatial distributions

Journal of Mathematical Biology volume 74,pages 1709–1727 (2017)Cite this article

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Abstract

The SIR-model is a basic epidemic model that classifies a population into three subgroups: susceptible S, infected I and removed R. This model does not take into consideration the spatial distribution of each subgroup, but considers the total number of individuals belonging to each subgroup. There are many variants of the SIR-model. For studying the spatial distribution, stochastic processes have often been introduced to describe the dispersion of individuals. Such assumptions do not seem to be applicable to humans, because almost everyone moves within a small fixed radius in practice. Even if individuals do not disperse, the transmission of disease occurs. In this paper, we do not assume the dispersion of individuals, and instead use the infectious radius. Then, we propose simple continuous and discrete SIR-models that show spatial distributions. The results of our simulations show that the propagation speed and size of an epidemic depend on the population density and the infectious radius.

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Acknowledgments

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03931459).

Author information

Authors and Affiliations

  1. Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul, 143-701, Korea

    Seong-Hun Paeng & Jonggul Lee

Corresponding author

Correspondence to Seong-Hun Paeng.

Appendix: Map of the Seoul metropolitan area and table for populations

Appendix: Map of the Seoul metropolitan area and table for populations

Figure 10 displays the map and the induced graph of the Seoul metropolitan area consisting of 33 districts, respectively. Table 1 concerns the populations of 33 districts in the Seoul metropolitan area. Figure 11 concerns the numbers of the infected between our discrete model and the spatial-temporal model of Lee and Jung (2015) for whole 33 districts.

Fig. 10
figure 10

Map of Seoul metropolitan area and the induced graph

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Table 1 Populations of 33 districts in Seoul metropolitan area

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Fig. 11
figure 11

Comparison of the cumulative numbers of the infected between our discrete model and the spatial-temporal model of Lee and Jung (2015) for 33 districts

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Paeng, SH., Lee, J. Continuous and discrete SIR-models with spatial distributions. J. Math. Biol. 74, 1709–1727 (2017). https://doi.org/10.1007/s00285-016-1071-8

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  • DOI : https://doi.org/10.1007/s00285-016-1071-8

Keywords

  • SIR-model
  • Spatial distribution
  • Infectious radius

Mathematics Subject Classification

  • 92D30

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