Research article

Modeling epidemic in metapopulation networks with heterogeneous diffusion rates

  • Received: 23 March 2019 Accepted: 29 July 2019 Published: 05 August 2019
  • In this paper, the process of the infectious diseases among cities is studied in metapopulation networks. Based on the heterogeneous diffusion rate, the epidemic model in metapopulation networks is established. The factors affecting diffusion rate are discussed, and the relationship among diffusion rate, connectivity of cities and the heterogeneity parameter of traffic flow is obtained. The existence and stability of the disease-free equilibrium and the endemic equilibrium are analyzed, and epidemic threshold is also obtained. It is shown that the more developed traffic of the city, the greater the diffusion rate, which resulting in the large number of infected individuals; the stronger the heterogeneity of the traffic flow, the greater the threshold of the disease outbreak. Finally, numerical simulations are performed to illustrate the analytical results.

    Citation: Maoxing Liu, Jie Zhang, Zhengguang Li, Yongzheng Sun. Modeling epidemic in metapopulation networks with heterogeneous diffusion rates[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7085-7097. doi: 10.3934/mbe.2019355

    Related Papers:

  • In this paper, the process of the infectious diseases among cities is studied in metapopulation networks. Based on the heterogeneous diffusion rate, the epidemic model in metapopulation networks is established. The factors affecting diffusion rate are discussed, and the relationship among diffusion rate, connectivity of cities and the heterogeneity parameter of traffic flow is obtained. The existence and stability of the disease-free equilibrium and the endemic equilibrium are analyzed, and epidemic threshold is also obtained. It is shown that the more developed traffic of the city, the greater the diffusion rate, which resulting in the large number of infected individuals; the stronger the heterogeneity of the traffic flow, the greater the threshold of the disease outbreak. Finally, numerical simulations are performed to illustrate the analytical results.


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    [1] L. A. Rvachev and I. M. J. Longini, A mathematical model for global spread of influenza, Math. Bio., 75 (1985), 3–23.
    [2] O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation, New York: Wiley, 2000.
    [3] W. Gu, R. Heikkilä and I. Hanski, Estimating the consequences of habitat fragmentation on extinction risk in dynamic landscapes, Landscape Ecol., 17 (2002), 699–710.
    [4] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Entomol. Soc. Amer. Bull., 15 (1969), 237–240.
    [5] B. Grenfell and B. M. Bolker, Cities and villages: Infection hierarchies in am easles metapopulation, Ecol. Lett., 1 (1998), 63–70.
    [6] B. Grenfell and J. Harwood, (Meta) population dynamics of infectious diseases, Trends Ecol. Evol., 12 (1997), 395–399.
    [7] M. J. Keeling and P. Rohani, Estimating spatial coupling in epidemiological systems: A mechanistic approach, Ecol. Lett., 5 (2002), 20–29.
    [8] R. M. May and R. M. Anderson, Population biology of infectious diseases: Part II, Nature, 280 (1979), 455–461.
    [9] R. M. May and R. M. Anderson, Spatial heterogeneity and the design of immunization programs, Math. Biosci., 72 (1984), 83–111.
    [10] V. Colizza, R. Pastorsatorras and A. Vespignani, Reaction-diffusion processes and metapopulation models in heterogeneous networks, Nat. Phys., 3 (2007), 276–282.
    [11] V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, J. Theor. Biol., 251 (2007), 450–467.
    [12] V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks, Phys. Rev. Lett., 99 (2007), 148701.
    [13] V. Colizza, A. Barrat, M. Barthélemy, et al., Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions, Plos Med., 4 (2007), e13.
    [14] V. Colizza, A. Barrat, M. Barthélemy, et al., The modeling of global epidemics: Stochastic dynamics and predictability, Bulletin Math. Biol., 68 (2006), 1893–1921.
    [15] V. Colizza, A. Barrat, M. Barthélemy, et al., The role of the airline transportation network in the prediction and predictability of global epidemics, Proc. Natl. Acad. Sci., 103 (2006), 2015–2020.
    [16] V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, J. Theor. Biol., 251 (2008), 450–467.
    [17] D. Juher, J. Ripoll and J. Saldaña, Analysis and Monte Carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulations, Phys. Rev. E, 80 (2009), 041920.
    [18] D. Juher and V. Mañosa, Spectral properties of the connectivity matrix and the, SIS-epidemic threshold for mid-size metapopulations, Math. Model. Natl. Phenomena, 9 (2014), 108–120.
    [19] J. Saldaña, Continuous-time formulation of reaction-diffusion processes on heterogeneous metapopulations, Phys. Rev. E, 78 (2008), 139–143.
    [20] A. Baronchelli, M. Catanzaro and R. Pastor-Satorras, Bosonic reaction-diffusion processes on scale-free networks, Phys. Rev. E, 78 (2008), 1302–1314.
    [21] K. Kuga and J. Tanimoto, Impact of imperfect vaccination and defense against contagion on vaccination behavior in complex networks, J. Stat. Mech.: Theory. Exp., 2018 (2018), 113402.
    [22] K. Kuga, J. Tanimoto and M. Jusup, To vaccinate or not to vaccinate: A comprehensive study of vaccination-subsidizing policies with multi-agent simulations and mean-field modeling, J. Theoretical Biol., 469 (2019), 107–126.
    [23] K. M. A. Kabir and J. Tanimoto, Evolutionary vaccination game approach in metapopulation migration model with information spreading on different graphs, Chaos, Solitons Fractals, 120 (2019), 41–55.
    [24] K. M. A. Kabir and J. Tanimoto, Dynamical Behaviors for Vaccination can Suppress Infectious Disease-A Game Theoretical Approach, Chaos, Solitons Fractals, 123 (2019), 229–239.
    [25] M. Alam, K. Kuga and J. Tanimoto, Three-strategy and four-strategy model of vaccination game introducing an intermediate protecting measure, Appl. Math. Comput. 346 (2019), 408–422.
    [26] N. Masuda, Effects of diffusion rates on epidemic spreads in metapopulation networks, New J. Phys., 12 (2010), 093009.
    [27] G. Tanaka, C. Urabe and K. Aihara, Random and targeted interventions for epidemic control in metapopulation models, Sci. Rep., 4 (2013), 5522.
    [28] V. Batagelj and A. Mrvar, Pajek dajek datasets, 2006. Available from: http://vlado.fmf.uni-lj.si/pub/networks/data/.
    [29] M. Zanin, On alternative formulations of the small-world metric in complex networks, Comput. Sci., preprint, arXiv1505.03689.
    [30] A. Barrat, M. Barthélemy, R. Pastor-Satorras, et al., The architecture of complex weighted networks, Proc. Natl. Acad. Sci. U. S. A., 101 (2004), 3747–3752.
    [31] S. Meloni, A. Arenas, Y. Moreno, et al., Traffic-driven epidemic spreading in finite-Size scale-free networks, Proc. Natl. Acad. Sci. U. S. A., 106 (2009), 16897–16902.
    [32] B. Wang, G. Tanaka, H. Suzuki, et al., Epidemic spread on interconnected metapopulation networks, Phys. Rev. E, 90 (2014), 032806.
    [33] C. Poletto, M. Tizzoni and V. Colizza, Heterogeneous length of stay of hosts' movements and spatial epidemic spread, Sci. Rep., 2 (2012), 476.
    [34] Y. W. Gong, Y. R. Song and G. P. Jiang, Epidemic spreading in metapopulation networks with heterogeneous infection rates, Phys. A Stat. Mechan. Its Appl., 416 (2014), 208–218.
    [35] J. Anderson, A secular equation for the eigenvalues of a diagonal matrix perturbation, Linear Algebra Its Appl., 246 (1996), 49–70.
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