The risk index for an SIR epidemic model and spatial spreading of the infectious disease

  • Received: 05 May 2016 Accepted: 19 September 2016 Published: 01 October 2017
  • MSC : Primary: 35K51, 35R35; Secondary: 35B40, 92D25

  • In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number $R_0^{DA}$ for an associated model with Dirichlet boundary condition, we introduce the risk index $R^F_0(t)$ for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if $R^F_0(t_0)≥q 1$ for some $t_0$ and the disease is vanishing if $R^F_0(∞) \lt 1$, while if $R^F_0(0) \lt 1$, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

    Citation: Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1565-1583. doi: 10.3934/mbe.2017081

    Related Papers:

  • In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number $R_0^{DA}$ for an associated model with Dirichlet boundary condition, we introduce the risk index $R^F_0(t)$ for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if $R^F_0(t_0)≥q 1$ for some $t_0$ and the disease is vanishing if $R^F_0(∞) \lt 1$, while if $R^F_0(0) \lt 1$, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.
    加载中
    [1] [ I. Ahn,S. Baek,Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016): 7082-7101.
    [2] [ L. J. S. Allen,B. M. Bolker,Y. Lou,A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008): 1-20.
    [3] [ A. L. Amadori,J. L. Vázquez, Singular free boundary problem from image processing, Math. Model. Methods. Appl. Sci., 15 (2005): 689-715.
    [4] [ A. Bezuglyy,Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010): 983-1004.
    [5] [ R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003.
    [6] [ Center for Disease Control and Prevention (CDC), Update: West Nile-like viral encephalitis-New York, 1999, Morb. Mortal Wkly. Rep., 48 (1999): 890-892.
    [7] [ X. F. Chen,A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000): 778-800.
    [8] [ Y. H. Du,Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010): 377-405.
    [9] [ Y. H. Du,Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014): 3105-3132.
    [10] [ Y. H. Du,B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015): 2673-2724.
    [11] [ A. M. Elaiw,N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. Real World Appl., 26 (2015): 161-190.
    [12] [ X. M. Feng,S. G. Ruan,Z. D. Teng,K. Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015): 52-64.
    [13] [ D. Z. Gao, Y. J. Lou and D. H. He, et al., Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis, Scientific Reports 6, Article number: 6 (2016), 28070.
    [14] [ J. Ge,K. I. Kim,Z. G. Lin,H. P. Zhu, A SIS reaction-diffusion model in a low-risk and high-risk domain, J. Differential Equation, 259 (2015): 5486-5509.
    [15] [ H. Gu,B. D. Lou,M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015): 1714-1768.
    [16] [ Globalization and disease Available from: https://en.wikipedia.org/wiki/Globalization_and_disease.
    [17] [ West African Ebola virus epidemic Available from: https://en.wikipedia.org/wiki/West_African_Ebola_virus_epidemic.
    [18] [ Epidemic situation of dengue fever in Guangdong, 2014 (Chinese) Available from: http://www.rdsj5.com/guonei/1369.html.
    [19] [ S. Iwami,Y. Takeuchi,X. N. Liu, Avian-human influenza epidemic model, Math. Biosci., 207 (2007): 1-25.
    [20] [ W. O. Kermack,A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A115 (1927): 700-721.
    [21] [ W. O. Kermack,A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A138 (1932): 55-83.
    [22] [ K. I. Kim,Z. G. Lin,L. Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Anal. Real World Appl., 11 (2010): 313-322.
    [23] [ K. I. Kim,Z. G. Lin,Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013): 1992-2001.
    [24] [ O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc, Providence, RI, 1968.
    [25] [ C. X. Lei,Z. G. Lin,Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014): 145-166.
    [26] [ C. Z. Li,J. Q. Li,Z. E. Ma,H. P. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014): 987-1004.
    [27] [ M. Li,Z. G. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015): 2089-2105.
    [28] [ Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007): 1883-1892.
    [29] [ Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., (2017), 1–29, http://link.springer.com/article/10.1007/s00285-017-1124-7?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst.
    [30] [ N. A. Maidana,H. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009): 403-417.
    [31] [ W. Merz,P. Rybka, A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004): 571-588.
    [32] [ L. I. Rubinstein, The Stefan Problem, Amer. Math. Soc, Providence, RI, 1971.
    [33] [ C. Shekhar, Deadly dengue: New vaccines promise to tackle this escalating global menace, Chem. Biol., 14 (2007): 871-872.
    [34] [ S. Side,S. M. Noorani, A SIR model for spread of Dengue fever disease (simulation for South Sulawesi, Indonesia and Selangor, Malaysia), World Journal of Modelling and Simulation, 9 (2013): 96-105.
    [35] [ Y. S. Tao,M. J. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006): 419-440.
    [36] [ J. Wang,J. F. Cao, The spreading frontiers in partially degenerate reaction-diffusion systems, Nonlinear Anal., 122 (2015): 215-238.
    [37] [ J. Y. Wang,Y. N. Xiao,Z. H. Peng, Modelling seasonal HFMD infections with the effects of contaminated environments in mainland China, Appl. Math. Comput., 274 (2016): 615-627.
    [38] [ M. X. Wang,J. F. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014): 655-672.
    [39] [ World Health Organization, World Health Statistics 2006-2012.
    [40] [ P. Zhou,D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014): 1927-1954.

    © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(565) PDF downloads(680) Cited by(4)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog