October 2017, 14(5&6): 1565-1583. doi: 10.3934/mbe.2017081

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

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

2. 

Department of Mathematics, Anhui Normal University, Wuhu 241000, China

* Corresponding author: zglin68@hotmail.com (Z. G. Lin)

Received  May 05, 2016 Accepted  September 19, 2016 Published  May 2017

Fund Project: The first author is supported by Graduate Research and Innovation Projects of Jiangsu Province KYZZ16−0489, and the third author is supported by NSFC of China 11371311 and 11626019

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 $R^F_0(∞)<1$, while if $R^F_0(0)<1$, 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. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1565-1583. doi: 10.3934/mbe.2017081
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Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

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Y. H. Du and 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. doi: 10.3934/dcdsb.2014.19.3105.

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Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568.

[11]

A. M. Elaiw and 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. doi: 10.1016/j.nonrwa.2015.05.007.

[12]

X. M. FengS. G. RuanZ. D. Teng and 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. doi: 10.1016/j.mbs.2015.05.005.

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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. doi: 10.1038/srep28070.

[14]

J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion model in a low-risk and high-risk domain, J. Differential Equation, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035.

[15]

H. GuB. D. Lou and M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768. doi: 10.1016/j.jfa.2015.07.002.

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West African Ebola virus epidemic Available from: https://en.wikipedia.org/wiki/West_African_Ebola_virus_epidemic.

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Epidemic situation of dengue fever in Guangdong, 2014 (Chinese) Available from: http://www.rdsj5.com/guonei/1369.html.

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S. IwamiY. Takeuchi and X. N. Liu, Avian-human influenza epidemic model, Math. Biosci., 207 (2007), 1-25. doi: 10.1016/j.mbs.2006.08.001.

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W. O. Kermack and A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A115 (1927), 700-721.

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W. O. Kermack and A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A138 (1932), 55-83.

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K. I. KimZ. G. Lin and L. Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Anal. Real World Appl., 11 (2010), 313-322. doi: 10.1016/j.nonrwa.2008.11.015.

[23]

K. I. KimZ. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[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. LeiZ. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015.

[26]

C. Z. LiJ. Q. LiZ. E. Ma and H. P. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987-1004. doi: 10.1016/j.jmaa.2014.06.035.

[27]

M. Li and Z. G. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089-2105. doi: 10.3934/dcdsb.2015.20.2089.

[28]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[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. doi: 10.1007/s00285-017-1124-7.

[30]

N. A. Maidana and H. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417. doi: 10.1016/j.jtbi.2008.12.032.

[31]

W. Merz and 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. doi: 10.1016/j.jmaa.2003.12.025.

[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. doi: 10.1016/j.chembiol.2007.08.004.

[34]

S. Side and 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 and M. J. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440. doi: 10.1088/0951-7715/19/2/010.

[36]

J. Wang and J. F. Cao, The spreading frontiers in partially degenerate reaction-diffusion systems, Nonlinear Anal., 122 (2015), 215-238. doi: 10.1016/j.na.2015.04.003.

[37]

J. Y. WangY. N. Xiao and Z. H. Peng, Modelling seasonal HFMD infections with the effects of contaminated environments in mainland China, Appl. Math. Comput., 274 (2016), 615-627. doi: 10.1016/j.amc.2015.11.035.

[38]

M. X. Wang and 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 and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.

show all references

References:
[1]

I. AhnS. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016), 7082-7101. doi: 10.1016/j.apm.2016.02.038.

[2]

L. J. S. AllenB. M. BolkerY. Lou and 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. doi: 10.3934/dcds.2008.21.1.

[3]

A. L. Amadori and J. L. Vázquez, Singular free boundary problem from image processing, Math. Model. Methods. Appl. Sci., 15 (2005), 689-715. doi: 10.1142/S0218202505000509.

[4]

A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010), 983-1004. doi: 10.1080/00036810903479723.

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003. doi: 10.1002/0470871296.

[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 and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[8]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[9]

Y. H. Du and 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. doi: 10.3934/dcdsb.2014.19.3105.

[10]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724. doi: 10.4171/JEMS/568.

[11]

A. M. Elaiw and 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. doi: 10.1016/j.nonrwa.2015.05.007.

[12]

X. M. FengS. G. RuanZ. D. Teng and 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. doi: 10.1016/j.mbs.2015.05.005.

[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. doi: 10.1038/srep28070.

[14]

J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion model in a low-risk and high-risk domain, J. Differential Equation, 259 (2015), 5486-5509. doi: 10.1016/j.jde.2015.06.035.

[15]

H. GuB. D. Lou and M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768. doi: 10.1016/j.jfa.2015.07.002.

[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. IwamiY. Takeuchi and X. N. Liu, Avian-human influenza epidemic model, Math. Biosci., 207 (2007), 1-25. doi: 10.1016/j.mbs.2006.08.001.

[20]

W. O. Kermack and A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A115 (1927), 700-721.

[21]

W. O. Kermack and A. G. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc., A138 (1932), 55-83.

[22]

K. I. KimZ. G. Lin and L. Zhang, Avian-human influenza epidemic model with diffusion, Nonlinear Anal. Real World Appl., 11 (2010), 313-322. doi: 10.1016/j.nonrwa.2008.11.015.

[23]

K. I. KimZ. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[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. LeiZ. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015.

[26]

C. Z. LiJ. Q. LiZ. E. Ma and H. P. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987-1004. doi: 10.1016/j.jmaa.2014.06.035.

[27]

M. Li and Z. G. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089-2105. doi: 10.3934/dcdsb.2015.20.2089.

[28]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[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. doi: 10.1007/s00285-017-1124-7.

[30]

N. A. Maidana and H. Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417. doi: 10.1016/j.jtbi.2008.12.032.

[31]

W. Merz and 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. doi: 10.1016/j.jmaa.2003.12.025.

[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. doi: 10.1016/j.chembiol.2007.08.004.

[34]

S. Side and 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 and M. J. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440. doi: 10.1088/0951-7715/19/2/010.

[36]

J. Wang and J. F. Cao, The spreading frontiers in partially degenerate reaction-diffusion systems, Nonlinear Anal., 122 (2015), 215-238. doi: 10.1016/j.na.2015.04.003.

[37]

J. Y. WangY. N. Xiao and Z. H. Peng, Modelling seasonal HFMD infections with the effects of contaminated environments in mainland China, Appl. Math. Comput., 274 (2016), 615-627. doi: 10.1016/j.amc.2015.11.035.

[38]

M. X. Wang and 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 and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.

Figure 1.  $\mu=20$. The left graph shows that the solution $I$ decays to zero quickly. The right graph is the corresponding contour graph, which shows the free boundaries expand slowly and will be limited in a long run
Figure 2.  $\mu=40$. The solution $I$ in the left graph keeps positive and stabilizes to an equilibrium. The right contour graph shows that the free boundaries expand fast
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