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September  2017, 22(7): 2763-2776. doi: 10.3934/dcdsb.2017134

A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

2. 

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

3. 

Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

4. 

Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187, Umeå, Sweden

* Corresponding author

Received  August 2016 Revised  November 2016 Published  April 2017

Fund Project: The work is partially supported by the NSF of China (Grant No. 11501494,11571301), NSF of Jiangsu Province (BK20151305, BK20151288).

To explore the impact of media coverage and spatial heterogeneity of environment on the prevention and control of infectious diseases, a spatial-temporal SIS reaction-diffusion model with the nonlinear contact transmission rate is proposed. The nonlinear contact transmission rate is spatially dependent and introduced to describe the impact of media coverage on the transmission dynamics of disease. The basic reproduction number associated with the disease in the heterogeneous environment is established. Our results show that the degree of mass media attention plays an important role in preventing the spreading of infectious diseases. Numerical simulations further confirm our analytical findings.

Citation: Jing Ge, Ling Lin, Lai Zhang. A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2763-2776. doi: 10.3934/dcdsb.2017134
References:
[1]

A. S. AcklehK. Deng and Y. X. Wu, Competitive exclusion and coexistence in a two-strain pathogen model with diffusion, Math. Biosci. Eng., 13 (2016), 1-18.  doi: 10.3934/mbe.2016.13.1.

[2]

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.

[3]

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.

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, UK, 1991.

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Springer, 2011. doi: 10.1007/978-1-4614-1686-9.

[6]

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

[7]

J. A. CuiX. Tao and H. P. Zhu, An SIS infection model incorporating media coverage, Rocky Mountain J. Math., 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323.

[8]

R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[9]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley series in mathematical and computational biology. John Wiley Sons, West Sussex, England, 2000.

[10]

J. GeK. I. KimZ. G. Lin and H. 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.

[11]

J. GeC. X. Lei and Z. G. Lin, Reproduction numbers and the expanding fronts for a diffusion-advection SIS model in heterogeneous time-periodic environment, Nonlinear Anal. Real World Appl., 33 (2017), 100-120.  doi: 10.1016/j.nonrwa.2016.06.005.

[12]

H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.

[13]

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). doi: 10.1007/s00285-017-1124-7.

[14]

Y. P. Liu and J. A. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.

[15]

R. S. LiuJ. H. Wu and H. P. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. Math. Methods Med., 8 (2007), 153-164.  doi: 10.1080/17486700701425870.

[16]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003.

[17]

A. K. MisraA. Sharma and J. Li, A mathematical model for control of vector borne diseases through media campaigns, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1909-1927.  doi: 10.3934/dcdsb.2013.18.1909.

[18]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[19]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[20]

Y. K. Sampei and A. U. Midori, Mass-media coverage, its influence on public awareness of climate-change issues, and implications for Japans national campaign to reduce greenhouse gas emissions, Global Environ.Change, 19 (2009), 203-212. 

[21]

G. P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl., 421 (2015), 1651-1672.  doi: 10.1016/j.jmaa.2014.08.019.

[22]

C. J. SunW. YangJ. Arino and K. Khan, Effect of media induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005.

[23]

Q. L. TangJ. Ge and Z. G. Lin, An SEI-SI avian-human influenza model with diffusion and nonlocal delay, Appl. Math. Comput., 247 (2014), 753-761.  doi: 10.1016/j.amc.2014.09.042.

[24]

J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomath., 2012 (2012), 1-10.  doi: 10.5402/2012/581274.

[25]

A. L. Wang and Y. N. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Anal. Hybrid Syst., 11 (2014), 84-97.  doi: 10.1016/j.nahs.2013.06.005.

[26]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[27]

Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[28]

World Health Organization, World Health Statistics 2005-2011.

[29]

Y. N. Xiao, S. Y. Tang and J. H. Wu, Media impact switching surface during an infectious disease outbreak Scientific Reports 5 (2015), p7838. doi: 10.1038/srep07838.

[30]

Q. L. YanS. Y. TangS. Gabriele and J. H. Wu, Media coverage and hospital notifications: Correlation analysis and optimal media impact duration to manage a pandemic, J. Theoret. Biol., 390 (2016), 1-13.  doi: 10.1016/j.jtbi.2015.11.002.

show all references

References:
[1]

A. S. AcklehK. Deng and Y. X. Wu, Competitive exclusion and coexistence in a two-strain pathogen model with diffusion, Math. Biosci. Eng., 13 (2016), 1-18.  doi: 10.3934/mbe.2016.13.1.

[2]

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.

[3]

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.

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control Oxford University Press, Oxford, UK, 1991.

[5]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Springer, 2011. doi: 10.1007/978-1-4614-1686-9.

[6]

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

[7]

J. A. CuiX. Tao and H. P. Zhu, An SIS infection model incorporating media coverage, Rocky Mountain J. Math., 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323.

[8]

R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[9]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley series in mathematical and computational biology. John Wiley Sons, West Sussex, England, 2000.

[10]

J. GeK. I. KimZ. G. Lin and H. 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.

[11]

J. GeC. X. Lei and Z. G. Lin, Reproduction numbers and the expanding fronts for a diffusion-advection SIS model in heterogeneous time-periodic environment, Nonlinear Anal. Real World Appl., 33 (2017), 100-120.  doi: 10.1016/j.nonrwa.2016.06.005.

[12]

H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.

[13]

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). doi: 10.1007/s00285-017-1124-7.

[14]

Y. P. Liu and J. A. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.

[15]

R. S. LiuJ. H. Wu and H. P. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. Math. Methods Med., 8 (2007), 153-164.  doi: 10.1080/17486700701425870.

[16]

C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003.

[17]

A. K. MisraA. Sharma and J. Li, A mathematical model for control of vector borne diseases through media campaigns, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1909-1927.  doi: 10.3934/dcdsb.2013.18.1909.

[18]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[19]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[20]

Y. K. Sampei and A. U. Midori, Mass-media coverage, its influence on public awareness of climate-change issues, and implications for Japans national campaign to reduce greenhouse gas emissions, Global Environ.Change, 19 (2009), 203-212. 

[21]

G. P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl., 421 (2015), 1651-1672.  doi: 10.1016/j.jmaa.2014.08.019.

[22]

C. J. SunW. YangJ. Arino and K. Khan, Effect of media induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005.

[23]

Q. L. TangJ. Ge and Z. G. Lin, An SEI-SI avian-human influenza model with diffusion and nonlocal delay, Appl. Math. Comput., 247 (2014), 753-761.  doi: 10.1016/j.amc.2014.09.042.

[24]

J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomath., 2012 (2012), 1-10.  doi: 10.5402/2012/581274.

[25]

A. L. Wang and Y. N. Xiao, A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Anal. Hybrid Syst., 11 (2014), 84-97.  doi: 10.1016/j.nahs.2013.06.005.

[26]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[27]

Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differential Equations, 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[28]

World Health Organization, World Health Statistics 2005-2011.

[29]

Y. N. Xiao, S. Y. Tang and J. H. Wu, Media impact switching surface during an infectious disease outbreak Scientific Reports 5 (2015), p7838. doi: 10.1038/srep07838.

[30]

Q. L. YanS. Y. TangS. Gabriele and J. H. Wu, Media coverage and hospital notifications: Correlation analysis and optimal media impact duration to manage a pandemic, J. Theoret. Biol., 390 (2016), 1-13.  doi: 10.1016/j.jtbi.2015.11.002.

Figure 1.  For $m=1$, the solution $(S(x,t),I(x,t))$ stabilizes to a positive equilibrium.
Figure 2.  For $m=5$, the infected individuals $I(x,t)$ with the given initial condition decays to zero quickly(left); the susceptible individuals $S(x,t)$ stabilizes to a positive steady-state (right).
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