• Previous Article
    Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems
  • DCDS-B Home
  • This Issue
  • Next Article
    Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

World Health Organization, World Health Statistics 2005-2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

World Health Organization, World Health Statistics 2005-2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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).
[1]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[2]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[3]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[4]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[5]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[7]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356

[8]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[9]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304

[10]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[11]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[12]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[13]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[14]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[15]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[16]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[17]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[18]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[19]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[20]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (131)
  • HTML views (72)
  • Cited by (1)

Other articles
by authors

[Back to Top]