• 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]

Chengxia Lei, Fujun Li, Jiang Liu. Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173

[2]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[3]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[4]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173

[5]

Tomás Caraballo, Mohamed El Fatini, Roger Pettersson, Regragui Taki. A stochastic SIRI epidemic model with relapse and media coverage. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3483-3501. doi: 10.3934/dcdsb.2018250

[6]

Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks & Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025

[7]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[8]

Mahin Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 185-202. doi: 10.3934/dcdsb.2006.6.185

[9]

Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

[10]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455

[11]

Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016

[12]

Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545

[13]

Mariusz Bodzioch, Marcin Choiński, Urszula Foryś. SIS criss-cross model of tuberculosis in heterogeneous population. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2169-2188. doi: 10.3934/dcdsb.2019089

[14]

Jing Ge, Zhigui Lin, Huaiping Zhu. Environmental risks in a diffusive SIS model incorporating use efficiency of the medical resource. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1469-1481. doi: 10.3934/dcdsb.2016007

[15]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[16]

Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989

[17]

Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114

[18]

Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1479-1494. doi: 10.3934/mbe.2018068

[19]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166

[20]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (24)
  • HTML views (22)
  • Cited by (0)

Other articles
by authors

[Back to Top]