October  2017, 14(5&6): 1317-1335. doi: 10.3934/mbe.2017068

An SEI infection model incorporating media impact

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China

3. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

1 Author to whom correspondence should be addressed

Received  July 08, 2016 Revised  November 07, 2016 Published  May 2017

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number $\Re_0$ for the model, and show that the modeled disease dies out regardless of initial infections when $\Re_0 < 1$, and becomes uniformly persistently endemic if $\Re_0>1$. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided $\Re_0 $ is greater than and near one. However, if $\Re_0 $ is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger $\Re_0$, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.

Citation: Xuejuan Lu, Shaokai Wang, Shengqiang Liu, Jia Li. An SEI infection model incorporating media impact. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1317-1335. doi: 10.3934/mbe.2017068
References:
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R. J. BlendonJ. M. BensonC. M. DesRochesE. Raleigh and K. Taylor-Clark, The public's response to severe acute respiratory syndormw in Toronto and the United States, Clinic Infectious Diseases, 38 (2004), 925-931.   Google Scholar

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S. Collinson and J. M. Heffernan, Modelling the effects of media during an influenza epidemic, BMC Public Health, 14 (2014), 376. doi: 10.1186/1471-2458-14-376.  Google Scholar

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J. CuiY. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar

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K. A. FantiE. VanmanC. C. Henrich and M. N. Avraamide, Desensitization to media violence over a short period of time, Aggressive Behavior, 35 (2009), 179-187.  doi: 10.1002/ab.20295.  Google Scholar

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K. FrostE. Frank and E. Maibach, Relative risk in the news media: A quantification of misrepresentation, American Journal of Public Health, 87 (1997), 842-845.  doi: 10.2105/AJPH.87.5.842.  Google Scholar

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X. LaiS. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect, Applicable Analysis, 89 (2010), 1271-1292.  doi: 10.1080/00036811.2010.483557.  Google Scholar

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B. Li, S. Liu, J. Cui and J. Li, A simple predator-prey population model with rich dynamics, App. Sci., 6 (2016), 151. doi: 10.3390/app6050151.  Google Scholar

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R. LinS. Liu and X. Lai, Bifurcations of a Predator-prey System with Weak Allee effects, J. Korean Math. Soc., 50 (2013), 695-713.  doi: 10.4134/JKMS.2013.50.4.695.  Google Scholar

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R. LiuJ. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar

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W. LiuH. Hethcote and S. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

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W. LiuS. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[26]

Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar

[27]

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

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A. K. MisraA. Sharma and J. B. Shukla, Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Math. Comput. Model, 53 (2011), 1221-1228.  doi: 10.1016/j.mcm.2010.12.005.  Google Scholar

[29]

A. K. MisraA. Sharma and V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Sys., 19 (2011), 389-402.  doi: 10.1142/S0218339011004020.  Google Scholar

[30]

A. Mummert and H. Weiss, Get the news out loudly and quickly: The Influence of the media on limiting emerging infectious disease outbreaks, PloS One, 8 (2013), e71692. doi: 10.1371/journal.pone.0071692.  Google Scholar

[31]

J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[32]

C. 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

[33]

J. Tchuenche, N. Dube, C. Bhunu, R. Smith and C. Bauch, The impact of media coverage on the transmission dynamics of human inuenza, BMC Public Health, 11 (2011), S5. Google Scholar

[34]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[35]

A. A. F. Wahlberg and L. Sjoberg, Risk perception and the media, Journal of Risk Research, 3 (2001), 31-50.  doi: 10.1080/136698700376699.  Google Scholar

[36]

Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak, Scientific Reports, 5 (2015), 7838. doi: 10.1038/srep07838.  Google Scholar

[37]

Y. XiaoT. Zhao and S. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. and Eng., 10 (2013), 445-461.  doi: 10.3934/mbe.2013.10.445.  Google Scholar

[38]

M. E. Young, G. R. Norman and K. R. Humphreys, Medicine in the popular press: The Influence of the media on perceptions of disease, PloS One, (3) (2008), e3552. doi: 10.1371/journal.pone.0003552.  Google Scholar

[39]

X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.   Google Scholar

show all references

References:
[1]

M. Becker and J. Joseph, AIDS and behavioral change to reduce risk: A review, Amer. J. Public Health, 78 (1988), 394-410.  doi: 10.2105/AJPH.78.4.394.  Google Scholar

[2]

R. J. BlendonJ. M. BensonC. M. DesRochesE. Raleigh and K. Taylor-Clark, The public's response to severe acute respiratory syndormw in Toronto and the United States, Clinic Infectious Diseases, 38 (2004), 925-931.   Google Scholar

[3]

S. M. Blower and A. R. McLean, Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco, Science, 265 (1994), 1451-1454.  doi: 10.1126/science.8073289.  Google Scholar

[4]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[5]

S. Busenberg and K. Cooke, Vertically Transmitted Diseases, Springer-Verlag, New York, 1993. doi: 10.1007/978-3-642-75301-5.  Google Scholar

[6]

L. Cai and X. Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Applied Mathematical Modelling, 33 (2009), 2919-2926.  doi: 10.1016/j.apm.2008.01.005.  Google Scholar

[7]

V. Capasso and G. Serio, A generalisation of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[8]

R. ChunaraJ. R. Andrews and J. S. Brownstein, Social and news media enable estimation of epidemiological patterns early in the 2010 Haitian cholera outbreak, Am. J. Trop. Med. Hyg., 86 (2012), 39-45.  doi: 10.4269/ajtmh.2012.11-0597.  Google Scholar

[9]

S. Collinson and J. M. Heffernan, Modelling the effects of media during an influenza epidemic, BMC Public Health, 14 (2014), 376. doi: 10.1186/1471-2458-14-376.  Google Scholar

[10]

J. CuiY. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar

[11]

J. CuiX. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mountain Journal of Mathematics, 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323.  Google Scholar

[12]

O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000.  Google Scholar

[13]

K. A. FantiE. VanmanC. C. Henrich and M. N. Avraamide, Desensitization to media violence over a short period of time, Aggressive Behavior, 35 (2009), 179-187.  doi: 10.1002/ab.20295.  Google Scholar

[14]

K. FrostE. Frank and E. Maibach, Relative risk in the news media: A quantification of misrepresentation, American Journal of Public Health, 87 (1997), 842-845.  doi: 10.2105/AJPH.87.5.842.  Google Scholar

[15]

H. Hethcote, The mathematics of infectious diseases, SIAM Revi., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[16]

X. LaiS. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect, Applicable Analysis, 89 (2010), 1271-1292.  doi: 10.1080/00036811.2010.483557.  Google Scholar

[17]

J. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, 1976.  Google Scholar

[18]

J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961.  Google Scholar

[19]

G. M. Leung, T. H. Lam, L. M. Ho, S. Y. Ho, B. H. Chang, I. O. Wong and A. J. Hedley, The impac of community sphychological response on outbreak control for severe acute respiratory syndrome in Hong Kong, J. Epid. Comm. Health 2003, p995. Google Scholar

[20]

S. Levin, T. Hallam and L. Gross, Applied Mathematical Ecology, Lecture Notes in Biomathematics, vol. 18, Springer, Berlin, Germany, 1989. doi: 10.1007/978-3-642-61317-3_1.  Google Scholar

[21]

B. Li, S. Liu, J. Cui and J. Li, A simple predator-prey population model with rich dynamics, App. Sci., 6 (2016), 151. doi: 10.3390/app6050151.  Google Scholar

[22]

R. LinS. Liu and X. Lai, Bifurcations of a Predator-prey System with Weak Allee effects, J. Korean Math. Soc., 50 (2013), 695-713.  doi: 10.4134/JKMS.2013.50.4.695.  Google Scholar

[23]

R. LiuJ. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar

[24]

W. LiuH. Hethcote and S. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

[25]

W. LiuS. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[26]

Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar

[27]

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

[28]

A. K. MisraA. Sharma and J. B. Shukla, Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Math. Comput. Model, 53 (2011), 1221-1228.  doi: 10.1016/j.mcm.2010.12.005.  Google Scholar

[29]

A. K. MisraA. Sharma and V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Sys., 19 (2011), 389-402.  doi: 10.1142/S0218339011004020.  Google Scholar

[30]

A. Mummert and H. Weiss, Get the news out loudly and quickly: The Influence of the media on limiting emerging infectious disease outbreaks, PloS One, 8 (2013), e71692. doi: 10.1371/journal.pone.0071692.  Google Scholar

[31]

J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[32]

C. 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

[33]

J. Tchuenche, N. Dube, C. Bhunu, R. Smith and C. Bauch, The impact of media coverage on the transmission dynamics of human inuenza, BMC Public Health, 11 (2011), S5. Google Scholar

[34]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[35]

A. A. F. Wahlberg and L. Sjoberg, Risk perception and the media, Journal of Risk Research, 3 (2001), 31-50.  doi: 10.1080/136698700376699.  Google Scholar

[36]

Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak, Scientific Reports, 5 (2015), 7838. doi: 10.1038/srep07838.  Google Scholar

[37]

Y. XiaoT. Zhao and S. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. and Eng., 10 (2013), 445-461.  doi: 10.3934/mbe.2013.10.445.  Google Scholar

[38]

M. E. Young, G. R. Norman and K. R. Humphreys, Medicine in the popular press: The Influence of the media on perceptions of disease, PloS One, (3) (2008), e3552. doi: 10.1371/journal.pone.0003552.  Google Scholar

[39]

X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.   Google Scholar

Figure 1.  Effects of media impact $a$ on the value of $S(t), I(t) $ under different media impacts. Here, $\gamma=0.05$ and the initial point are all ($5\times 10^{6}$, 1, 1); $\Re_0=1.1765$ and $R_{H_0}=5.5206$
Figure 2.  Effects of media impact $a$ on the value of $S(t), I(t) $ under different media impacts. Here $\gamma=0.02$, the initial point is ($5\times 10^{6}$, 1, 1); $\Re_0=3$ and $R_{H_0}=8.2523$
Figure 3.  The peak value of the infective number $I_{max}$ when $a$ from 0 to $a=1 \times 10^{-8}$
Table 1.  Endemic equilibrium $ (S^*, E^*, I^*)$ when $a>0$ is varied. In the table, expect for the parameters given in Table 2, here we have $\gamma=0.05.$ In this case, $\Re_0=1.1765 $ and $R_{H_{0}}=5.5206 $
Parameter a $S^*$ $E^*$$I^*$
$a=0$4208333659713194
$a=1 \times 10^{-11}$4215551654813096
$a=1 \times 10^{-10}$4272153615712314
Parameter a $S^*$ $E^*$$I^*$
$a=0$4208333659713194
$a=1 \times 10^{-11}$4215551654813096
$a=1 \times 10^{-10}$4272153615712314
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