# American Institute of Mathematical Sciences

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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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. [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. [8] R. Chunara, J. 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. [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. [10] J. Cui, Y. 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. [11] J. Cui, X. Tao and H. 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[17] J. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, 1976. [18] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961. [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. [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. [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. [22] R. Lin, S. 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. [23] R. Liu, J. 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. [24] W. Liu, H. Hethcote and S. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. [25] W. Liu, S. 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. [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. [27] A. K. Misra, A. 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. [28] A. K. Misra, A. 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. [29] A. K. Misra, A. 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. [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. [31] J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [32] C. Sun, W. Yang, J. 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. [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. [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. [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. [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. [37] Y. Xiao, T. 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. [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. [39] X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.

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##### 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. [2] R. J. Blendon, J. M. Benson, C. M. DesRoches, E. 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. [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. [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. [5] S. Busenberg and K. Cooke, Vertically Transmitted Diseases, Springer-Verlag, New York, 1993. doi: 10.1007/978-3-642-75301-5. [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. [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. [8] R. Chunara, J. 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. [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. [10] J. Cui, Y. 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. [11] J. Cui, X. 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. [12] O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000. [13] K. A. Fanti, E. Vanman, C. 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. [14] K. Frost, E. 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. [15] H. Hethcote, The mathematics of infectious diseases, SIAM Revi., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [16] X. Lai, S. 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. [17] J. LaSalle, The Stability of Dynamical Systems, Reg. Conf. Ser. Appl. Math., SIAM, Philadelphia, 1976. [18] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961. [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. [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. [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. [22] R. Lin, S. 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. [23] R. Liu, J. 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. [24] W. Liu, H. Hethcote and S. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. [25] W. Liu, S. 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. [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. [27] A. K. Misra, A. 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. [28] A. K. Misra, A. 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. [29] A. K. Misra, A. 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. [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. [31] J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [32] C. Sun, W. Yang, J. 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. [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. [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. [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. [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. [37] Y. Xiao, T. 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. [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. [39] X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.
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 $R_{H_0}=5.5206$
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 $R_{H_0}=8.2523$
The peak value of the infective number $I_{max}$ when $a$ from 0 to $a=1 \times 10^{-8}$
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 $S^*$ $E^*$ $I^*$ $a=0$ 4208333 6597 13194 $a=1 \times 10^{-11}$ 4215551 6548 13096 $a=1 \times 10^{-10}$ 4272153 6157 12314
 Parameter a $S^*$ $E^*$ $I^*$ $a=0$ 4208333 6597 13194 $a=1 \times 10^{-11}$ 4215551 6548 13096 $a=1 \times 10^{-10}$ 4272153 6157 12314
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