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December  2018, 15(6): 1315-1343. doi: 10.3934/mbe.2018061

Modeling the control of infectious diseases: Effects of TV and social media advertisements

1. 

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi - 221005, India

2. 

College of Science and Engineering, Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 252-5258, Japan

* Corresponding author: Arvind Kumar Misra

Received  September 26, 2017 Revised  April 25, 2018 Published  September 2018

Public health information through media plays an important role to curb the spread of various infectious diseases as most of the populations rely on what media projects to them. Social media and TV advertisements are important mediums to communicate people regarding the spread of any infectious disease and methods to prevent its spread. Therefore, in this paper, we propose a mathematical model to see how TV and social media advertisements impact the dynamics of an infectious disease. The susceptible population is assumed vulnerable to infection as well as information (through TV and social media ads). It is also assumed that the growth rate of TV and social media ads is proportional to the number of infected individuals with decreasing function of aware individuals. The feasibility of possible equilibria and their stability properties are discussed. It is shown that the increment in growth rate of TV and social media ads destabilizes the system and periodic oscillations arise through Hopf-bifurcation. It is also found that the increase in dissemination rate of awareness among susceptible population also gives rise interesting dynamics about the stability of endemic equilibrium and causes stability switch. It is observed that TV and social media advertisements regarding the spread of infectious diseases have the potential to bring behavioral changes among the people and control the spread of diseases. Numerical simulations also support analytical findings.

Citation: Arvind Kumar Misra, Rajanish Kumar Rai, Yasuhiro Takeuchi. Modeling the control of infectious diseases: Effects of TV and social media advertisements. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1315-1343. doi: 10.3934/mbe.2018061
References:
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M. O. AdibeJ. M. Okonta and P. O. Udeogaranya, The effects of television and radio commercials on behavior and attitude changes towards the campaign against the spread of HIV/AIDS, Int. J. Drug. Dev. Res., 2 (2010), 975-9344.   Google Scholar

[2]

G. O. AgabaY. N. Kyrychko and K. B. Blyuss, Mathematical model for the impact of awareness on the dynamics of infectious diseases, Math. Biosci., 286 (2017), 22-30.  doi: 10.1016/j.mbs.2017.01.009.  Google Scholar

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F. B. AgustoS. Del ValleK. W. BlaynehC. N. NgonghalaM. J. GoncalvesN. LiR. Zhao and H. Gong, The impact of bed-net use on malaria prevalence, J. Theor. Biol., 320 (2013), 58-65.  doi: 10.1016/j.jtbi.2012.12.007.  Google Scholar

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J. AminielD. Kajunguri and E. A. Mpolya, Mathematical modeling on the spread of awareness information to infant vaccination, Appl. Math., 5 (2015), 101-110.   Google Scholar

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B. BuonomoA. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16.  doi: 10.1016/j.mbs.2008.07.011.  Google Scholar

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

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J. CuiX. Tao and H. 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

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S. Del ValleA. M. EvangelistaM. C. VelascoC. M. Kribs-Zaleta and S. H. Schmitz, Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity, Math. Biosci., 187 (2004), 111-133.  doi: 10.1016/j.mbs.2003.11.004.  Google Scholar

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S. Del ValleH. HethcoteJ. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Math. Biosci., 195 (2005), 228-251.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

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

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B. DubeyP. Dubey and U. S. Dubey, Role of media and treatment on an SIR model, Nonlinear Anal. Model. Control., 21 (2016), 185-200.   Google Scholar

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N. Ferguson, Capturing human behaviour, Nature., 446 (2007), 733. doi: 10.1038/446733a.  Google Scholar

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S. FunkE. GiladC. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proc. Natl. Acad. Sci. USA., 106 (2009), 6872-6877.  doi: 10.1073/pnas.0810762106.  Google Scholar

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S. FunkE. Gilad and V. A. A. Jansen, Endemic disease, awareness, and local behavioural response, J. Theor. Biol., 264 (2010), 501-509.  doi: 10.1016/j.jtbi.2010.02.032.  Google Scholar

[15]

D. GreenhalghS. RanaS. SamantaT. SardarS. Bhattacharya and J. Chattopadhyay, Awareness programs control infectious disease multiple delay induced mathematical model, Appl. Math. Comput., 251 (2015), 539-563.   Google Scholar

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[18]

H. H. Hyman and P. B. Sheatsley, Some reasons why information campaigns fail, Pub. Opin. Quart., 11 (1947), 412-423.   Google Scholar

[19]

T. F. JosephScD LauX. YangH. Y. Tsui and J. H. Kim, Impacts of SARS on health-seeking behaviors in general population in Hong Kong, Prev. Med., 41 (2005), 454-462.   Google Scholar

[20]

H. JoshiS. LenhartK. Albright and K. Gipson, Modeling the effect of information campaigns on the HIV epidemic in Uganda, Math. Biosci. Eng., 5 (2008), 557-570.  doi: 10.3934/mbe.2008.5.757.  Google Scholar

[21]

M. Kim and B. K. Yoo, Cost-effectiveness analysis of a television campaign to promote seasonal influenza vaccination among the elderly, Value in Health, 18 (2015), 622-630.  doi: 10.1016/j.jval.2015.03.1794.  Google Scholar

[22]

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[24]

A. KumarP. K. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevalence, J. Theor. Biol., 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016.  Google Scholar

[25]

V. Lakshmikantham and S. Leela, Differential and Integral Ineualities; Theory and Applications, Acedemic press New Yark and Landan, 1969.  Google Scholar

[26]

R. LiuJ. Wu and H. 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

[27]

Y. Liu and J. Cui, The impact of media convergence on the dynamics of infectious diseases, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar

[28]

X. LuS. WangS. Liu and J. Li, An SEI infection model incorporating media impact, Math. Biosci. Eng., 14 (2017), 1317-1335.  doi: 10.3934/mbe.2017068.  Google Scholar

[29]

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

[30]

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

[31]

A. K. MisraA. Sharma and J. B. Shukla, Stability analysis and optimal control of an epidemic model with awareness programs by media, BioSystems, 138 (2015), 53-62.  doi: 10.1016/j.biosystems.2015.11.002.  Google Scholar

[32]

A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, Int. J. Biomath, 11 (2018), 1850027, 20 pp. doi: 10.1142/S1793524518500274.  Google Scholar

[33]

C. N. NgonghalaS. Del ValleR. Zhao and J. M. Awel, Quantifying the impact of decay in bed-net efficacy on malaria transmission, J. Theor. Biol., 363 (2014), 247-261.  doi: 10.1016/j.jtbi.2014.08.018.  Google Scholar

[34]

F. NyabadzaC. ChiyakaZ. Mukandavire and S. D. Hove-Musekwa, Analysis of an HIV/AIDS model with public-health information campaigns and individual withdrawal, J. Biol. Syst., 18 (2010), 357-375.  doi: 10.1142/S0218339010003329.  Google Scholar

[35]

P. PolettiB. CaprileM. AjelliA. Pugliese and S. Merler, Spontaneous behavioural changes in response to epidemics, J. Theor. Biol., 260 (2009), 31-40.  doi: 10.1016/j.jtbi.2009.04.029.  Google Scholar

[36]

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

[37]

S. SamantaS. RanaA. SharmaA. K. Misra and J. Chattopadhyay, Effect of awareness programs by media on the epidemic outbreaks: A mathematical model, Appl. Math. Comput., 219 (2013), 6965-6977.  doi: 10.1016/j.amc.2013.01.009.  Google Scholar

[38]

S. Samanta and J. Chattopadhyay, Effect of awareness program in disease outbreak-A slowast dynamics, Appl. Math. Comput., 237 (2014), 98-109.  doi: 10.1016/j.amc.2014.03.109.  Google Scholar

[39]

A. Sharma and A. K. Misra, Modeling the impact of awareness created by media campaigns on vacination coverage in a variable population, J. biol. syst., 22 (2014), 249-270.  doi: 10.1142/S0218339014400051.  Google Scholar

[40]

Statista, Number of mobile phone users in India from 2013 to 2019, statista.com, https://www.statista.com/statistics/274658/forecast-of-mobile-phone-users-in-india/. Google Scholar

[41]

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

[42]

J. TchuencheN. DubeC. Bhunu and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), 1-5.   Google Scholar

[43]

J. Tchuenche and C. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomath., 2012 (2012), Article ID 581274, 10 pages. doi: 10.5402/2012/581274.  Google Scholar

[44]

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

show all references

References:
[1]

M. O. AdibeJ. M. Okonta and P. O. Udeogaranya, The effects of television and radio commercials on behavior and attitude changes towards the campaign against the spread of HIV/AIDS, Int. J. Drug. Dev. Res., 2 (2010), 975-9344.   Google Scholar

[2]

G. O. AgabaY. N. Kyrychko and K. B. Blyuss, Mathematical model for the impact of awareness on the dynamics of infectious diseases, Math. Biosci., 286 (2017), 22-30.  doi: 10.1016/j.mbs.2017.01.009.  Google Scholar

[3]

F. B. AgustoS. Del ValleK. W. BlaynehC. N. NgonghalaM. J. GoncalvesN. LiR. Zhao and H. Gong, The impact of bed-net use on malaria prevalence, J. Theor. Biol., 320 (2013), 58-65.  doi: 10.1016/j.jtbi.2012.12.007.  Google Scholar

[4]

J. AminielD. Kajunguri and E. A. Mpolya, Mathematical modeling on the spread of awareness information to infant vaccination, Appl. Math., 5 (2015), 101-110.   Google Scholar

[5]

B. BuonomoA. d'Onofrio and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16.  doi: 10.1016/j.mbs.2008.07.011.  Google Scholar

[6]

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

[7]

J. CuiX. Tao and H. 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]

S. Del ValleA. M. EvangelistaM. C. VelascoC. M. Kribs-Zaleta and S. H. Schmitz, Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity, Math. Biosci., 187 (2004), 111-133.  doi: 10.1016/j.mbs.2003.11.004.  Google Scholar

[9]

S. Del ValleH. HethcoteJ. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Math. Biosci., 195 (2005), 228-251.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

[10]

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

[11]

B. DubeyP. Dubey and U. S. Dubey, Role of media and treatment on an SIR model, Nonlinear Anal. Model. Control., 21 (2016), 185-200.   Google Scholar

[12]

N. Ferguson, Capturing human behaviour, Nature., 446 (2007), 733. doi: 10.1038/446733a.  Google Scholar

[13]

S. FunkE. GiladC. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proc. Natl. Acad. Sci. USA., 106 (2009), 6872-6877.  doi: 10.1073/pnas.0810762106.  Google Scholar

[14]

S. FunkE. Gilad and V. A. A. Jansen, Endemic disease, awareness, and local behavioural response, J. Theor. Biol., 264 (2010), 501-509.  doi: 10.1016/j.jtbi.2010.02.032.  Google Scholar

[15]

D. GreenhalghS. RanaS. SamantaT. SardarS. Bhattacharya and J. Chattopadhyay, Awareness programs control infectious disease multiple delay induced mathematical model, Appl. Math. Comput., 251 (2015), 539-563.   Google Scholar

[16]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf-bifurcation, Cambridge University Press, Cambridge, 1981.  Google Scholar

[17]

H. HethcoteZ. Ma and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.  Google Scholar

[18]

H. H. Hyman and P. B. Sheatsley, Some reasons why information campaigns fail, Pub. Opin. Quart., 11 (1947), 412-423.   Google Scholar

[19]

T. F. JosephScD LauX. YangH. Y. Tsui and J. H. Kim, Impacts of SARS on health-seeking behaviors in general population in Hong Kong, Prev. Med., 41 (2005), 454-462.   Google Scholar

[20]

H. JoshiS. LenhartK. Albright and K. Gipson, Modeling the effect of information campaigns on the HIV epidemic in Uganda, Math. Biosci. Eng., 5 (2008), 557-570.  doi: 10.3934/mbe.2008.5.757.  Google Scholar

[21]

M. Kim and B. K. Yoo, Cost-effectiveness analysis of a television campaign to promote seasonal influenza vaccination among the elderly, Value in Health, 18 (2015), 622-630.  doi: 10.1016/j.jval.2015.03.1794.  Google Scholar

[22]

I. Z. KissJ. CassellM. Recker and P. L. Simon, The impact of information transmission on epidemic outbreaks, Math. Biosci., 255 (2010), 1-10.  doi: 10.1016/j.mbs.2009.11.009.  Google Scholar

[23]

I. S. KristiansenP. A. Halvorsen and D. G. Hansen, Influenza pandemic: Perception of risk and individual precautions in a general population: Cross sectional study, BMC Public Health., 7 (2007), 48-54.  doi: 10.1186/1471-2458-7-48.  Google Scholar

[24]

A. KumarP. K. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevalence, J. Theor. Biol., 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016.  Google Scholar

[25]

V. Lakshmikantham and S. Leela, Differential and Integral Ineualities; Theory and Applications, Acedemic press New Yark and Landan, 1969.  Google Scholar

[26]

R. LiuJ. Wu and H. 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

[27]

Y. Liu and J. Cui, The impact of media convergence on the dynamics of infectious diseases, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar

[28]

X. LuS. WangS. Liu and J. Li, An SEI infection model incorporating media impact, Math. Biosci. Eng., 14 (2017), 1317-1335.  doi: 10.3934/mbe.2017068.  Google Scholar

[29]

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

[30]

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

[31]

A. K. MisraA. Sharma and J. B. Shukla, Stability analysis and optimal control of an epidemic model with awareness programs by media, BioSystems, 138 (2015), 53-62.  doi: 10.1016/j.biosystems.2015.11.002.  Google Scholar

[32]

A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, Int. J. Biomath, 11 (2018), 1850027, 20 pp. doi: 10.1142/S1793524518500274.  Google Scholar

[33]

C. N. NgonghalaS. Del ValleR. Zhao and J. M. Awel, Quantifying the impact of decay in bed-net efficacy on malaria transmission, J. Theor. Biol., 363 (2014), 247-261.  doi: 10.1016/j.jtbi.2014.08.018.  Google Scholar

[34]

F. NyabadzaC. ChiyakaZ. Mukandavire and S. D. Hove-Musekwa, Analysis of an HIV/AIDS model with public-health information campaigns and individual withdrawal, J. Biol. Syst., 18 (2010), 357-375.  doi: 10.1142/S0218339010003329.  Google Scholar

[35]

P. PolettiB. CaprileM. AjelliA. Pugliese and S. Merler, Spontaneous behavioural changes in response to epidemics, J. Theor. Biol., 260 (2009), 31-40.  doi: 10.1016/j.jtbi.2009.04.029.  Google Scholar

[36]

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

[37]

S. SamantaS. RanaA. SharmaA. K. Misra and J. Chattopadhyay, Effect of awareness programs by media on the epidemic outbreaks: A mathematical model, Appl. Math. Comput., 219 (2013), 6965-6977.  doi: 10.1016/j.amc.2013.01.009.  Google Scholar

[38]

S. Samanta and J. Chattopadhyay, Effect of awareness program in disease outbreak-A slowast dynamics, Appl. Math. Comput., 237 (2014), 98-109.  doi: 10.1016/j.amc.2014.03.109.  Google Scholar

[39]

A. Sharma and A. K. Misra, Modeling the impact of awareness created by media campaigns on vacination coverage in a variable population, J. biol. syst., 22 (2014), 249-270.  doi: 10.1142/S0218339014400051.  Google Scholar

[40]

Statista, Number of mobile phone users in India from 2013 to 2019, statista.com, https://www.statista.com/statistics/274658/forecast-of-mobile-phone-users-in-india/. Google Scholar

[41]

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

[42]

J. TchuencheN. DubeC. Bhunu and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), 1-5.   Google Scholar

[43]

J. Tchuenche and C. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomath., 2012 (2012), Article ID 581274, 10 pages. doi: 10.5402/2012/581274.  Google Scholar

[44]

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

Figure 1.  Effect of changing the values of $\lambda$ and $M_0$ on $R_1$
Figure 2.  Non-linear stability behavior of model system (2) in $I-A-M$ space for $\lambda = 0.00003$, keeping rest of parameter values same as given in Table 1, which shows that all solution trajectory attains their equilibrium $E^*$ inside the region of attraction $\Omega$
Figure 3.  Variation of $S(t)$, $I(t)$, $A(t)$ and $M(t)$ with respect to time $t$ for $r = 0.005$, which shows that the equilibrium $E^*$ is stable and we have damped oscillation
Figure 4.  Phase portrait of model system (2) for $r = 0.005$ in $I-A-M$ space, which shows that the equilibrium $E^*$ is stable
Figure 5.  Variation of $S(t)$, $I(t)$, $A(t)$ and $M(t)$ with respect to time $t$ for $r = 0.011$, which shows that the equilibrium $E^*$ is unstable and we have undamped sustained oscillation
Figure 6.  Appearance of limit cycle of model system (2) for $r = 0.011$ in $I-A-M$ space, which shows that the equilibrium $E^*$ is unstable
Figure 7.  Bifurcation diagram of infected population $I(t)$, aware population $A(t)$ and the cumulative number of TV and social media ads $M(t)$ with respect to $r$, keeping rest of parameters same as given in Table 1
Figure 8.  Bifurcation diagram of infected population $I(t)$, aware population $A(t)$ and the cumulative number of TV and social media ads $M(t)$ with respect to $\lambda$ for $r = 0.05$, $\omega = 60$, keeping rest of parameters same as given in Table 1
Table 1.  Parameter values for the model system (2)
ParameterValuesParameterValues
$\Lambda$ $5$ ${\rm{day}}^{-1}$ $\beta$ $0.0000030$ ${\rm{day}}^{-1}$
$\lambda$ $0.012$ ${\rm{day}}^{-1}$ $\lambda_0$ $0.008$ ${\rm{day}}^{-1}$
$\nu$ $0.2$ ${\rm{day}}^{-1}$ $\alpha$ $0.00001$ ${\rm{day}}^{-1}$
$d$ $0.00004$ ${\rm{day}}^{-1}$ $r$ $0.006$ ${\rm{day}}^{-1}$
$r_0$ $0.005$ ${\rm{day}}^{-1}$ $\theta$ $0.0005$
$\omega$ $6000$ $p$ $1200$
$M_0$ $500$
ParameterValuesParameterValues
$\Lambda$ $5$ ${\rm{day}}^{-1}$ $\beta$ $0.0000030$ ${\rm{day}}^{-1}$
$\lambda$ $0.012$ ${\rm{day}}^{-1}$ $\lambda_0$ $0.008$ ${\rm{day}}^{-1}$
$\nu$ $0.2$ ${\rm{day}}^{-1}$ $\alpha$ $0.00001$ ${\rm{day}}^{-1}$
$d$ $0.00004$ ${\rm{day}}^{-1}$ $r$ $0.006$ ${\rm{day}}^{-1}$
$r_0$ $0.005$ ${\rm{day}}^{-1}$ $\theta$ $0.0005$
$\omega$ $6000$ $p$ $1200$
$M_0$ $500$
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