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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:
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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/.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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/.

[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.

[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.

[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.

[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.

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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