COVID-19 propagation and the usefulness of awareness-based control measures: A mathematical model with delay

Chandan Maji; Fahad Al Basir; Debasis Mukherjee; Kottakkaran Sooppy Nisar; Chokkalingam Ravichandran; Chandan Maji; Fahad Al Basir; Debasis Mukherjee; Kottakkaran Sooppy Nisar; Chokkalingam Ravichandran

doi:10.3934/math.2022672

AIMS Mathematics

2022, Volume 7, Issue 7: 12091-12105. doi: 10.3934/math.2022672

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Research article

COVID-19 propagation and the usefulness of awareness-based control measures: A mathematical model with delay

1.
Department of Mathematics, Jagannath Kishore College, Purulia, West Bengal-723101, India
2.
Department of Mathematics, Asansol Girls' College, Asansol-4, West Bengal-713304, India
3.
Department of Mathematics, Vivekananda Collage, Thakurpukur, Kolkata-700063, India
4.
Department of Mathematics, College of Arts & Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser-11991, Saudi Arabia
5.
Department of Mathematics, Kongunadu Arts and Science College (Autonomous), Coimbatore-641029, India

Received: 25 February 2022 Revised: 27 March 2022 Accepted: 30 March 2022 Published: 21 April 2022
MSC : 34C23, 93A30

The current emergence of coronavirus (SARS-CoV-2 or COVID-19) has put the world in threat. Social distancing, quarantine and governmental measures such as lockdowns, social isolation, and public hygiene are helpful in fighting the pandemic, while awareness campaigns through social media (radio, TV, etc.) are essential for their implementation. On this basis, we propose and analyse a mathematical model for the dynamics of COVID-19 transmission influenced by awareness campaigns through social media. A time delay factor due to the reporting of the infected cases has been included in the model for making it more realistic. Existence of equilibria and their stability, and occurrence of Hopf bifurcation have been studied using qualitative theory. We have derived the basic reproduction number ($ R_0 $) which is dependent on the rate of awareness. We have successfully shown that public awareness has a significant role in controlling the pandemic. We have also seen that the time delay destabilizes the system when it crosses a critical value. In sum, this study shows that public awareness in the form of social distancing, lockdowns, testing, etc. can reduce the pandemic with a tolerable time delay.
- mathematical model,
- basic reproduction number,
- stability analysis,
- time delay,
- Hopf bifurcation,
- sensitivity analysis
Citation: Chandan Maji, Fahad Al Basir, Debasis Mukherjee, Kottakkaran Sooppy Nisar, Chokkalingam Ravichandran. COVID-19 propagation and the usefulness of awareness-based control measures: A mathematical model with delay[J]. AIMS Mathematics, 2022, 7(7): 12091-12105. doi: 10.3934/math.2022672

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Abstract

The current emergence of coronavirus (SARS-CoV-2 or COVID-19) has put the world in threat. Social distancing, quarantine and governmental measures such as lockdowns, social isolation, and public hygiene are helpful in fighting the pandemic, while awareness campaigns through social media (radio, TV, etc.) are essential for their implementation. On this basis, we propose and analyse a mathematical model for the dynamics of COVID-19 transmission influenced by awareness campaigns through social media. A time delay factor due to the reporting of the infected cases has been included in the model for making it more realistic. Existence of equilibria and their stability, and occurrence of Hopf bifurcation have been studied using qualitative theory. We have derived the basic reproduction number ($ R_0 $) which is dependent on the rate of awareness. We have successfully shown that public awareness has a significant role in controlling the pandemic. We have also seen that the time delay destabilizes the system when it crosses a critical value. In sum, this study shows that public awareness in the form of social distancing, lockdowns, testing, etc. can reduce the pandemic with a tolerable time delay.

References

[1]	Reported cases and deaths by country or territory, Available from: https://www.worldometers.info/coronavirus
[2]	World Health Organization, WHO Director-General's opening remarks at the media briefing on COVID-19-11 March 2020, Geneva, Switzerland, 2020.
[3]	J. Cai, W. J. Sun, J. P. Huang, M. Gamber, J. Wu, G. Q. He, Indirect virus transmission in cluster of COVID-19 cases, Wenzhou, China, 2020, Emerg. Infect. Dis., 26 (2020), 1343–1345. https://doi.org/10.3201/eid2606.200412 doi: 10.3201/eid2606.200412
[4]	Z. Yong, C. Chen, S. L. Zhu, C. Shu, D. Y. Wang, J. D. Song, et al., Isolation of 2019-nCoV from a stool specimen of a laboratory-confirmed case of the coronavirus disease 2019 (COVID-19), China CDC Weekly, 2 (2020), 123–124. https://doi.org/10.46234/ccdcw2020.033 doi: 10.46234/ccdcw2020.033
[5]	Y. P. Liu, J. A. Cui, The impact of media coverage on the dynamics of infectious diseases. Int. J. Biomath., 1 (2008), 65–74. https://doi.org/10.1142/S1793524508000023 doi: 10.1142/S1793524508000023
[6]	L. R. Zou, F. Ruan, M. X. Huang, L. J. Liang, H. T. Huang, Z. S. Hong, et al., SARS-CoV-2 viral load in upper respiratory specimens of infected patients, N. Engl. J. Med., 382 (2020), 1177–1179. https://doi.org/10.1056/NEJMc2001737 doi: 10.1056/NEJMc2001737
[7]	N. Zhu, D. Y. Zhang, W. L. Wang, X. W. Li, B. Yang, J. D. Song, et al., A novel coronavirus from patients with pneumonia in China, 2019, New Engl. J. Med., 382 (2020), 727–733.
[8]	B. L. Zhong, W. Luo, H. M. Li, Q. Q. Zhang, X. G. Liu, W. T. Li, et al., Knowledge, attitudes, and practices towards COVID-19 among Chinese residents during the rapid rise period of the COVID-19 outbreak: A quick online cross-sectional survey, Int. J. Biol. Sci., 16 (2020), 1745–1752. https://doi.org/10.7150/ijbs.45221 doi: 10.7150/ijbs.45221
[9]	M. Mandal S. Jana, S. K. Nandi, A. Khatua, S. Adak, T. K. Kar, A model based study on the dynamics of COVID-19: Prediction and control, Chaos Soliton. Fract., 136 (2020), 109889. https://doi.org/10.1016/j.chaos.2020.109889 doi: 10.1016/j.chaos.2020.109889
[10]	A. Yousefpour, H. Jahanshahi, S. Bekiros, Optimal policies for control of the novel coronavirus disease (COVID-19) outbreak, Chaos Soliton. Fract., 136 (2020), 109883. https://doi.org/10.1016/j.chaos.2020.109883 doi: 10.1016/j.chaos.2020.109883
[11]	A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, et al., , Early dynamics of transmission and control of COVID-19: A mathematical modelling study, Lancet Infect. Diseases, 20 (2020), 553–558. https://doi.org/10.1016/S1473-3099(20)30144-4 doi: 10.1016/S1473-3099(20)30144-4
[12]	F. Ndariou, I. Area, J. J. Nieto, D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos Soliton. Fract., 135 (2020), 109846. https://doi.org/10.1016/j.chaos.2020.109846 doi: 10.1016/j.chaos.2020.109846
[13]	J. Hellewell, S. Abbott, A. Gimma, N. I. Bosse, C. I. Jarvis, T. W. Russell, et al., Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, Lancet Glob. Health, 8 (2020), e488–e496. https://doi.org/10.1016/S2214-109X(20)30074-7 doi: 10.1016/S2214-109X(20)30074-7
[14]	V. Volpert, M. Banerjee, S. Petrovskii, On a quarantine model of coronavirus infection and data analysis, Math. Model. Nat. Phenom., 15 (2020), 24. https://doi.org/10.1051/mmnp/2020006 doi: 10.1051/mmnp/2020006
[15]	D. Fanelli, F. Piazza, Analysis and forecast of COVID-19 spreading in China, Italy and France, Chaos Soliton. Fract., 134 (2020), 109761. https://doi.org/10.1016/j.chaos.2020.109761 doi: 10.1016/j.chaos.2020.109761
[16]	M. H. D. M. Ribeiro, R. G. da. Silva, V. C. Mariani, L. dos Santos Coelhoa, Short-term forecasting COVID-19 cumu-lative confirmed cases: Perspectives for Brazil, Chaos Soliton. Fract., 135 (2020), 109853. https://doi.org/10.1016/j.chaos.2020.109853 doi: 10.1016/j.chaos.2020.109853
[17]	B. Buonomo, R. D. Marca, Effects of information-induced behavioural changes during the COVID-19 lockdowns: The case of Italy, R. Soc. Open Sci., 7 (2020), 201635. https://doi.org/10.1098/rsos.201635 doi: 10.1098/rsos.201635
[18]	M. S. Mahmud, M. Kamrujjaman, J. Jubyrea, M. S. Islam, M. S. Islam, Quarantine vs social consciousness: A prediction to control COVID-19 infection, J. Appl. Life Sci. Int., 23 (2020), 20–27. https://doi.org/10.9734/jalsi/2020/v23i330150 doi: 10.9734/jalsi/2020/v23i330150
[19]	G. O. Agaba, Y. N. Kyrychko, K. B. Blyuss, Mathematical model for the impact of awareness on the dynamics of infectious diseases, Math. Biosci., 286 (2017), 22–30. https://doi.org/10.1016/j.mbs.2017.01.009 doi: 10.1016/j.mbs.2017.01.009
[20]	F. A. Basir, S. Ray, E. Venturino, Role of media coverage and delay in controlling infectious diseases: A mathematical model, Appl. Math. Comput., 337 (2018), 372–385. https://doi.org/10.1016/j.amc.2018.05.042 doi: 10.1016/j.amc.2018.05.042
[21]	F. A. Basir, Dynamics of infectious diseases with media coverage and two time delay, Math. Models Comput. Simul., 10 (2018), 770–783. https://doi.org/10.1134/S2070048219010071 doi: 10.1134/S2070048219010071
[22]	V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosc., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
[23]	J. G. Cui, Y. H. Sun, H. P. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Eqn., 20 (2008), 31–53. https://doi.org/10.1007/s10884-007-9075-0 doi: 10.1007/s10884-007-9075-0
[24]	C. J. Sun, W. Yang, J. Arino, K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87–95. https://doi.org/10.1016/j.mbs.2011.01.005 doi: 10.1016/j.mbs.2011.01.005
[25]	J. K. Hale, Theory of functional differential equations, New York: Springe, 1977, https://doi.org/10.1007/978-1-4612-9892-2
[26]	M. Bodnar, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91–95. https://doi.org/10.1016/S0893-9659(00)00061-6 doi: 10.1016/S0893-9659(00)00061-6
[27]	X. Yang, L. S. Chen, J. F. Chen, Permanence and positive periodic solution for the single species nonautonomus delay diffusive model, Comput. Math. Appl., 32 (1996), 109–116. https://doi.org/10.1016/0898-1221(96)00129-0 doi: 10.1016/0898-1221(96)00129-0
[28]	J. D. Murray, Mathematical biology, Berlin: Springer Verlag, 1989.
[29]	H. L. Freedman, V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol., 45 (1983), 991–1004. https://doi.org/10.1007/BF02458826 doi: 10.1007/BF02458826
[30]	A. Saltelli, M. Scott, Guest editorial: The role of sensitivity analysis in the corroboration of models and its link to model structural and parametric uncertainty, Reliab. Eng. Syst. Safe., 57 (1997), 1–4.
[31]	A. Saltelli, S. Tarantola, K. S. Chan, A quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39–56. https://doi.org/10.1080/00401706.1999.10485594 doi: 10.1080/00401706.1999.10485594
[32]	W. C. Roda, M. B. Varughese, D. L. Han, M. Y. Li, Why is it difficult to accurately predict the COVID-19 epidemic? Infect. Disease Model., 5 (2020), 271–281. https://doi.org/10.1016/j.idm.2020.03.001 doi: 10.1016/j.idm.2020.03.001

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