Global dynamics for a Filippov system with media effects

Cunjuan Dong; Changcheng Xiang; Wenjin Qin; Yi Yang; Cunjuan Dong; Changcheng Xiang; Wenjin Qin; Yi Yang

doi:10.3934/mbe.2022130

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 3: 2835-2852. doi: 10.3934/mbe.2022130

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Research article

Global dynamics for a Filippov system with media effects

1.
School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China
2.
School of Mathematics and Computer Science, Yunnan Minzu University, Yunnan 650031, China
3.
College of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing 404020, China

Academic Editor: Yang Kuang

Received: 14 November 2021 Revised: 16 December 2021 Accepted: 26 December 2021 Published: 13 January 2022

In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.
- SIR epidemic model,
- media coverage,
- filippov system,
- stability
Citation: Cunjuan Dong, Changcheng Xiang, Wenjin Qin, Yi Yang. Global dynamics for a Filippov system with media effects[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2835-2852. doi: 10.3934/mbe.2022130

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Abstract

In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.

References

[1]	K. I. Bos, V. J. Schuenemann, G. B. Golding, A draft genome of yersinia pestis from victims of the black death, Nature, 478 (2011), 506–510. https://doi.org/10.1038/nature10549 doi: 10.1038/nature10549
[2]	Z. B. Zhang, The outbreak pattern of SARS cases in China as revealed by a mathematical model, Ecol. Model., 204 (2007), 420–426. https://doi.org/10.1016/j.ecolmodel.2007.01.020 doi: 10.1016/j.ecolmodel.2007.01.020
[3]	L. Hailong, R. X. Yu, L. Shuang, Analysis of the efficiency of the preventing and isolating treatments of SARS based on mathematical model, Int. J. Biomath., 19 (2004), 72–76. https://doi.org/10.2116/analsci.20.717 doi: 10.2116/analsci.20.717
[4]	X. S. Zhang, E. Vynnycky, A. Charlett, Transmission dynamics and control measures of COVID-19 outbreak in China: a modelling study, Sci. Rep., 11 (2021), 1–12. https://doi.org/10.1038/s41598-021-81985-z doi: 10.1038/s41598-021-81985-z
[5]	J. W. Deng, S. Y. Tang, H. Y. Shu, Joint impacts of media, vaccination and treatment on an epidemic Filippov model with application to COVID-19, J. Theor. Biol., 523 (2021), 110698. https://doi.org/10.1016/j.jtbi.2021.110698 doi: 10.1016/j.jtbi.2021.110698
[6]	S. He, S. Y. Tang, L. B. Rong, A discrete stochastic model of the COVID-19 outbreak: forecast and control, Math. Biosci. Eng., 17 (2020), 2792–2804. https://doi.org/10.3934/mbe.2020153 doi: 10.3934/mbe.2020153
[7]	A. Ibeas, M. D. L. Sen, S. A. Quesada, Robust sliding control of SEIR epidemic models, Math. Biosci. Eng., 2014 (2014), 11. https://doi.org/10.1155/2014/104764 doi: 10.1155/2014/104764
[8]	M. Sharifi, H. Moradi, Nonlinear robust adaptive sliding mode control of influenza epidemic in the presence of uncertainty, J. Process. Contr., 56 (2017), 48–57. https://doi.org/10.1016/j.jprocont.2017.05.010 doi: 10.1016/j.jprocont.2017.05.010
[9]	A. Wang, Y. Xiao, Sliding bifurcation and global dynamics of a filippov epidemic model with vaccination, Int. J. Bifurcat. Chaos, 23 (2013), 1350144. https://doi.org/10.1142/S0218127413501447 doi: 10.1142/S0218127413501447
[10]	J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith, C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), 1–16. https://doi.org/10.1186/1471-2458-11-S1-S5 doi: 10.1186/1471-2458-11-S1-S5
[11]	J. M. Tchuenche, C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission, Int. Schol. Res. Not., 2012 (2012), 1–10. https://doi.org/10.5402/2012/581274 doi: 10.5402/2012/581274
[12]	J. G. Cui, Y. H. Sun, H. P. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equations, 20 (2008), 31–53. https://doi.org/10.1007/s10884-007-9075-0 doi: 10.1007/s10884-007-9075-0
[13]	Y. Liu, J. A. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65–74. https://doi.org/10.1142/S1793524508000023 doi: 10.1142/S1793524508000023
[14]	Y. N. Xiao, X. X. Xu, S. Y. Tang, Sliding mode control of outbreaks of emerging infectious diseases, B. Math. Biol., 74 (2012), 2403–2422. https://doi.org/10.1007/s11538-012-9758-5 doi: 10.1007/s11538-012-9758-5
[15]	Y. N. Xiao, S. Y. Tang, J. H. Wu, Media impact switching surface during an infectious disease outbreak, Sci. Rep., 5 (2015), 1–9. https://doi.org/10.1038/srep07838 doi: 10.1038/srep07838
[16]	Y. H. Zhang, Y. N. Xiao, Global dynamics for a filippov epidemic system with imperfect vaccination, Nonlinear Anal. Hybri., 38 (2020), 100932. https://doi.org/10.1016/j.nahs.2020.100932 doi: 10.1016/j.nahs.2020.100932
[17]	Y. H. Zhang, P. F. Song, Dynamics of the piecewise smooth epidemic model with nonlinear incidence, Chaos Soliton. Fract., 146 (2020), 110903. https://doi.org/10.1016/j.chaos.2021.110903 doi: 10.1016/j.chaos.2021.110903
[18]	Y. Yang, X. F. Liao, Filippov hindmarsh-rose neuronal model with threshold policy control, IEEE T. Neur. Net. Lear., 30 (2019), 306–311. https://doi.org/10.1109/TNNLS.2018.2836386 doi: 10.1109/TNNLS.2018.2836386
[19]	T. Carvalho, L. F. Gonçalves, Combing the hairy ball using a vector field without equilibria, J. Dyn. Control Syst., 26 (2020), 233–242. https://doi.org/10.1007/s10883-019-09446-5 doi: 10.1007/s10883-019-09446-5
[20]	D. C. Vicentin, P. F. A. Mancera, T. Carvalho, Mathematical model of an antiretroviral therapy to HIV via Filippov theory, Appl. Math. Comput., 387 (2020), 125179. https://doi.10.1016/j.amc.2020.125179
[21]	M. D. Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, Bifurcations in nonsmooth dynamical systems, Siam. Rev., 50 (2008), 629–701. https://doi.10.1137/050625060
[22]	M. Guardia, T. M. Seara, M. A. Teixeira, Generic bifurcations of low codimension of planar filippov systems, J. Differ. Equations, 250 (2011), 1967–2023. https://doi.10.1016/j.jde.2010.11.016
[23]	W. Qin, S. Tang, The selection pressures induced non-smooth infectious disease model and bifurcation analysis, Chaos Solition. Fract., 69 (2014), 160–171. https://doi.10.1016/j.chaos.2014.09.014
[24]	A. Wang, Y. Xiao, R. A. Cheke, Global dynamics of a piece-wise epidemic model with switching vaccination strategy, Discrete. Cont. Dyn.-B., 19 (2014), 2915–2940. https://doi.10.3934/dcdsb.2014.19.2915
[25]	S. Tang, Y. Xiao, N. Wang, H. Wu, Piecewise HIV virus dynamic model with CD4(+) T cell count-guided therapy: I, J. Theor. Biol., 308 (2012), 123–134. https://doi.org/10.1016/j.jtbi.2012.05.022 doi: 10.1016/j.jtbi.2012.05.022
[26]	P. V. D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[27]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[28]	A. F. Filippov, Differential equations with discontinuous righthand sides, J. Math. Anal. Appl., 154 (1991), 377–390. https://doi.org/10.1016/0022-247X(91)90044-Z doi: 10.1016/0022-247X(91)90044-Z
[29]	V. Utkin, J. Guldner, J. X. Shi, Sliding mode control in electro-mechanical systems, 2$^{nd}$ edition, CRC Press, Boca Raton, 2009. https://doi.org/10.1201/9781420065619
[30]	Y. A. Kuznetsov, S. Rinaldi, A. Gragnani, Non-smooth ecological systems with a switching threshold depending on the pest density and its rate of change, Nonlinear Anal. Hybri., 42 (2021), 101094. https://doi.org/10.1016/j.nahs.2021.101094 doi: 10.1016/j.nahs.2021.101094
[31]	Y. A. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar filippov systems, Int. J. Bifurcat. Chaos, 13 (2003), 2157–2188. https://doi.org/10.1142/S0218127403007874 doi: 10.1142/S0218127403007874
[32]	A. A. Arafa, S. A. A. Hamdallah, S. Tang, Dynamics analysis of a filippov pest control model with time delay, Commun. Nonlinear Sci., 101 (2021), 105865. https://doi.org/10.1016/j.cnsns.2021.105865 doi: 10.1016/j.cnsns.2021.105865

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