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Dynamics of a non-smooth epidemic model with three thresholds

Theory in Biosciences volume 139, pages 47–65 (2020)Cite this article

Abstract

A non-smooth epidemic model with piecewise incidence rate dependent on the derivative of the case number is proposed for the transmission dynamics of an infectious disease with media coverage, enhanced vaccination and treatment policy. This is an implicitly defined system, which is converted into an explicit system with three thresholds by employing the properties of the Lambert W function. We first analyze the dynamics of the proposed model for the limiting case, which induces two non-smooth but continuous models. The dynamic analysis of the model demonstrates that either one of the two generalized equilibria or the pseudo-equilibrium is globally asymptotically stable if the disease does not die out. This suggests that the case number can be contained either at an a priori level or at a high/low level, depending on the threshold, which governs whether the enhanced vaccination and treatment policies are implemented. Media coverage cannot help eradicate the disease, but it significantly delays the epidemic peak and lowers the peak case number. Hence, a good threshold policy and continuously updating the awareness of case numbers are required to combat the disease successfully.

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References

  • Abdelrazec A, Bélair J, Shan C et al (2016) Modeling the spread and control of dengue with limited public health resources. Math Biosci 271:136–145

    Article  Google Scholar 

  • Al Basir F, Ray S, Venturino E (2018) Role of media coverage and delay in controlling infectious diseases: a mathematical model. Appl Math Comput 337:372–385

    Google Scholar 

  • Berrhazi B, El Fatini M, Laaribi A et al (2017) A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise. Chaos Solitons Fractals 105:60–68

    Article  Google Scholar 

  • Brewer NT, Chapman GB, Gibbons FX et al (2007) Meta-analysis of the relationship between risk perception and health behavior: the example of vaccination. Health Psychol 26(2):136

    Article  Google Scholar 

  • Capasso V, Serio G (1978) A generalization of the Kermack–McKendrick deterministic epidemic model. Math Biosci 42(1–2):0-61

    Google Scholar 

  • Chen C, Chong NS, Smith R (2018) A Filippov model describing the effects of media coverage and quarantine on the spread of human influenza. Math Biosci 296:98–112

  • Chong NS, Dionne B, Smith R (2016) An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza. J Math Biol 73:751–784

  • Clarke F, Ledyaev Y, Stern R, Wolenski P (1998) Nonsmooth analysis and control theory. Springer, New York

    Google Scholar 

  • Claudio AB, Paulo PDS, Marco AT (2006) A singular approach to discontinuous vector fields on the plane. J Differ Equ 231:633–655

    Article  Google Scholar 

  • Corless RM, Gonnet GH, Hare DE et al (1996) On the Lambert \(W\) function. Adv Comput Math 5(1):329–359

    Article  Google Scholar 

  • Cui JA, Sun YH, Zhu HP (2008a) The impact of media on the control of infectious diseases. J Dyn Differ Equ 20:31–53

    Article  Google Scholar 

  • Cui JA, Tao X, Zhu HP (2008b) An SIS infection model incorporating media coverage. Rocky Mt J Math 38(5):1323–1334

    Article  Google Scholar 

  • Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  • Funk S, Gilad E, Watkins C, Jansen VAA (2010) Modelling the influence of human behaviour on the spread of infectious diseases: a review. J R Soc Interface 7:1247–1256

    Article  PubMed  PubMed Central  Google Scholar 

  • Hörmander L (1990) The analysis of linear partial differential operators, I. Springer, Berlin

    Google Scholar 

  • Jiang J, Zhou T (2018) Resource control of epidemic spreading through a multilayer network. Sci Rep 8(1):1629

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Jones JH, Salathe M (2009) Early assessment of anxiety and behavioral response to novel swine-origin influenza A (H1N1). PLoS ONE 4(12):e8032

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Kabineh AK, Carr W, Motevalli M et al (2018) Operationalizing international regulatory standards in a limited-resource setting during an epidemic: the Sierra Leone trial to introduce a vaccine against Ebola (STRIVE) experience. J Infect Dis 217(suppl 1):S56–S59

    Article  PubMed  PubMed Central  Google Scholar 

  • Khan MA, Islam S, Zaman G (2018) Media coverage campaign in Hepatitis B transmission model. Appl Math Comput 331:378–393

    Google Scholar 

  • Kristiansen IS, Halvorsen PA, Gyrd-Hansen D (2007) Influenza pandemic: perception of risk and individual precautions in a general population. Cross sectional study. BMC Public Health 7(1):48

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Leine RI (2006) Bifurcations of equilibria in non-smooth continuous systems. Phys D 223:121–137

    Article  Google Scholar 

  • Li YF, Cui JA (2009) The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage. Commun Nonlinear Sci Numer Simul 14:2353–2365

    Article  Google Scholar 

  • Liu RS, Wu JH, Zhu HP (2007) Media/psychological impact on multiple outbreaks of emerging infectious diseases. Comput Math Methods Med 8(3):153–164

    Article  CAS  Google Scholar 

  • Melin J (2004) Does distribution theory contain means for extending Poincar\(\acute{e}\)–Bendixson theory? J Math Anal Appl 303:81–89

    Article  Google Scholar 

  • Misra AK, Rai RK, Takeuchi Y (2018) Modeling the control of infectious diseases: effects of TV and social media advertisements. Math Biosci Eng 15(6):1315–1343

    Article  PubMed  PubMed Central  Google Scholar 

  • Qin WJ, Tang SY, Xiang CC et al (2016) Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure. Appl Math Comput 283(C):339–354

    Google Scholar 

  • Rahman MS, Rahman ML (2007) Media and education play a tremendous role in mounting AIDS awareness among married couples in Bangladesh. AIDS Res Ther 4(1):10

    Article  PubMed  PubMed Central  Google Scholar 

  • Sahu GP, Dhar J (2015) Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity. J Math Anal Appl 421(2):1651–1672

    Article  Google Scholar 

  • Song PF, Xiao YN (2018) Global Hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease. J Math Biol 76(5):1249–1267

    Article  PubMed  PubMed Central  Google Scholar 

  • Sun CJ, Yang W, Arino J et al (2011) Effect of media-induced social distancing on disease transmission in a two patch setting. Math Biosci 230(2):87–95

    Article  Google Scholar 

  • Tang SY, Liang JH, Xiao YN et al (2012) Sliding bifurcations of Filippov two stage pest control models with economic thresholds. SIAM J Appl Math 72(72):1061–1080

    Article  Google Scholar 

  • Tang B, Xiao YN, Wu JH (2016) A piecewise model of virus-immune system with two thresholds. Math Biosci 278:63–76

    Article  CAS  Google Scholar 

  • Tchuenche JM, Bauch CT (2012) Dynamics of an infectious disease where media coverage influences transmission. ISRN Biomath 2012:581274

    Article  Google Scholar 

  • Tchuenche JM, Dube N, Bhunu CP et al (2011) The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health 11(Suppl 1):S5

    Article  PubMed  PubMed Central  Google Scholar 

  • Tracy CS, Rea E, Upshru REG (2009) Public perceptions of quarantine: community-based telephone survey following an infectious disease outbreak. BMC Public Health 9(1):470

    Article  PubMed  PubMed Central  Google Scholar 

  • Utkin VI (1992) Sliding modes in control and optimization. Springer, Berlin

    Book  Google Scholar 

  • Wandeler G, Coffie PA, Kuniholm MH et al (2018) Issues with measuring hepatitis prevalence in resource-limited settings. Lancet 391(10123):835–836

    Article  PubMed  PubMed Central  Google Scholar 

  • Wang WD (2006) Backward bifurcation of an epidemic model with treatment. Math Biosci 201(1–2):58–71

    Article  PubMed  PubMed Central  Google Scholar 

  • Wang AL, Xiao YN (2013) Sliding bifurcation and global dynamics of a Filippov epidemic model with vaccination. Int J Bifurc Chaos 23(8):1350144

    Article  Google Scholar 

  • Wang AL, Xiao YN (2014) A Filippov system describing media effects on the spread of infectious diseases. Nonlinear Anal Hybrid 11:84–97

    Article  Google Scholar 

  • Wang LW, Liu ZJ, Zhang XA (2016) Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination. Nonlinear Anal Real World Appl 32:136–158

    Article  CAS  Google Scholar 

  • Wang AL, Xiao YN, Smith R (2019) Multiple equilibria in a non-smooth epidemic model with medical-resource constraints. Bull Math Biol 81(4):963–994

  • Wang AL, Xiao YN, Zhu HP (2018) Dynamics of a Filippov epidemic model with limited hospital beds. Math Biosci Eng 15(3):739–764

    Article  PubMed  PubMed Central  Google Scholar 

  • Xiao YN, Xu XX, Tang SY (2012) Sliding mode control of outbreaks of emerging infectious diseases. Bull Math Biol 74:2403–2422

    Article  PubMed  PubMed Central  Google Scholar 

  • Xiao YN, Zhao TT, Tang SY (2013) Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Math Biosci Eng 10(2):445–461

    Article  PubMed  PubMed Central  Google Scholar 

  • Xiao Y, Tang S, Wu J (2015) Media impact switching surface during an infectious disease outbreak. Sci Rep 5(4):7838

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Zhou LH, Fan M (2012) Dynamics of an SIR epidemic model with limited medical resources revisited. Nonlinear Anal Real World Appl 13(1):312–324

    Article  Google Scholar 

  • Zhou WK, Xiao YN, Cheke RA (2016) A threshold policy to interrupt transmission of West Nile Virus to birds. Appl Math Model 40(19–20):8794–8809

    Article  Google Scholar 

Download references

Acknowledgements

AW was supported by the National Natural Science Foundation of China (NSFC, 11801013) and the funding from Baoji University of Arts and Sciences (ZK1048). YX was supported by the National Natural Science Foundation of China (NSFC, 11571273 and 11631012) and Fundamental Research Funds for the Central Universities (GK 08143042). RS? was supported by an Discovery Grant. For citation purposes, note that the question mark in “Smith?” is part of his name.

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Affiliations

  1. School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic of China

    Aili Wang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yanni Xiao

  3. Department of Mathematics and Faculty of Medicine, The University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Robert Smith

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  1. Aili Wang
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  2. Yanni Xiao
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  3. Robert Smith
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Correspondence to Robert Smith.

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Wang, A., Xiao, Y. & Smith, R. Dynamics of a non-smooth epidemic model with three thresholds. Theory Biosci. 139, 47–65 (2020). https://doi.org/10.1007/s12064-019-00297-z

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  • DOI: https://doi.org/10.1007/s12064-019-00297-z

Keywords

  • Media coverage
  • Multiple thresholds
  • Non-smooth model
  • Sliding dynamics
  • Global dynamics