# American Institute of Mathematical Sciences

2013, 10(2): 445-461. doi: 10.3934/mbe.2013.10.445

## Dynamics of an infectious diseases with media/psychology induced non-smooth incidence

 1 Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049 2 Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049, China 3 College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China

Received  May 2012 Revised  October 2012 Published  January 2013

This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).
Citation: Yanni Xiao, Tingting Zhao, Sanyi Tang. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Mathematical Biosciences & Engineering, 2013, 10 (2) : 445-461. doi: 10.3934/mbe.2013.10.445
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control," Oxford University, Oxford, 1991. [2] J. Aubin and A. Cellina, "Differential Inclusion," Springer-Verlag, Berlin, 1984. [3] A. Banaszuk, On the existence and uniqueness of solutions for implicit linear systems on finite interval, Circ. Syst. Signal. Pr., 12 (1993), 375-390. doi: 10.1007/BF01223316. [4] S. Banerjee and G. Verghese, "Nonlinear Phenomena in Power Electronics," IEEE Press, New York, 2001. [5] M. D. Bernardo, C. J. Budd, A. R. Champneys, et al., Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701. doi: 10.1137/050625060. [6] M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, Int. J. Bifurcat. Chaos, 11 (2001), 1121-1140. [7] M. P. Brinn, K. V. Carson, A. J. Esterman, A. B. Chang and B. J. Smith, Mass media interventions for preventing smoking in young people, Cochrane Db. Syst. Rev., 11 (2010) , 1-47. [8] B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," Springer-Verlag, London, 1999. [9] R. M. Corless, G. H. Gonnet, et al., On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [10] J. Cui, Y. Sun and H. Zhu, The impact of media on the spreading and control of infectious disease. J. Dynam. Diff. Eqns., 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0. [11] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mt. J. Math., 38 (2008), 1323-1334. doi: 10.1216/RMJ-2008-38-5-1323. [12] K. Deimling, "Multivalued Differential Equations," Walter De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228. [13] A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer Academic, Dordrecht, The Netherlands, 1988. [14] S. Funk, M. Salathe and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. R. Soc. Interface, 7 (2010), 1247-1256 [15] S. Funk, E. Gilad, C. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, PNAS, 106 (2009), 6872-6877. [16] A. Handel, Jr I. M. Longini and R. Antia, What is the best control strategy for multiple infectious disease outbreaks?, Proc. R. Soc. B., 274 (2007), 833-837. [17] W. Heemels and B. Brogliato, The complementarity class of hybrid dynamical systems, European J. Control., 9 (2003), 311-319. [18] J. H. Jones and M. Salathé, Early assessment of anxiety and behavioral response to novel swine-origin influenza A(H1N1), PLoS ONE, 4 (2009), e8032. doi: 10.1371/journal.pone.0008032. [19] R. Liu, J. 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. [20] Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74. doi: 10.1142/S1793524508000023. [21] J. Melin, Does distribution theory contain means for extending Poincaré-Bendixson theory, J. Math. Anal. Appl., 303 (2004), 81-89. doi: 10.1016/j.jmaa.2004.06.069. [22] G. J. Rubin, H. W. W. Potts and S. Michie, The impact of communications about swine flu (influenza A H1N1v) on public responses to the outbreak: Results from 36 national telephone surveys in the UK, Health Technol. Assess., 14 (2010), 183-266. [23] C. Sun, W. Yang, J. 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. [24] J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith and C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), S5. [25] J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomathematics, (2012). doi: 10.5402/2012/581274. [26] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022. [27] Y. Xiao, Y. Zhou and S. Tang, Modelling disease spread in dispersal networks at two levels, IMA. J. Math. appl. Med. Biol., 28 (2010), 227-244. doi: 10.1093/imammb/dqq007. [28] Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonl. Anal. RWA., 11 (2010), 4154-4163. doi: 10.1016/j.nonrwa.2010.05.002. [29] Y. Xiao, X. Xu and S. Tang, Sliding mode control of outbreaks of emerging infectious diseases, Bull. Math. Biol., 74 (2012), 2403-2422.

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##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control," Oxford University, Oxford, 1991. [2] J. Aubin and A. Cellina, "Differential Inclusion," Springer-Verlag, Berlin, 1984. [3] A. Banaszuk, On the existence and uniqueness of solutions for implicit linear systems on finite interval, Circ. Syst. Signal. Pr., 12 (1993), 375-390. doi: 10.1007/BF01223316. [4] S. Banerjee and G. Verghese, "Nonlinear Phenomena in Power Electronics," IEEE Press, New York, 2001. [5] M. D. Bernardo, C. J. Budd, A. R. Champneys, et al., Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701. doi: 10.1137/050625060. [6] M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, Int. J. Bifurcat. Chaos, 11 (2001), 1121-1140. [7] M. P. Brinn, K. V. Carson, A. J. Esterman, A. B. Chang and B. J. Smith, Mass media interventions for preventing smoking in young people, Cochrane Db. Syst. Rev., 11 (2010) , 1-47. [8] B. Brogliato, "Nonsmooth Mechanics: Models, Dynamics and Control," Springer-Verlag, London, 1999. [9] R. M. Corless, G. H. Gonnet, et al., On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [10] J. Cui, Y. Sun and H. Zhu, The impact of media on the spreading and control of infectious disease. J. Dynam. Diff. Eqns., 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0. [11] J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mt. J. Math., 38 (2008), 1323-1334. doi: 10.1216/RMJ-2008-38-5-1323. [12] K. Deimling, "Multivalued Differential Equations," Walter De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228. [13] A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Sides," Kluwer Academic, Dordrecht, The Netherlands, 1988. [14] S. Funk, M. Salathe and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. R. Soc. Interface, 7 (2010), 1247-1256 [15] S. Funk, E. Gilad, C. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, PNAS, 106 (2009), 6872-6877. [16] A. Handel, Jr I. M. Longini and R. Antia, What is the best control strategy for multiple infectious disease outbreaks?, Proc. R. Soc. B., 274 (2007), 833-837. [17] W. Heemels and B. Brogliato, The complementarity class of hybrid dynamical systems, European J. Control., 9 (2003), 311-319. [18] J. H. Jones and M. Salathé, Early assessment of anxiety and behavioral response to novel swine-origin influenza A(H1N1), PLoS ONE, 4 (2009), e8032. doi: 10.1371/journal.pone.0008032. [19] R. Liu, J. 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. [20] Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74. doi: 10.1142/S1793524508000023. [21] J. Melin, Does distribution theory contain means for extending Poincaré-Bendixson theory, J. Math. Anal. Appl., 303 (2004), 81-89. doi: 10.1016/j.jmaa.2004.06.069. [22] G. J. Rubin, H. W. W. Potts and S. Michie, The impact of communications about swine flu (influenza A H1N1v) on public responses to the outbreak: Results from 36 national telephone surveys in the UK, Health Technol. Assess., 14 (2010), 183-266. [23] C. Sun, W. Yang, J. 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. [24] J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith and C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), S5. [25] J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomathematics, (2012). doi: 10.5402/2012/581274. [26] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022. [27] Y. Xiao, Y. Zhou and S. Tang, Modelling disease spread in dispersal networks at two levels, IMA. J. Math. appl. Med. Biol., 28 (2010), 227-244. doi: 10.1093/imammb/dqq007. [28] Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonl. Anal. RWA., 11 (2010), 4154-4163. doi: 10.1016/j.nonrwa.2010.05.002. [29] Y. Xiao, X. Xu and S. Tang, Sliding mode control of outbreaks of emerging infectious diseases, Bull. Math. Biol., 74 (2012), 2403-2422.
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