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The impact of media on the control of infectious diseases. (English) Zbl 1160.34045
This paper uses a compartmental model to address the impact of media coverage on the transmission of infectious diseases. The mathematical model is a variant of the standard SIE model governed by ODEs in which the usual $SI$ term is multiplied by a factor $\mu {e}^{-mI}$ which decreases exponentially in $I$ and the parameter $m$ reflects the impact of media coverage to the contact transmission. The studies reveals that the model has a disease free equilibrium which is globally asymptotically stable if the basic reproduction number ${R}_{0}$ is less than the unity. Conversely, if ${R}_{0}>1$, then a unique endemic equilibrium appears and a Hopf bifurcation can occur which leads to oscillatory phenomena. Numerical studies show that, if ${R}_{0}>1$ and the effect of the media coverage is sufficiently strong, the model exhibits multiple positive equilibria which gives rise to challenge to the prediction and control of the outbreaks of infectious diseases.
MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 92D30 Epidemiology 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE)
References:
 [1] Brauer F., Castillo-Chavez C.(2000). Mathematical Models in Population Biology and Epidemics. Springer-Verlag, New York [2] Busenberg S., Cooke K.(1993). Vertically Transmitted Diseases. Springer-Verlag, New York [3] Capasso V.(1993). Mathematical Structure of Epidemic System, Lecture Note in Biomathematics, Vol. 97. Springer, Berlin [4] Capasso V., Serio G. (1978). A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42: 43 · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 [5] Diekmann O., Heesterbeek J.A.P.(2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York [6] Dumortier F., Roussarie R., Sotomayor J.(1987). Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3. Ergodic Theory Dynamical Systems 7(3): 375–413 [7] Health Canada: http://www.hc-sc.gc.ca/pphb-dgspsp/sars-sras/prof-e.html [8] Hethcote H.W.(2000). The mathematics of infectious diseases. SIAM Revi. 42, 599–653 · Zbl 0993.92033 · doi:10.1137/S0036144500371907 [9] Levin S.A., Hallam T.G., Gross L.J. (1989). Applied Mathematical Ecology. Springer, New York [10] Liu W.M., Hethcote H.W., Levin S.A.(1987). Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25: 359 · Zbl 0621.92014 · doi:10.1007/BF00277162 [11] Liu W.M., Levin S.A., Iwasa Y.(1986). Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23: 187 · Zbl 0582.92023 · doi:10.1007/BF00276956 [12] Liu, R., Wu, J., and Zhu, H. (2005). Media/Psychological Impact on Multiple Outbreaks of Emerging Infectious Diseases, preprint [13] Murray J.D. (1998). Mathematical Biology. Springer-Verlag, Berlin [14] Ruan S., Wang W.(2003). Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Diff. Equs. 188: 135 · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X [15] SARS EXPRESS: http://www.syhao.com/sars/20030623.htm [16] Shen Z. et al. (2004). Superspreading SARS events, Beijing, 2003. Emerg. Infect. Dis. 10(2): 256–260 [17] van den Driessche P., Watmough J.(2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180: 29–48 · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 [18] Wang W., Ruan S.(2004). Simulating SARS outbreak in Beijing with limit data. J. Theor. Biol. 227: 369 · doi:10.1016/j.jtbi.2003.11.014 [19] WHO. Epidemic curves: Serve Acute Respiratory Syndrome (SARS) http://www.who.int/csr/sars/epicurve/epiindex/en/print.html [20] Yorke J.A., London W.P.(1973). Recurrent outbreaks of measles, chickenpox and mumps II. Am. J. Epidemiol. 98: 469 [21] Zhu H., Campbell S.A., Wolkowicz G.S.(2002). Bifurcation analysis of a predator-prey system with nonmonotonic function response. SIAM J. Appl. Math. 63(2): 636–682