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Zuo, Lixia; Liu, Maoxing

Effect of awareness programs on the epidemic outbreaks with time delay. (English) Zbl 1406.92651

Abstr. Appl. Anal. 2014, Article ID 940841, 8 p. (2014).
Summary: An epidemic model with time delay has been proposed and analyzed. In this model the effect of awareness programs driven by media on the prevalence of an infectious disease is studied. It is assumed that pathogens are transmitted via direct contact between the susceptible and the infective populations and further assumed that the growth rate of cumulative density of awareness programs increases at a rate proportional to the infective population. The model is analyzed by using stability theory of differential equations and numerical simulations. Both equilibria have been proved to be globally asymptotically stable. The results we obtained and numerical simulations suggest the increasing of the dissemination rate and implementation rate can reduce the proportion of the infective population.

Cited in 16 Documents

MSC:

92D30 Epidemiology

Keywords:

epidemic outbreaks; time delay; awareness programs
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References:

[1] World Health Organization, Global burden of disease: 2004 update
[2] World Health Organization, Report on the Global AIDS Epidemic
[3] Breman, J. G.; Alilio, M. S.; Mills, A., Conquering the intolerable burden of malaria: what’s new, what’s needed: a summary, American Journal of Tropical Medicine and Hygiene, 71, 2, 1-15 (2004)
[4] Laxminarayan, R.; Mills, A. J.; Breman, J. G.; Measham, A. R.; Alleyne, G.; Claeson, M.; Jha, P.; Musgrove, P.; Chow, J.; Shahid-Salles, S.; Jamison, D. T., Advancement of global health: key messages from the Disease Control Priorities Project, Lancet, 367, 9517, 1193-1208 (2006) · doi:10.1016/S0140-6736(06)68440-7
[5] Vasterman, P. L. M.; Ruigrok, N., Pandemic alarm in the Dutch media: media coverage of the 2009 influenza A (H1N1) pandemic and the role of the expert sources, European Journal of Communication, 28, 4, 436-453 (2013) · doi:10.1177/0267323113486235
[6] Gao, D.; Ruan, S., An SIS patch model with variable transmission coefficients, Mathematical Biosciences, 232, 2, 110-115 (2011) · Zbl 1218.92064 · doi:10.1016/j.mbs.2011.05.001
[7] Tchuenche, J. M.; Dube, N.; Bhunu, C. P.; Smith, R. J.; Bauch, C. T., The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11, 1, article S5 (2011) · doi:10.1186/1471-2458-11-S1-S5
[8] Cui, J.; Tao, X.; Zhu, H., An {SIS} infection model incorporating media coverage, The Rocky Mountain Journal of Mathematics, 38, 5, 1323-1334 (2008) · Zbl 1170.92024 · doi:10.1216/RMJ-2008-38-5-1323
[9] Li, Y.; Cui, J., The effect of constant and pulse vaccination on {SIS} epidemic models incorporating media coverage, Communications in Nonlinear Science and Numerical Simulation, 14, 5, 2353-2365 (2009) · Zbl 1221.34034 · doi:10.1016/j.cnsns.2008.06.024
[10] Mukandavire, Z.; Garira, W.; Tchuenche, J. M., Modelling effects of public health educational campaigns on {HIV}/{AIDS} transmission dynamics, Applied Mathematical Modelling, 33, 4, 2084-2095 (2009) · Zbl 1205.34092 · doi:10.1016/j.apm.2008.05.017
[11] Samanta, S.; Rana, S.; Sharma, A.; Misra, A. K.; Chattopadhyay, J., Effect of awareness programs by media on the epidemic outbreaks: a mathematical model, Applied Mathematics and Computation, 219, 12, 6965-6977 (2013) · Zbl 1316.34052 · doi:10.1016/j.amc.2013.01.009
[12] Misra, A. K.; Sharma, A.; Shukla, J. B., Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Mathematical and Computer Modelling, 53, 5-6, 1221-1228 (2011) · Zbl 1217.34097 · doi:10.1016/j.mcm.2010.12.005
[13] Joshi, H.; Lenhart, S.; Albright, K., Modeling the effect of information campaigns on the HIV epidemic in Uganda, Mathematical Biosciences and Engineering, 5, 4, 757-770 (2008) · Zbl 1154.92037 · doi:10.3934/mbe.2008.5.757
[14] Nyabadza, F.; Chiyaka, C.; Mukandavire, Z.; Hove-Musekwa, S. D., Analysis of an HIV/AIDS model with public-health information campaigns and individual withdrawal, Journal of Biological Systems, 18, 2, 357-375 (2010) · Zbl 1342.92269 · doi:10.1142/S0218339010003329
[15] Yorke, J. A.; London, W. P., Recurrent outbreaks of measles, chickenpox and mumps. II. Systematic differences in contact rates and stochastic effects, American Journal of Epidemiology, 98, 6, 469-482 (1973)
[16] Kiss, I. Z.; Cassell, J.; Recker, M.; Simon, P. L., The impact of information transmission on epidemic outbreaks, Mathematical Biosciences, 225, 1, 1-10 (2010) · Zbl 1188.92033 · doi:10.1016/j.mbs.2009.11.009
[17] Capasso, V., Mathematical Structure of Epidemic System. Mathematical Structure of Epidemic System, Lecture Notes in Biomathematics, 97 (1993), Berlin, Germany: Springer, Berlin, Germany · Zbl 0798.92024 · doi:10.1007/978-3-540-70514-7
[18] Hethcote, H. W.; Levin, S. A., Periodicity in epidemiological models, Applied Mathematical Ecology. Applied Mathematical Ecology, Biomathematics, 18, 193-211 (1989) · doi:10.1007/978-3-642-61317-3_8
[19] Collinson, S.; Heffernan, J. M., Modelling the effects of media during an influenza epidemic, Collinson and Heffernan BMC Public Health, 14, article 376 (2014) · doi:10.1186/1471-2458-14-376
[20] Pang, J. H.; Cui, J. A., An SIRS epidemiological model with nonlinear incidence rate incorporating media coverage, Proceedings of the 2nd International Conference on Information and Computing Science (ICIC ’09) · doi:10.1109/ICIC.2009.235
[21] Liu, M.; Röst, G.; Vas, G., SIS model on homogeneous networks with threshold type delayed contact reduction, Computers & Mathematics with Applications, 66, 9, 1534-1546 (2013) · Zbl 1346.92074 · doi:10.1016/j.camwa.2013.02.009
[22] Elenbaas, M.; Boomgaarden, H. G.; Schuck, A. R. T.; de Vreese, C. H., The impact of media coverage and motivation on performance-relevant information, Political Communication, 30, 1, 1-16 (2013) · doi:10.1080/10584609.2012.737411
[23] Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, Journal of Mathematical Biology, 25, 4, 359-380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162
[24] Liu, W. M.; Levin, S. A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23, 2, 187-204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956
[25] Liu, Y.; Cui, J., The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1, 1, 65-74 (2008) · Zbl 1155.92343 · doi:10.1142/S1793524508000023
[26] Zhao, H.; Lin, Y.; Dai, Y., An SIRS epidemic model incorporating media coverage with time delay, Computational and Mathematical Methods in Medicine, 2014 (2014) · Zbl 1307.92359 · doi:10.1155/2014/680743
[27] Tchuenche, J. M.; Bauch, C. T., Dynamics of an infectious disease where media coverage influences transmission, ISRN Biomathematics, 2012 (2012) · Zbl 1269.92052 · doi:10.5402/2012/581274
[28] Cui, J.; Sun, Y.; Zhu, H., The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20, 1, 31-53 (2008) · Zbl 1160.34045 · doi:10.1007/s10884-007-9075-0
[29] Liu, R. S.; Wu, J. H.; Zhu, H. P., Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8, 3, 153-164 (2007) · Zbl 1121.92060 · doi:10.1080/17486700701425870
[30] Liu, Y. Y.; Xiao, Y. N., An epidemic model with saturated media/psychological impact, Applied Mathematics and Mechanics, 34, 4, 99-407 (2013)
[31] Wang, A.; Xiao, Y., A Filippov system describing media effects on the spread of infectious diseases, Nonlinear Analysis: Hybrid Systems, 11, 84-97 (2014) · Zbl 1323.92215 · doi:10.1016/j.nahs.2013.06.005
[32] Bhunu, C. P.; Mushayabasa, S.; Kojouharov, H., Mathematical analysis of an {HIV}/{AIDS} model: impact of educational programs and abstinence in sub-Saharan Africa, Journal of Mathematical Modelling and Algorithms, 10, 1, 31-55 (2011) · Zbl 1235.92041 · doi:10.1007/s10852-010-9134-0
[33] Wang, Y.; Cao, J. D.; Jin, Z.; Zhang, H. F.; Sun, G. Q., Impact of media coverage on epidemic spreading in complex networks, Physica A: Statistical Mechanics and Its Applications, 392, 23, 5824-5835 (2013) · Zbl 1395.34055 · doi:10.1016/j.physa.2013.07.067
[34] Yuan, X.; Xue, Y.; Liu, M., Analysis of an epidemic model with awareness programs by media on complex networks, Chaos, Solitons & Fractals, 48, 1, 1-11 (2013) · Zbl 1262.92042 · doi:10.1016/j.chaos.2012.12.001
[35] Funk, S.; Gilad, E.; Watkins, C.; Jansen, V. A. A., The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences of the United States of America, 106, 16, 6872-6877 (2009) · Zbl 1203.91242 · doi:10.1073/pnas.0810762106
[36] Liu, W., A SIRS epidemic model incorporating media coverage with random perturbation, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.34132 · doi:10.1155/2013/792308
[37] Wang, L.; Huang, H. L.; Xu, A. C.; Wang, W. M., Stochastic extinction in an SIRS epidemic model incorporating media coverage, Abstract and Applied Analysis, 2013 (2013) · Zbl 1420.92113 · doi:10.1155/2013/891765
[38] Wei, H. M.; Li, X. Z.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342, 2, 895-908 (2008) · Zbl 1146.34059 · doi:10.1016/j.jmaa.2007.12.058
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