Lévy noise perturbation for an epidemic model with impact of media coverage

References

• M.F. Abakar, H.Y. Azami, P.J. Bless, L. Crump, P. Lohmann, M. Laager, N. Chitnis, and J. Zinstag, Transmission dynamics and elimination potential of zoonotic tuberculosis in Morocco, PLOS Neglected Trop. Dis. 11(2) (2017), pp. 1–17. doi: 10.1371/journal.pntd.0005214
   Web of Science ®Google Scholar
• E. Allen, Modeling with Itô, Stochastic Differential Equations, Springer, Dordrecht, 2007.
   Google Scholar
• H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics Vol. 151, Springer, New York, 2000.
   Google Scholar
• R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
   Google Scholar
• D. Applebaum, Lévy Process and Stochastic Calculus, Cambridge Press, New York, 2009.
   Google Scholar
• D Applebaum and M Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Prob. 46 (2009), pp. 1116–1129. doi: 10.1239/jap/1261670692
   Web of Science ®Google Scholar
• N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 2nd ed., Hafner, New York, 1975.
   Google Scholar
• M.S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuens Monographs on Applied Probability and Statistics Vol. 4, Spottiswoode, Ballantyne, London, 1960.
   Google Scholar
• B. Berrhazi, El Fatini M., T. Caraballo, and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), pp. 2415–2431. doi: 10.3934/dcdsb.2018057
   Web of Science ®Google Scholar
• B. Berrhazi, El Fatini M., A. Laaribi, R. Pettersson, and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals 105 (2017), pp. 60–68. doi: 10.1016/j.chaos.2017.10.007
   Web of Science ®Google Scholar
• S. Blower, Modelling the genital herpes epidemic, Herpes 11 (Suppl 3) (2004), pp. 138A–146A.
   PubMedGoogle Scholar
• F. Brauer and Castillo-Chavez C., Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics Vol. 40, Springer, New York, 2001.
   Google Scholar
• Y. Cai, Y. Kang, M. Banerjee, and W. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci. 14 (2016), pp. 839–910. doi: 10.4310/CMS.2016.v14.n4.a1
   Web of Science ®Google Scholar
• T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure. Appl. Anal 16 (2017), pp. 151–162. doi: 10.3934/cpaa.2017007
   Web of Science ®Google Scholar
• Castillo-Chavez C., S. Blower, van den Driessche P., D. Kirschner, and A.-A. Yakubu, eds., Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, The IMA Volumes in Mathematics and Its Applications Vol. 125, Springer, New York, 2002.
   Google Scholar
• Castillo-Chavez C., S. Blower, van den Driessche P., D. Kirschner, and A-A. Yakubu, eds., Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, The IMA Volumes in Mathematics and Its Applications Vol. 126, Springer, New York, 2002.
   Google Scholar
• C. Chen and Y. Kang, The asymptotic behavior of a stochastic vaccination model with backward bifurcation, Appl. Math. Model. 40 (2016), pp. 6051–6068. doi: 10.1016/j.apm.2016.01.045
   Web of Science ®Google Scholar
• C. Chen and Y. Kang, Dynamics of a stochastic multi-strain SIS epidemic model driven by Lévy noise, Commun. Nonlinear Sci. Numer. Simulat. 42 (2017), pp. 379–395. doi: 10.1016/j.cnsns.2016.06.012
   Web of Science ®Google Scholar
• D.J. Daley and J. Gani, Epidemic Modelling, an Introduction, Cambridge: Studies in Mathematical Biology Vol. 15, Cambridge University Press, Cambridge, 1999.
   Google Scholar
• O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, Wiley, New York, 2000.
   Google Scholar
• El Fatini M., A. Lahrouz, R. Pettersson, A. Settati, and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput. 316 (2018), pp. 326–341.
   Web of Science ®Google Scholar
• P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput. 219 (2013), pp. 8496–8507.
   Web of Science ®Google Scholar
• A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011), pp. 876–902. doi: 10.1137/10081856X
   Web of Science ®Google Scholar
• H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), pp. 599–653. doi: 10.1137/S0036144500371907
   Web of Science ®Google Scholar
• D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001), pp. 525–546. doi: 10.1137/S0036144500378302
   Web of Science ®Google Scholar
• L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Diff. Equ. 217 (2005), pp. 26–53. doi: 10.1016/j.jde.2005.06.017
   Web of Science ®Google Scholar
• C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38 (2014), pp. 5067–5079. doi: 10.1016/j.apm.2014.03.037
   Web of Science ®Google Scholar
• W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London Ser. A 115 (1927), pp. 700–721. doi: 10.1098/rspa.1927.0118
   Google Scholar
• R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012.
   Google Scholar
• P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
   Google Scholar
• A. Korobeinikov and P.K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004), pp. 57–60. doi: 10.3934/mbe.2004.1.57
   PubMed Web of Science ®Google Scholar
• A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett 83 (2013), pp. 960–968. doi: 10.1016/j.spl.2012.12.021
   Web of Science ®Google Scholar
• A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput. 233 (2014), pp. 10–19.
   Web of Science ®Google Scholar
• Y. Li and J. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 2353–2365. doi: 10.1016/j.cnsns.2008.06.024
   Web of Science ®Google Scholar
• Y. Lin, D. Jiang, and S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A. 394 (2014), pp. 187–197. doi: 10.1016/j.physa.2013.10.006
   Web of Science ®Google Scholar
• S. Liu, S. Wang, and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear Anal.: Real World Appl. 12 (2011), pp. 119–127. doi: 10.1016/j.nonrwa.2010.06.001
   Web of Science ®Google Scholar
• W. Liu and Q. Zheng, A stochastic SIS epidemic model incorporating media coverage in a two patch setting, Appl. Math. Comput. 262 (2015), pp. 160–168.
   Web of Science ®Google Scholar
• X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood, Cambridge, 2007.
   Google Scholar
• X. Mao, G. Marion, and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stoch. Process Appl. 97 (2002), pp. 95–110. doi: 10.1016/S0304-4149(01)00126-0
   Web of Science ®Google Scholar
• H.N. Moreira and Y. Wang, Global stability in an SIRI model, SIAM Rev. 39 (1997), pp. 496–502. doi: 10.1137/S0036144595295879
   Web of Science ®Google Scholar
• ∅ksendal B. and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer-Verlag, Berlin, Heidelberg, 2005.
   Google Scholar
• 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), pp. 87–95. doi: 10.1016/j.mbs.2011.01.005
   PubMed Web of Science ®Google Scholar
• J. Tchuenche, N. Dube, C. Bhunu, R. Smith, and C. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health 11(S5) (2011), pp. 1–14.
   PubMedGoogle Scholar
• D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev. 32 (1990), pp. 136–139. doi: 10.1137/1032003
   Web of Science ®Google Scholar
• van den Driessche P. and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48. doi: 10.1016/S0025-5564(02)00108-6
   PubMed Web of Science ®Google Scholar
• Van den Driessche P. and X. Zou, Modeling relapse in infectious diseases, Math. Biosci. 207 (2007), pp. 89–103. doi: 10.1016/j.mbs.2006.09.017
   PubMed Web of Science ®Google Scholar
• Van den Driessche P., L. Wang, and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng. 4 (2007), pp. 205–219. doi: 10.3934/mbe.2007.4.205
   PubMed Web of Science ®Google Scholar
• Vargas-De-León C., On the global stability of infectious disease models with relapse, Abstraction Appl. 9 (2013), pp. 50–61.
   Google Scholar
• P. Wildy, H.J. Field, and A.A. Nash, Classical herpes latency revisited, in Virus Persistence Symposium, vol. 33, B.W.J. Mahy, A.C. Minson, and G.K. Darby, eds., Cambridge University Press, Cambridge, 1982, pp. 133–168.
   Google Scholar
• R. Xu, Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Anal.: Model. Control 18 (2013), pp. 250–263.
   Web of Science ®Google Scholar
• X. Zhang and K. Wang, Stochastic model for spread of AIDS driven by Lévy noise, J. Dyn. Diff. Equ. 27 (2015), pp. 215–236. doi: 10.1007/s10884-015-9459-5
   Web of Science ®Google Scholar
• X. Zhang and K. Wang, Stochastic SEIR model with jumps, Appl. Math. Comput 239 (2014), pp. 133–149.
   Web of Science ®Google Scholar
• X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett. 26 (2013), pp. 867–874. doi: 10.1016/j.aml.2013.03.013
   Web of Science ®Google Scholar
• M. Zhao and H. Zhao, Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage, Adv. Differ. Equ. 149 (2016), pp. 1–17.
   Google Scholar