Dynamics of a Stochastic SIS Epidemic Model with Birth Pulses and Pulse Treatments

In this paper, a stochastic SIS epidemic model with birth pulses and pulse treatments is constructed and investigated. By applying Floquet theory and qualitative theory of stochastic differential equations, the existence and stability of trivial solution and infection-free periodic solution are discussed. A special endemic solution and the boundedness of infective individuals are investigated. Numerical results for phase portraits, which are illustrated with two examples, are in good agreement with the theoretical analysis.

[1]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[2]  J. Koenderink.  Q… , 2014, Les noms officiels des communes de Wallonie, de Bruxelles-Capitale et de la communaute germanophone.

[3]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Margherita Carletti,et al.  Mean-square stability of a stochastic model for bacteriophage infection with time delays. , 2007, Mathematical biosciences.

[6]  R R Kao,et al.  The dynamics of an infectious disease in a population with birth pulses. , 1998, Mathematical biosciences.

[7]  Jianjun Jiao,et al.  Analysis of a stage-structured predator–prey system with birth pulse and impulsive harvesting at different moments☆ , 2011 .

[8]  Jiajia Yu,et al.  Asymptotic behavior of global positive solution to a stochastic SIR model , 2011, Math. Comput. Model..

[9]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[10]  Martin Bohner,et al.  Impulsive differential equations: Periodic solutions and applications , 2015, Autom..

[11]  Shujing Gao,et al.  An SIR epidemic model with time-varying pulse control schemes and saturated infectious force , 2013 .

[12]  Pasquale Vetro,et al.  Stability of a stochastic SIR system , 2005 .

[13]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[14]  Chuanzhi Bai,et al.  Asymptotic stability of a two-group stochastic SEIR model with infinite delays , 2014, Commun. Nonlinear Sci. Numer. Simul..

[15]  Guirong Jiang,et al.  Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination , 2009, Appl. Math. Comput..

[16]  Xinzhi Liu,et al.  Pulse and constant control schemes for epidemic models with seasonality , 2011 .

[17]  G. Caughley,et al.  Analysis of vertebrate populations , 1977 .

[18]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[19]  Graeme Caughley,et al.  Analysis of vertebrate populations , 1977 .