Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence

The deterministic and stochastic SIQS models with non-linear incidence are introduced. For deterministic model, the basic reproductive rate R"0 is derived. Moreover, if R"0= 1, there exists a unique endemic equilibrium which is globally asymptotically stable. For stochastic model, sufficient condition for extinction of the disease that is regardless of the value of R"0 is presented. In addition, if the intensities of the white noises are sufficiently small and R"0>1, then there exists a unique stationary distribution to stochastic model. Numerical simulations are also carried out to confirm the analytical results.

[1]  Weiming Wang,et al.  Stability analysis of an epidemic model with diffusion and stochastic perturbation , 2012 .

[2]  Y. Iwasa,et al.  Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models , 1986, Journal of mathematical biology.

[3]  Dingshi Li,et al.  The stationary distribution and ergodicity of a stochastic generalized logistic system , 2013 .

[4]  Daqing Jiang,et al.  Global stability of two-group SIR model with random perturbation☆ , 2009 .

[5]  Xuerong Mao,et al.  Stationary distribution of stochastic population systems , 2011, Syst. Control. Lett..

[6]  Aadil Lahrouz,et al.  Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence , 2013 .

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

[8]  Y. Takeuchi,et al.  Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise , 2004 .

[9]  Eric Renshaw,et al.  Asymptotic behaviour of the stochastic Lotka-Volterra model , 2003 .

[10]  Kai Liu.  Stochastic Stability of Differential Equations in Abstract Spaces , 2019 .

[11]  Shigui Ruan,et al.  Dynamical behavior of an epidemic model with a nonlinear incidence rate , 2003 .

[12]  Gang George Yin,et al.  Asymptotic Properties of Hybrid Diffusion Systems , 2007, SIAM J. Control. Optim..

[13]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[14]  H. Hethcote,et al.  Effects of quarantine in six endemic models for infectious diseases. , 2002, Mathematical biosciences.

[15]  Daqing Jiang,et al.  Stationary distribution of a stochastic SIS epidemic model with vaccination , 2014 .

[16]  Shigui Ruan,et al.  Global analysis of an epidemic model with nonmonotone incidence rate , 2006, Mathematical Biosciences.

[17]  Ravi P. Agarwal,et al.  Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation , 2012 .

[18]  Daqing Jiang,et al.  Multigroup SIR epidemic model with stochastic perturbation , 2011 .

[19]  Dinh Van Huynh,et al.  Algebra and Its Applications , 2006 .

[20]  Ke Wang,et al.  Extinction and permanence in a stochastic non-autonomous population system , 2010, Appl. Math. Lett..

[21]  Qiuying Lu,et al.  Stability of SIRS system with random perturbations , 2009 .

[22]  Zhixia Han,et al.  Stochastic SIRS model under regime switching , 2013 .

[23]  Ke Wang,et al.  Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system , 2012, Appl. Math. Lett..

[24]  M. Carletti,et al.  On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. , 2002, Mathematical biosciences.

[25]  V. B. Kolmanovskii,et al.  Stability of epidemic model with time delays influenced by stochastic perturbations 1 This paper was , 1998 .

[26]  Donal O'Regan,et al.  The extinction and persistence of the stochastic SIS epidemic model with vaccination , 2013 .

[27]  Daqing Jiang,et al.  The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence , 2012 .

[28]  K. Sato,et al.  Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment , 2006 .

[29]  S. Zacks,et al.  Introduction to stochastic differential equations , 1988 .

[30]  Daqing Jiang,et al.  Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation☆ , 2008 .

[31]  Liangjian Hu,et al.  A Stochastic Differential Equation SIS Epidemic Model , 2011, SIAM J. Appl. Math..

[32]  G. Serio,et al.  A generalization of the Kermack-McKendrick deterministic epidemic model☆ , 1978 .

[33]  Daqing Jiang,et al.  A note on nonautonomous logistic equation with random perturbation , 2005 .

[34]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[35]  Aadil Lahrouz,et al.  Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination , 2012, Appl. Math. Comput..

[36]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[37]  Xuerong Mao,et al.  The SIS epidemic model with Markovian switching , 2012 .