The article investigates a SIQR epidemic model with specific nonlinear incidence rate and stochastic model based on the former, respectively. For deterministic model, we study the existence and stability of the equilibrium points by controlling threshold parameter R0 which determines whether the disease disappears or prevails. Then by using Routh-Hurwitz criteria and constructing suitable Lyapunov function, we get that the disease-free equilibrium is globally asymptotically stable if R0<1 or unstable if R0>1. In addition, the endemic equilibrium point is globally asymptotically stable in certain region when R0>1. For the corresponding stochastic model, the existence and uniqueness of the global positive solution are discussed and some sufficient conditions for the extinction of the disease and the persistence in the mean are established by defining its related stochastic threshold R0s. Moreover, our analytical results show that the introduction of random fluctuations can suppress disease outbreak. And numerical simulations are used to confirm the theoretical results.
References
[1]
Jiang, D.Q., Yu, J.J. and Ji, C.Y. (2011) Asymptotic Behavior of Global Positive Solution to a Stochastic SIR Model. Mathematical and Computer Modelling, 54, 221-232. https://doi.org/10.1016/j.mcm.2011.02.004
[2]
Hattaf, K., Lashari, A.A., Louartassi, Y. and Yousfi, N. (2013) A Delayed SIR Epidemic Model with General Incidence Rate. Electronic Journal of Qualitative Theory of Differential Equations, 3, 1-9. https://doi.org/10.14232/ejqtde.2013.1.3
[3]
Fan, K.G., Zhang, Y., Gao, S.J. and Wei, X. (2017) A Class of Stochastic Delayed SIR Epidemic Models with Generalized Nonlinear Incidence Rate and Temporary Immunity. Physica A: Statistical Mechanics and Its Applications, 481, 198-208.
https://doi.org/10.1016/j.physa.2017.04.055
[4]
Alakes, M., Prosenjit, S. and Samanta, G.P. (2018) Analysis of an SIQR Model. Journal of Ultra Scientist of Physical Sciences, 30, 218-226.
https://doi.org/10.22147/jusps-A/300307
[5]
Lan, G.J., Chen, Z.W., Wei, C.J. and Zhang, S.W. (2018) Stationary Distribution of a Stochastic SIQR Epidemic Model with Saturated Incidence and Degenerate Diffusion. Physica A: Statistical Mechanics and Its Applications, 511, 61-77.
https://doi.org/10.1016/j.physa.2018.07.041
[6]
Chahrazed, L. and Lazhar, R.F. (2013) Stability of a Delayed SIQRS Model with Temporary Immunity. Advances in Pure Mathematics, 3, 240-245.
https://doi.org/10.4236/apm.2013.32034
[7]
Zhang, X.B. and Huo, H.F. (2014) Dynamics of the Deterministic and Stochastic SIQS Epidemic Model with Nonlinear Incidence. Applied Mathematics and Computation, 243, 546-558. https://doi.org/10.1016/j.amc.2014.05.136
[8]
Joshi, H. and Sharma, R.K. (2017) Global of an SIQR Epidemic Model with Saturated Incidence Rate. Asian Journal of Mathematics and Computer Research, 21, 156-166.
[9]
Adnani, J., Hattaf, K. and Yousfi, N. (2013) Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate. International Journal of Stochastic Analysis, 2013, Article ID: 431257. https://doi.org/10.1155/2013/431257
[10]
Ji, C.Y. and Jiang, D.Q. (2011) Dynamics of a Stochastic Density Dependent Predator-Prey System with Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 381, 441-453.
https://doi.org/10.1016/j.jmaa.2011.02.037
[11]
Crowley, P.H. and Martin, E.K. (1989) Functional Responses and Interference within and between Year Classes of a Dragonfly Population. Freshwater Science, 8, 211-221. https://doi.org/10.2307/1467324
[12]
Adnani, J., Hattaf, K. and Yousfi, N. (2016) Analysis of a Stochastic SIRS Epidemic Model with Specific Functional Response. Applied Mathematical Sciences, 10, 301-314. https://doi.org/10.12988/ams.2016.511697
[13]
Hattaf, K., Mahrouf, K. and Adnani, J. (2018) Qualitative Analysis of a Stochastic Epidemic Model with Specific Functional Response and Temporary Immunity. Physica A: Statistical Mechanics and its Applications, 490, 591-600.
https://doi.org/10.1016/j.physa.2017.08.043
[14]
Li, D., Cui, J.A., Liu, M. and Liu, S.Q. (2015) The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate. Bulletin of Mathematical Biology, 77, 1705-1743. https://doi.org/10.1007/s11538-015-0101-9
[15]
Cai, Y.L., Kang, Y. and Wang, W.M. (2017) A Stochastic SIRS Epidemic Model with Nonlinear Incidence Rate. Applied Mathematics and Computation, 305, 221-240.
https://doi.org/10.1016/j.amc.2017.02.003
[16]
Ji, C.Y. and Jiang, D.Q. (2014) Threshold Behaviour of a Stochastic SIR Model. Applied Mathematical Modelling, 38, 5067-5079.
https://doi.org/10.1016/j.apm.2014.03.037
[17]
Chang, Z.B., Meng, X.Z. and Lu, X. (2017) Analysis of a Novel Stochastic SIRS Epidemic Model with Two Different Saturated Incidence Rates. Physica A: Statistical Mechanics and Its Applications, 472, 103-116.
https://doi.org/10.1016/j.physa.2017.01.015
[18]
Higham, D.J. (2001) An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. Society for Industrial and Applied Mathematics, 43, 525-546. https://doi.org/10.1137/S0036144500378302
