Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate.

*(English)*Zbl 1358.92093Summary: In this paper, we formulate an epidemic model for the spread of an infectious disease in a population of varying size. The total population is divided into three distinct epidemiological subclass of individuals (susceptible, infectious and recovered) and we study a deterministic and stochastic models with saturated incidence rate. The stochastic model is obtained by incorporating a random noise into the deterministic model. In the deterministic case, we briefly discuss the global asymptotic stability of the disease free equilibrium by using a Lyapunov function. For the stochastic version, we study the global existence and positivity of the solution. Under suitable conditions on the intensity of the white noise perturbation, we prove that there are a \(p\)-th moment exponential stability and almost sure exponential stability of the disease free equilibrium. Furthermore, sufficient conditions for the extinction of the disease are obtained and the asymptotic behavior around the endemic equilibrium is studied. Finally, we give some numerical simulations to illustrate our theoretical results.

##### MSC:

92D30 | Epidemiology |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

##### Keywords:

stochastic epidemic model; saturated incidence rate; global stability; moment exponentialstability; Lyapunov function
PDF
BibTeX
XML
Cite

\textit{M. N'zi} and \textit{J. Tano}, Random Oper. Stoch. Equ. 25, No. 1, 11--26 (2017; Zbl 1358.92093)

Full Text:
DOI

##### References:

[1] | Abta A., Kaddar A. and Alaoui H. T., Global stability for delay SIR and SEIR epidemic models with saturated incidence rates, Electron. J. Differential Equations 2012 (2012), no. 23, 1-13. · Zbl 1243.34115 |

[2] | Agarwal M. and Verma V., Stability and Hopf bifurcation analysis of a SIRS Epidemic models with time delay, Int. J. Appl. Math. Mech. 8 (2012), no. 9, 1-16. |

[3] | Anderson R. M. and May R. M., Regulation and stability of host-parasite population interactions. I: Regulatory processes, J. Anim. Ecol. 47 (1978), no. 1, 219-267. |

[4] | Capasso V. and Serio G., A generalization of Kermack-McKendrick deterministic epidemic model, Math. Biosci. 42 (1978), no. 1-2, 43-61. · Zbl 0398.92026 |

[5] | Gabriela M., Gomes M., White L. J. and Medley G. F., The reinfection threshold, J. Theoret. Biol. 236 (2005), no. 1, 111-113. |

[6] | Gao S., Chen L., Nieto J. J. and Torres A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24 (2006), no. 35-36, 6037-6045. |

[7] | Kaddar A., Abta A. and Talibi Alaoui H., A comparaison of delayed SIR and SEIR epidemic models, Nonlinear Anal. Model. Control 16 (2011), no. 2, 181-190. · Zbl 1322.92073 |

[8] | Krychko Y. and Blyuss B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. Real World Appl. 6 (2005), no. 3, 495-507. · Zbl 1144.34374 |

[9] | Lahrouz A., Omari L. and Kiouach D., Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control 16 (2001), no. 1, 59-76. · Zbl 1271.93015 |

[10] | Lahrouz L., Omari L., Kiouach D. and Belmaâti A., Deterministic and stochastic stability of a mathematical model of smoking, Statist. Probab. Lett. 81 (2011), 1276-1284. · Zbl 1219.92043 |

[11] | Liptser R., A strong law of large numbers for local martingales, Stochastics 3 (1980), no. 1-4, 217-228. · Zbl 0435.60037 |

[12] | Liptser R. S. and Shiryaev A. N., Theory of Martingales, Kluwer Academic, Dordrecht, 1989. · Zbl 0482.60030 |

[13] | Lu Q., Stability of SIRS system with random pertubations, Phys. A 388 (2009), no. 18, 3677-3686. |

[14] | Pathak S., Maiti A. and Samanta G. P., Rich dynamics of an SIR epidemic model, Nonlinear Anal. Model. Control 15 (2010), no. 1, 71-81. · Zbl 1217.93072 |

[15] | Van den Driessche P. and Watmough J., Reproduction numbers and sub-threshold endemic equilibria for compartimental models of disease transmission, Math. Biosci. 180 (2002), no. 1, 29-48. · Zbl 1015.92036 |

[16] | Xu R. and Ma Z., Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals 41 (2009), no. 5, 2319-2325. · Zbl 1198.34098 |

[17] | Zhang J.-Z., Jin Z. and Liu Q.-X., Analysis of delayed SIR model with nonlinear incidence rate, Discrete Syn. Nat. Soc. 2008 (2008), 10.1155/2008/636153. · Zbl 1159.92037 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.