Analysis of a delayed SIR model subject to multiple infectious stages and nonlinear incidence rate

Volume 10, Issue 11, pp 6071--6083 http://dx.doi.org/10.22436/jnsa.010.11.42

Publication Date: November 27, 2017 Submission Date: May 13, 2017

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Authors

Hong Zhang - School of Mathematical Science, Harbin Normal University, Harbin 150025, China. Chunming Li - School of Mathematical Science, Heilongjiang University, Harbin 150080, China. Hongquan Sun - School of Mathematical Science, Heilongjiang University, Harbin 150080, China.

Abstract

We investigate the threshold dynamics problem of a delayed Susceptible-Infected-Recovered (SIR) model with general nonlinear incidence and multiple parallel infectious stages. Biologically, the model contains the following aspects: (i) once infection occurs, a fraction of the infected individuals is detected and treated, while the rest of the infected remains undetected and untreated; (ii) distributed delays governed by a general nonlinear incidence function are included into the model due to the complexity of disease transmissions. Mathematically, under some suitable assumptions on nonlinear incidence rate, we prove that the reproduction number \(\Re_0 \) can be used to govern the the global dynamics of the model. The proofs of global attractivity of disease-free equilibrium (which means the extinction of disease) and endemic equilibrium (which means the persistence of the disease) are achieved by constructing suitable Lyapunov functionals.

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ISRP Style

Hong Zhang, Chunming Li, Hongquan Sun, Analysis of a delayed SIR model subject to multiple infectious stages and nonlinear incidence rate, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 11, 6071--6083

AMA Style

Zhang Hong, Li Chunming, Sun Hongquan, Analysis of a delayed SIR model subject to multiple infectious stages and nonlinear incidence rate. J. Nonlinear Sci. Appl. (2017); 10(11):6071--6083

Chicago/Turabian Style

Zhang, Hong, Li, Chunming, Sun, Hongquan. "Analysis of a delayed SIR model subject to multiple infectious stages and nonlinear incidence rate." Journal of Nonlinear Sciences and Applications, 10, no. 11 (2017): 6071--6083

Keywords

SIR epidemic model
nonlinear incidence
global attractivity
Lyapunov functional

MSC

34D23
34K20
92D30

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