Dynamics of an epidemic model with delays and stage structure
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Dynamics of an epidemic model with delays and stage structure Juan Liu1 · Kai Wang2
Received: 21 April 2016 / Revised: 24 March 2017 / Accepted: 17 April 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, dynamics of a stage-structured epidemic model with delays and nonlinear incidence rate is analyzed. Local stability and existence of Hopf bifurcation is discussed by choosing possible combination of the delays as the bifurcation parameter. It is proved that the unique endemic equilibrium is locally asymptotically stable when the delay is suitably small and a bifurcating periodic solution will be caused once the delay passes through the corresponding critical value of the delay. We make use of the normal form theory and center manifold theorem to obtain the explicit formulas for determining the properties of the Hopf bifurcation. Numerical simulations supporting our obtained findings are carried out in the end. Keywords Delays · Hopf bifurcation · Nonlinear incidence · SIRS epidemic model Mathematics Subject Classification 34C15 · 34C23 · 37G15 · 37N25
1 Introduction Since the pioneering work of Kermark and Mckendrick (1927), various epidemic models (Liu 2013; Enatsu et al. 2012; Zhang et al. 2014, 2015) have been formulated and used to analyze
Communicated by Maria do Rosário de Pinho. This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2015A144) and Anhui Provincial Natural Science Foundation (1508085QA13, 1708085MA17).
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Juan Liu [email protected] Kai Wang [email protected]
1
Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China
2
School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China
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J. Liu, K. Wang
the spread of infectious disease by scholars at home and abroad. However, all the epidemic models above neglected the stage of infections. This is not consistent with reality. Because infected individuals may have different infectious ability in different stages of infection for some infectious diseases (Anderson and May 1991; Cai et al. 2009). Based on this, epidemic models with stage structure have been investigated by many authors recently (Cai et al. 2009; Li and Wang 2005; Rhodes and House 2013; Zhang et al. 2010; Glasser et al. 2012; Chen and Hong 2014). In Zhang et al. (2010), Zhang et al. investigated Hopf bifurcation of a delayed SIRS epidemic model with stage structure and bilinear incidence rate under assumption that the infectious disease spread only among mature individuals. Considering the limitation of bilinear incidence rate in epidemiological modeling and based on the epidemic model in Zhang et al. (2010), Chen and Hong proposed the following delayed SIRS epidemic model with stage structure and nonlinear incidence rate (Chen and Hong 2014): ⎧ dX (t) A ⎪ ⎪ = + B S(t) − (σ + μ0 )X (t), ⎪ dt S(t) ⎪ ⎪ ⎪ ⎨ dS(t) = σ X (t) − μS(t) − β S(t)I (t−τ ) + δ R(t), dt 1+α I (t−τ ) (1) ⎪ β S(t)I (
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