Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
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DOI：
10.3934/mbe.2014.11.785
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发表时间：
2014-03
影响因子：
2.6
通讯作者：
Yoichi Enatsu;Y. Nakata
中科院分区：
文献类型：
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作者：
Yoichi Enatsu;Y. Nakata
研究了一类具有分布时滞的SIRS传染病模型的局部渐近稳定性。发病率由感染个体数的一般饱和函数给出。我们的第一个目标是找到一类非单调的发病率，使得唯一的地方病平衡点总是渐近稳定的。我们建立了一个表征的发病率，这表明，非单调延迟的发病率是必要的地方病平衡的不稳定。我们进一步阐述了特定发生率的稳定性分析。这里我们改进了[Y. Yang和D.肖，潜伏期和非线性发病率对SIRS流行病学模型动力学的影响，Disc。续戴纳姆Sys. B 13（2010）195-211]，其在合适的参数平面中示出。双参数平面分析与隐函数定理的应用，使我们能够得到一个精确的稳定性条件。证明了随着参数的增加，饱和效应的度量，处于地方病平衡态的感染个体数减少，而平衡态可以通过Hopf分支而不稳定.这可以解释为，降低接触率可能会导致感染个体数量的周期性振荡，因此，尽管感染人群的地方病平衡水平降低，但疾病不能从宿主人群中完全根除。数值模拟来说明我们的理论结果。
We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals. Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable. We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.