MULTIPLE ENDEMIC STATES IN AGE-STRUCTURED SIR EPIDEMIC MODELS
MULTIPLE ENDEMIC STATES IN AGE-STRUCTURED SIR EPIDEMIC MODELS
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DOI：
10.3934/mbe.2012.9.577
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发表时间：
2012-07-01
影响因子：
2.6
通讯作者：
Breda, Dimitri
中科院分区：
文献类型：
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作者：
Franceschetti, Andrea;Pugliese, Andrea;Breda, Dimitri
SIR年龄结构模型经常被用作流行病传播的基本模型。然而，在对不同年龄组之间接触率的一般假设下，它们的行为尚不完全清楚，而且，在迄今为止最详细的分析中，Inaba（1990）仅能在相当严格的条件下证明地方性平衡的独特性。这里，我们给出了一个3 × 3接触矩阵形式的例子，其中存在多个非平凡稳态。这种正平衡的非唯一性实例不同于大多数流行病模型的非唯一性实例，因为它不是由无病平衡的向后跨临界分岔引起的，而是由正平衡的两个鞍节点分岔引起的。在存在多个地方性平衡的范围内，数值分析了模型的动力学行为；从多个吸引周期解的共存，一些具有极长的周期，到准周期和混沌吸引子，显示出许多其他特征。如果接触率是2 × 2的WAIFW矩阵形式，则非平凡稳态的唯一性总是成立，因此3是允许存在多个地方性平衡的接触矩阵的最小维数。
SIR age-structured models are very often used as a basic model of epidemic spread. Yet, their behaviour, under generic assumptions on contact rates between different age classes, is not completely known, and, in the most detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic equilibrium only under a rather restrictive condition.Here, we show an example in the form of a 3 x 3 contact matrix in which multiple non-trivial steady states exist. This instance of non-uniqueness of positive equilibria differs from most existing ones for epidemic models, since it arises not from a backward transcritical bifurcation at the disease free equilibrium, but through two saddle-node bifurcations of the positive equilibrium. The dynamical behaviour of the model is analysed numerically around the range where multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors.It is also shown that, if the contact rates are in the form of a 2 x 2 WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is the minimum dimension of the contact matrix to allow for multiple endemic equilibria.