Dept. Mathematics, Universita di Trento, Via Sommarive 14, 38123 Povo (TN), Italy.
Math Biosci Eng. 2012 Jul;9(3):577-99. doi: 10.3934/mbe.2012.9.577.
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 3x3 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 2x2 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.
SIR 年龄结构模型通常被用作传染病传播的基本模型。然而，在接触率的一般假设下，它们的行为还不完全清楚，在迄今为止最详细的分析中，Inaba（1990）仅在相当严格的条件下证明了地方病平衡点的唯一性。在这里，我们以 3x3 接触矩阵的形式展示了一个例子，其中存在多个非平凡的稳定状态。这种正平衡点的非唯一性与大多数现有的传染病模型不同，因为它不是从无病平衡点的向后跨临界分岔引起的，而是通过正平衡点的两个鞍结分岔引起的。在存在多个地方病平衡点的范围内，对模型的动力学行为进行了数值分析；还显示了许多其他特征的出现，包括多个有吸引力的周期解的共存，其中一些具有极长的周期，以及准周期和混沌吸引子。还表明，如果接触率的形式为 2x2WAIFW 矩阵，则非平凡稳定状态的唯一性始终成立，因此 3 是允许存在多个地方病平衡点的接触矩阵的最小维度。
