Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, USA.
J Biol Dyn. 2007 Oct;1(4):363-78. doi: 10.1080/17513750701605465.
A sharp threshold is established that separates disease persistence from the extinction of small disease outbreaks in an S→E→I→R→S type metapopulation model. The travel rates between patches depend on disease prevalence. The threshold is formulated in terms of a basic replacement ratio (disease reproduction number), ℛ(0), and, equivalently, in terms of the spectral bound of a transmission and travel matrix. Since frequency-dependent (standard) incidence is assumed, the threshold results do not require knowledge of a disease-free equilibrium. As a trade-off, for ℛ(0)>1, only uniform weak disease persistence is shown in general, while uniform strong persistence is proved for the special case of constant recruitment of susceptibles into the patch populations. For ℛ(0)<1, Lyapunov's direct stability method shows that small disease outbreaks do not spread much and eventually die out.
在 S→E→I→R→S 型集合种群模型中，存在一个明显的阈值，它将疾病持续存在与小型疾病爆发的灭绝区分开来。斑块之间的迁移率取决于疾病的流行程度。该阈值可通过基本替代率（疾病繁殖数）R₀表示，也可等效地通过传播和迁移矩阵的谱界表示。由于采用了频率依赖型（标准）发病率，因此该阈值结果不需要疾病无平衡点的知识。作为一种权衡，当 R₀>1 时，通常只显示出均匀的弱疾病持续存在，而在易感人群持续不断地补充到斑块种群的特殊情况下，则证明了均匀的强持续存在。对于 R₀<1，Lyapunov 直接稳定性方法表明，小型疾病爆发不会传播得太远，最终会灭绝。
