Moghadas Seyed M, Alexander Murray E
Institute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba, Canada R3B 1Y6.
Math Med Biol. 2006 Sep;23(3):231-54. doi: 10.1093/imammb/dql011. Epub 2006 Apr 28.
An epidemic model with a generalized non-linear incidence is extended to incorporate the effect of an infection-dependent removal strategy, which is defined as a function of the number of infected individuals. It is assumed that the removal rate decreases from a maximum capacity for removing infected individuals as their number increases. The existence and stability of the associated equilibria are analysed, and the basic reproductive number (R0) is formulated. It is shown that R0 is independent of the functional form of the incidence, but depends on the removal rate. Normal forms are derived to show the different types of bifurcation the model undergoes, including transcritical, generalized Hopf (Bautin), saddle-node and Bogdanov-Takens. A degenerate Hopf bifurcation at the Bautin point, where the first Lyapunov coefficient vanishes, is discussed. Sotomayor's theorem is applied to establish a saddle-node bifurcation at the turning point of backward bifurcation. The Bogdanov-Takens normal form is derived, from which the local bifurcation curve for a family of homoclinic orbits is formulated. Bifurcation diagrams and numerical simulations, using parameter values estimated for some infectious diseases, are also presented to provide more intuition to the theoretical findings. The results show that sufficiently increasing the removal rate can reduce R0 below a subthreshold domain, which leads to disease eradication.
一个具有广义非线性发病率的流行病模型被扩展，以纳入依赖感染的清除策略的影响，该策略被定义为感染个体数量的函数。假设清除率随着感染个体数量的增加而从清除感染个体的最大能力下降。分析了相关平衡点的存在性和稳定性，并制定了基本再生数（R0）。结果表明，R0与发病率的函数形式无关，但取决于清除率。推导了范式以展示模型经历的不同类型的分岔，包括跨临界、广义霍普夫（鲍廷）、鞍结和博格达诺夫 - 塔克恩斯分岔。讨论了在鲍廷点处的退化霍普夫分岔，其中第一个李雅普诺夫系数消失。应用索托马约尔定理在反向分岔的转折点处建立鞍结分岔。推导了博格达诺夫 - 塔克恩斯范式，由此制定了一族同宿轨道的局部分岔曲线。还给出了使用针对某些传染病估计的参数值的分岔图和数值模拟，以便为理论结果提供更多直观理解。结果表明，充分提高清除率可以将R0降低到亚阈值域以下，从而导致疾病根除。
