Safan Muntaser, Heesterbeek Hans, Dietz Klaus
Department of Medical Biometry, Eberhard-Karls-University Tübingen, Westbahnhofstr. 55, 72070, Tübingen, Germany.
J Math Biol. 2006 Oct;53(4):703-18. doi: 10.1007/s00285-006-0028-8. Epub 2006 Aug 5.
We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (mu/alpha) this model shows backward bifurcations, where mu is the death rate and alpha is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant beta. This eradication effort is explicitly expressed in terms of alpha,r, and mu As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R0 is smaller than the eradication effort for r > 1 + (mu/alpha) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.
我们研究了一个流行病学模型，该模型假定初次感染后的易感性是初次感染前易感性的r倍。当r = 0（r = 1）时，这就是SIR（SIS）模型。当r > 1 +（μ/α）时，该模型呈现后向分岔，其中μ是死亡率，α是恢复率。我们首次表明，对于此类模型，如果我们专注于影响传播率常数β的控制措施，那么我们可以给出根除感染所需最小努力的表达式。这种根除努力可以用α、r和μ明确表示。与没有后向分岔的模型一样，它可以解释为一个繁殖数，但不一定是基本繁殖数。我们为此定义了相关的繁殖数。根除努力可以从地方病稳态进行估计。经典的基本繁殖数R0对于r > 1 +（μ/α）小于根除努力，而对于r的其他值则等于该努力。我们提出的方法适用于具有后向分岔的整个 compartmental 模型类别。
