Non-linear incidence and stability of infectious disease models
Non-linear incidence and stability of infectious disease models
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DOI：
10.1093/imammb/dqi001
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发表时间：
2005-06-01
影响因子：
1.1
通讯作者：
Maini, PK
中科院分区：
文献类型：
--
作者：
Korobeinikov, A;Maini, PK
本文考虑传染病发病率非线性形式对流行病学模型动力学的影响。我们考虑了非线性发病率的一种非常一般的形式（事实上，我们假设发病率是由一个任意函数f （S， I， N）给出的，该函数受一些生物学上可行的条件的约束）和各种流行病学模型。我们证明了在人口规模不变的假设下，这些模型表现出渐近稳定的稳定状态。确切地说，我们证明了发病率相对于感染个体数量的凹性是稳定性的充分条件。如果感染率在感染人数上呈凹形，则我们考虑的模型要么具有唯一且稳定的地方性平衡状态，要么根本没有地方性平衡状态；在后一种情况下，无感染的平衡状态是稳定的。对于形式g(I)h(S)的发生率，我们通过构造Lyapunov函数并使用直接Lyapunov方法证明了全局稳定性。值得注意的是，系统动力学与发病率如何取决于易感个体的数量无关。我们使用SIRS模型和SEIRS模型作为案例研究来证明这一结果。对于其他的隔室流行病模型，分析结果也非常相似，并且得出了相同的结论，即平衡状态的稳定性。
In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds.