Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates
Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates
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DOI：
10.1016/j.amc.2014.03.030
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发表时间：
2014-06-01
影响因子：
4
通讯作者：
Mak, M. K.
中科院分区：
文献类型：
--
作者：
Harko, Tiberiu;Lobo, Francisco S. N.;Mak, M. K.
本文给出了SIR流行病模型的参数形式的精确解析解。通过使用精确解，我们调查了一些显式模型对应的固定值的参数，并表明，数值解再现精确的解析解。我们还表明，广义的SIR模型，包括出生和死亡，描述了一个非线性系统的微分方程，可以减少到阿贝尔型方程。将具有生命动力学的复杂SIR模型约化为Abel型方程，可以大大简化其性质的分析。利用摄动方法得到了Abel方程的幂级数形式的通解，并证明了具有生命动力学的SIR模型的通解可以用精确的参数形式表示. (c)2014爱思唯尔公司All rights reserved.
In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form. (c) 2014 Elsevier Inc. All rights reserved.