An Analytic Approximate Solution of the SIR Model  

作　　者：I. Lazzizzera I. Lazzizzera(Department of Physics, University of Trento, via Sommarive 14 Povo (Trento), Italy;Trento Institute for Fundamental Physics and Applications (INFN), via Sommarive 14 Povo (Trento), Italy)

机构地区：[1]Department of Physics, University of Trento, via Sommarive 14 Povo (Trento), Italy [2]Trento Institute for Fundamental Physics and Applications (INFN), via Sommarive 14 Povo (Trento), Italy

出　　处：《Applied Mathematics》2021年第1期58-73,共16页应用数学（英文）

摘　　要：The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure <em>exact</em> (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the <em>exact</em> equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called <em>basic reproduction ratio</em>, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure <em>exact</em> (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the <em>exact</em> equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called <em>basic reproduction ratio</em>, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

关 键 词：SIR Epidemic Model Kermack-McKendrick Model Epidemic Dynamics Approximate Analytic Solution 

分 类 号：O17[理学—数学]