Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Version 1 : Received: 19 July 2020 / Approved: 19 July 2020 / Online: 19 July 2020 (15:35:44 CEST)

A peer-reviewed article of this Preprint also exists.

Journal reference: J. Phys. A Math. Theor. 2020, 53, 505601
DOI: 10.1088/1751-8121/abc65d

## Abstract

We revisit the Susceptible-Infectious-Recovered/Removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t=0, where data is assumed to be available. We address the question on how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be released in part B. Releasing this assumption allows to formulate initial conditions incompatible with the original SIR model.

## Keywords

coronavirus; statistical analysis; extrapolation; parameter estimation; epidemic spreading

## Subject

PHYSICAL SCIENCES, Mathematical Physics

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