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

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.

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

“…We further present a highly accurate analytical approximation for all SIRV-functions as a function of τ. Our new analytical solutions reduce in the appropriate limit to the earlier [2,3,11] solutions for the non-vaccination case b = 0. We also consider as special cases the non-recovery case k = 0 and the special case of equal values of k = b.…”

“…In the general case b = k the transcendental Eq. ( 46) is solved in terms of the realvalued Lambert functions [11]. We discuss below which of the two existing real-valued Lambert functions W 0 (principal) and W −1 (non-principal), respectively, applies in different parameter ranges.…”

“…As discussed before in Appendix G of Ref. [11] there is a single real-valued solution W 0 (y) of Lambert's equation for arguments y ≥ 0, referred to as the principal. There are two real valued solutions for y ∈ [−e −1 , 0]: the principal one W 0 ∈ [−1, 0] and the non-principal solution W −1 ≤ −1.…”

First Consistent Determination of the Basic Reproduction Number for the First Covid-19 Wave in 71 Countries from the SIR-Epidemics Model with a Constant Ratio of Recovery to Infection Rate

“…In terms of the reduced time , accounting for arbitrary but given time-dependent infection rates, and the medically interesting daily rate of new infections (5) where the dot denotes a derivative with respect to t, the SIR-model equations can be written as (6) with and the dimensionless For the special and important case of a time independent ratio K(t) = k = const., new analytical results of the SIR-model (6) have been recently derived. 4 For a growing epidemics with time the constant ratio k < 1 has to be smaller than unity corresponding to the initial infection rate at time t = 0 being larger than the initial recovery rate , both in units of days . The new analytical solutions assume that the SIR equations are valid for all times , and that the time refers to the 'observing time' when the existence of a pandemic wave in the society is realized and the monitoring of newly infected persons is started.…”

Epidemics forecast from SIR-modeling, verification and calculated effects of lockdown and lifting of interventions

“…where the dot denotes a derivative with respect to t. For the special and important case of a time-independent ratio K(t) = k = const. new analytical results of the SIRmodel (1) have been recently derived 17 -hereafter referred to * rsch@tp4.rub.de (R.S. ), mk@mat.ethz.ch (M.K.)…”