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Analytical Parameter Estimation of the Sir Epidemic Model Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2020 ANALYTICAL PARAMETER ESTIMATION OF THE SIR EPIDEMIC MODEL. APPLICATIONS TO THE COVID-19 PANDEMIC. DIMITER PRODANOV 1) Environment, Health and Safety, IMEC, Belgium 2) MMSIP, IICT, BAS, Bulgaria Abstract. The dramatic outbreak of the coronavirus disease 2019 (COVID- 19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modeling and forecasting. The SIR (Susceptible-Infected- Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive an explicit solu- tion. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega function. The present manuscript reports new analytical results and numerical routines suitable for parametric estimation of the SIR model. The manuscript intro- duces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The numerical approach is exemplified with data from the European Centre for Disease Prevention and Control (ECDC) for several European countries in the period Jan 2020 { Jun 2020. Keywords: SIR model; special functions; Lambert W function; Wright Omega function MSC: 92D30; 92C60; 26A36; 33F05; 65L09 1. Introduction The coronavirus 2019 (COVID-19) disease was reported to appear for the first time in Wuhan, China, and later it spread to Europe, which is the subject of the pre- sented case studies, and eventually worldwide. While there are individual clinical reports for COVID-19 re-infections, the present stage of the pandemic still allows for the application of a relatively simple epidemic model, which is the subject of the present report. The motivation behind the presented research was the inten- tion to accurately model the short-term dynamics of the outbreaks of COVID-19 pandemics, which by the time of writing (Sept 2020), has infected more than 27 million individuals worldwide, while by time of the latest submission { more than 43 million cases. The efforts to contain the spread of the pandemic induce sustained social and economic damage. Therefore, the ability to accurately forecast short to medium-term epidemic outbreak's dynamics is of substantial public interest and brings certain sense of urgency to the present work. The present manuscript gives a comprehensive analytical and numerical treat- ment of the SIR (Susceptible-Infected-Removed) epidemiological model. The SIR model was introduced in 1927 by Kermack and McKendrick in 1927 to study the 1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2020 2 ANALYTICAL SOLUTION OF THE SIR MODEL plague and cholera epidemics in London and Bombay [9]. To date the SIR model remains as a cornerstone of mathematical epidemiology. It is a deterministic model formulated in terms of ordinary differential equations (ODEs). The model has been extensively used to study the spread of various infectious diseases (see the monograph of Martcheva [11]). The objective of the present paper is to demonstrate numerical routines for curve- fitting allowing for estimation of the parameters of the SIR model from empirical data. To this end, the paper exhibits an algorithm, which can be used to compute the population variables as functions of time. In contrast to previous approaches, I do not consider the SIR model as an initial value problem but as a problem in the theory of special functions. This change of perspective allows for handling noisy data, e.g. time series having fluctuations caused by delays and accumulation of case reporting. The numerical approach is applied to the COVID-19 incidence and case fatality data in different European countries, having different population densities and dynamics of the epidemic outbreaks. 2. The SIR model The SIR model is formulated in terms of 3 populations of individuals. The S population consists of all individuals susceptible to the infection of concern. The I population comprises the infected individuals. These persons have the disease and can transmit it to the susceptible individuals. The R poulation cannot become infected and the individuals cannot transmit the disease to others. The model comprises a set of three ODEs: β S_(t) = − S(t)I(t) (1) N β I_(t) = S(t)I(t) − γI(t) (2) N R_ (t) = γI(t) (3) The model assumes a constant overall population N = S + I + R. An disease carrier infects on average β individuals per day, for an average time of 1/γ days. The β parameter is called disease transmission rate, while γ { recovery rate. The average number of infections arising from an infected individual is then modelled β by the number R0 = γ , the basic reproduction number. Typical initial conditions are S(0) = S0;I(0) = I0;R(0) = 0 [9]. The model can be re-parametrized using normalized variables as s_ = −si (4) γ 1 i_ = si − gi; g = = (5) β R0 r_ = gi; (6) subject to normalization s + i + r = 1 and time rescaling τ = βt. Therefore, since i(τ) is integrable on [0; 1) then i(1) = 0. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2020 ANALYTICAL SOLUTION OF THE SIR MODEL 3 3. The analytical solution The analytical solution will be formulated first in an implicit form. Since there is a first integral by construction the system can be reduced to two equations: di g = −1 + (7) ds s di s = − 1 (8) dr g Remark 1. From this formulation s S β R = N 0 = 0 ≥ 1 e g γ must hold for the infection to propagate. Re is called the effective reproductive number, while the basic reproduction number is R0 = ReN [21]. In order to solve the model we will consider the two equations separately. Direct integration of the equation 7 gives i = −s + g log s + c where the constant c can be determined from the initial conditions. In the present treatment, the constant c will be left indeterminate to be assigned by the different re-parametrization procedures. The s variable can be expressed explicitly in terms of the Lambert W function [4]: i−c ! e g s = −gW − (9) ± g where the signs denote the two different real-valued branches of the function. Note, that both branches are of interest since the argument of the Lambert W function is negative. Therefore, the ODE 5 can be reduced to the first-order autonomous system i−c ! ! e g i_ = −ig W − + 1 (10) ± g valid for two disjoined domains on the real line. The ODEs can be solved for the time τ as Z di − = gτ (11) i−c e g i W± − g + 1 Remark 2. There is another equivalent form of the system using the Wright Ω function [5] since i−c ! e g i − c W − = Ω − log (−g) g g so that i − c i_ = −gi Ω − log g ∓ iπ + 1 g Remarkably, Bronstein [1] gave a proof that the Lambert W and Wright Ω are not Liouvillian. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2020 4 ANALYTICAL SOLUTION OF THE SIR MODEL The s variable can be determined by substitution in equation 4, resulting in the autonomous system s_ = −s (−s + g log s + c) (12) which can be solved implicitly as Z ds = τ (13) s (s − g log s − c) Finally, the r variable can also be conveniently expressed in terms of i. For this purpose we solve the differential equation dr g −1 = = i−c di s − g e g 1 + W± − g Therefore, i−c !! e g r = c − g log −gW − = c − g log s 1 ± g 1 by Prop. 4. On the other hand, i−c !! i−c ! i−c !! e g e g e g g log −gW − = g log − W − = g g g i−c ! e g − gW − + i − g log g − c = s + i − g log g − c g So that i−c ! e g r = gW − − i + c g 1 For the purposes of curve fitting we assume that i(−∞) = r(−∞) = 0. Therefore, − c ! e g c = −gW − 1 − g Furthermore, ds s = − dr g r c1 whence log s = − g + g in accordance with the previous result. 3.1. Peak value parametrization. The upper terminal of integration can be determined by the requirement for the real-valuedness of i. This value of i is denoted as im; that is im−c ! e g W − = −1 ± g Therefore, im = c − g log g − g (14) The peak-value parametrization is supported by the following result. Proposition 1. i(t) attains a global maximum i = im = c − g log g − g. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 November 2020 ANALYTICAL SOLUTION OF THE SIR MODEL 5 Proof. We use a parametrization for which i(0) = im. Then i−im −1 i_(τ) = −gi W± −e g + 1 It follows that i_(0) = 0; i(0) = im so im is an extremum. In the most elementary way since W(z) should be real-valued then i−im −1 i − im −e g ≥ −1=e =) ≤ 0 g Hence, i ≤ im. z−im −1 If we consider formally the phase space z × y = −gz W± −e g + 1 the following argument allows for the correct branch identification. For i ! −∞ z−im −1 z−im −1 + W− −e g ! −∞ so y < 0; while W+ −e g ! 0 so y > 0. There- fore, if we move the origin as t(0) = im then conveniently Z i dz − = gτ; τ > 0 (15) z−im −1 im z W+ −e g + 1 Z i dz − = gτ; τ ≤ 0 (16) z−im −1 im z W− −e g + 1 Furthermore, the recovered population under this parametrization is i−im −1 −im −1 r = gW± −e g − gW− −e g − i (17) under the same choice of origin.
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