On an interval prediction of COVID-19 development based on a SEIR epidemic model Denis Efimov, Rosane Ushirobira To cite this version: Denis Efimov, Rosane Ushirobira. On an interval prediction of COVID-19 development basedona SEIR epidemic model. [Research Report] Inria. 2020. hal-02517866v6 HAL Id: hal-02517866 https://hal.inria.fr/hal-02517866v6 Submitted on 3 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 On an interval prediction of COVID-19 development based on a SEIR epidemic model Denis Efimov1 and Rosane Ushirobira1 Abstract—In this report, a revised version of the well-known the epidemic of COVID-19 in China (other similar SIR/SEIR- mathematical outbreak SEIR model is used to analyze the type models used recently for modeling SARS-CoV-2 virus epidemic’s course of COVID-19 in eight different countries. The are [8], [9], [10], [11]). The model we adopt in this work is as proposed model enhancements reflect the societal feedback on pandemic and confinement features. The parameters of the SEIR follows (we do not consider the influence of the natural birth model are identified by using publicly available data for France, and mortality, since for the short period of analysis considered Italy, Spain, Germany, Brazil, Russia, New York State (US), and here the population may be assumed quasi-constant): China. The identified model is then applied for the prediction of the SARS-CoV-2 virus propagation under different conditions of pt−t It + rt−t Et confinement. For this purpose, an interval predictor is designed S = S − b p r S ; (1a) allowing variations and uncertainties in the model parameters t+1 t N t to be taken into account. The code and the utilized data are pt−t It + rt−t Et available in Github. E = (1 − s − s 0)E + b p r S ; (1b) t+1 t N t It+1 = (1 − g − m)It + sEt ; (1c) I. INTRODUCTION R = R + gI + s 0E ; (1d) The SEIR model is one of the most elementary com- t+1 t t t partmental models of epidemics [1]. It is very popular and Dt+1 = Dt + mIt ; (1e) widely used in different contexts [2]. This model describes where t 2 N (the set of non-negative integers) is the time the evolution of the relative proportions of four classes of counted in days (t = 0 corresponds to the beginning of individuals in a population of constant size, see a general measurements or prediction), N = S + E + I + R + D denotes scheme given in Fig. 1. Namely, the susceptible individuals the total population, the parameter 0 < g < +¥ represents the S, capable of contracting the disease and becoming infectious; recovery rate, 0 < m < +¥ is the mortality rate, the parameter the asymptomatic E and symptomatic I infectious, capable of 0 < b < +¥ corresponds to the rate of the virus transmission transmitting the disease to susceptible; and the recovered R, from infectious/exposed to susceptible individuals during a permanently immune after healing or dying (if the number of contact, 0 < s;s 0 < +¥ are the incubation rates at which deaths is of special interest, then an additional compartment the exposed develop symptoms or directly become recovered D can be introduced). Such a simple model represents well a without a viral indication, 0 ≤ pt < +¥ corresponds to the generic behavior of epidemics (plainly as a series of transitions number of contacts for the infectious I (it is supposed that between these populations), and a related advantage consists in infected people with symptoms are in quarantine, then the a small number of parameters to be identified (three transition number of contacts is decreased), pt ≤ rt < +¥ is the number rates in Fig. 1: s, g and b). This is an important outcome of contacts per person per day for the exposed population E in the case of a virus attack with a limited amount of data (in the presence of confinement and depending on its severity, 1 available. That is the case of the current worldwide situation this number is time-varying), and tp;tr > 0 are the delays in under the presence of the SARS-CoV-2 virus. the reactions of the compartments on variations of quarantine There are many types and variations of SEIR models [1] conditions (we assume that if t < tp or t < tr, then pt−tp = p0 (e.g., in the simplest case the classes E and I are modeled or rt−tr = r0, respectively). Compared to the model in [7], the at once, leading to a SIR model). A specificity of COVID- inflow/outflow variables from/to other regions for each state 19 pandemics is the global confinement imposed by the are not considered in our analysis. majority countries, which influences the virus dynamics [3]. Remark 1. In the versions 1 − 4 of this report (they can In the literature, numerous suggestions are answering the be found in HAL), the compartment D was not explicitly question of how to reflect the confinement characteristics in considered in the model (it was a part of R) and we used the mathematical models [4], [5], [6]. 0 s = 0 and tp = 0. A. Considered model B. Societal feedback and confinement influence in the model This note is based on a modified SEIR discrete-time model proposed in [7], where it has been used to model the trend of To consider the reaction of society on confinement and virus propagation, we introduced the delays tp and tr in the 1 Denis Efimov and Rosane Ushirobira are with Inria, Univ. Lille, seclusion inputs p and r , respectively. CNRS UMR 9189 - CRIStAL, F-59000 Lille, France, Denis.Efimov, t t [email protected] The idea behind tr is that after the activation of the 1This note was initially written on March 23 and updated June 3, 2020. quarantine, several days pass before changes in the disease 2 propagation become detectable (such an effect can be easily observed in the data for all analyzed countries). Roughly speaking, the increase in the number of infected individuals E and I is predefined by the number of contacts in the previous days, when the confinement was not yet imposed, for example. We assume that during the phase of active lockdown, rt−t = pt−t always holds, i.e., the number of contacts for r p Fig. 1. A schematic representation of SEIR model asymptomatic E and symptomatic I infected populations is the same (it corresponds to the case of the society that follows well the government requirements). It also implies that the delays can be selected in the corre- The delay tp is used to model the clustering effect of sponding limits: the confinement: under restrictions on displacement activities, people are constrained to stay in their district and visit a 2 ≤ tr ≤ 12; tr + 2 ≤ tp; limited number of attractions (shops, pharmacies, hospitals, where the condition t > t entails that the clustering starts etc.). So the population can be considered to be divided into p r to be important after the effect of confinement becomes smaller groups, where after some time the chances to meet an significant (adding an incubation period). infected person start to decay (e.g., in such a group, there is no The numbers of contacts have to be selected separately for infected person, or the individual was isolated, or the whole each country. For example, we may take the values of [7] and group can be infected, but anyway the virus propagation is make some reduction related with a smaller population density basically stopped). in the considered countries: Remark 2. Another way of inclusion of societal feedback on virus development is the substitution: p = 3 (number of contacts in quarantine); b Q b −! ; pN = 15 (number of contacts in normal mode); 1 + hIt where 0 < h < +¥ is a tuning parameter. In this case, we pR = 10 (number of contacts in relaxed quarantine); model the effect of natural augmentation of confinement strict- pC = 0:1 (number of contacts under clustering). ness, when society becomes aware of the problem following Then the input p 2 fp ; p g and r 2 fp ; p ; p ; p g for all the increased number of infected or dead people (I implicitly t Q C t Q N R C t 2 . represent them, or it can also be explicitly replaced with D). N The identification of the model parameters must be per- To this end, we decrease the virus transmission rate b with formed using statistics published by authorities2. As a worthy the growth of the number of infected/dead individuals. This remark, many researches devoted to the estimation and iden- variant has been tested, but we prefer to use the delays t and p tification of SIR/SEIR models were developed by now, and t since the parameter identification is simpler in this case. r several in the last few years, like [13], [14], [15], [16], [17], to mention a few. C. Model parameters Therefore, the SEIR model (1) has seven parameters to be 0 D. Uncertainty and prediction identified or assigned: s, s , tp, tr, g, m and b. 1) Generic observations: The parameters s, g, m and b Since the measured data and parameters contain numerous represent, respectively, the rate of changes between the states E uncertainties and perturbations, it is then difficult to carry out to I, I to R, I to D and S to E +I (as in Fig.