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13AGENT-BASED MODELLING OF THE FIRST AND THE SECOND WAVES OF COVID-19 SPREADING IN RUSSIAN FEDERATION REGIONS
MIKHAIL KIRILLIN1, EKATERINA SERGEEVA1, ALEKSANDR KHILOV1, VALERIYA PEREKATOVA1, DARIA KURAKINA1, ILYA FIKS1, NIKOLAY SAPERKIN2, MING TANG3, AND ELBERT MACAU4
1 Institute of Applied Physics RAS, Nizhny Novgorod, Russia, 2 Privolzhsky Research Medical University, Nizhny Novgorod, Russia 3School of Physics and Electronic Science, East China Normal University, Shanghai, China 43Instituto de Ciencias e Tecnologia, Universidade Federal de Sao Paulo, Sao Paulo, Brasil
mkirinin@yandex. ru
ABSTRACT
The COVID-19 pandemics remains one of the largest worldwide challenges. Necessity of effective systemic aids for the minimization of losses leads to the requirement of adequate models allowing to predict the impact of different factors on the spread of disease. Traditionally employed simulation approaches are based on derivatives of a SIR model, which major drawback is not accounting for random factors. Agent-based simulation models provide a suitable solution with the possibility to accurately account for such factors as age structure of population, features of isolation and self-isolation strategies and testing strategies, presence of super-spreaders etc. In this paper we report on the results of modelling the first and the second waves of COVID-19 spreading in regions of Russian Federation. The simulations approach employs an agent-based model with a general pool with the inclusion of a model of the population testing strategy. The model accounts for key epidemiologic characteristics, such as population age distribution, spreading rate, isolation factors etc. It is demonstrated that the daily case curves for different regions are reproduced well with the same spreading rate parameter, while the initial number of infected agents, testing and isolation strategies, which serve as other tuning parameters of the model, are region-dependent.
INTRODUCTION
Prediction of further development of disease outbreaks for the purpose of timely introduction of preventive measures require reliable tools for pandemic spread simulations. Several classes of models are traditionally employed for the prognosis of the spread of infections.
Regression models allow to obtain rapid estimations of the diseases spread [1,2]. They include non-adaptive models, which ignore local perturbations of epidemical characteristics and are not suitable for short-term prognosis, and adaptive models, which are predominantly applied for short-term prognosis only. Short-term prognosis can be made with the application of autoregressive moving average models [3], while autoregressive integrated moving average model allows both for short-term and long-term prognosis [4]. Dynamic Bayesian networks [5], neural networks and other machine learning based methods are applicable for short-term prognosis only. Feedforward neural networks and backpropagation algorithm provide with prognosis of infections spread [6].
Long-term prognosis is traditionally made with the application of dynamic systems based on differential equations, which are the class of compartmental models. SIR model, firstly introduced by Kermack and McKendrick [7], is based on the division of the population into three groups — susceptible (S), infected (I), recovered (R) — and the description of their interaction with non-linear differential equations. Further development of compartmental models includes the accounting for bigger number of groups, such as exposed (E), hospitalized (H), critical (C), dead (D), and those at the quarantine (Q) or isolation (J). SEIR model has been applied for simulation of COVID-19 spread [8-11] and the estimation of the efficacy of governmental measures during early COVID-19 spread [12]. The major drawback of compartmental models is the fact that they do not account for random factors and individual characteristics of population members.
For both short-term and long-term prognosis, it is reasonable to employ individually-oriented models, which include so-called agent-based models. Agent-based model consists of the description of every member of population (agent) by a set of characteristics with the determination of rules of interactions between the agents. Agent-based models showed their efficiency in the description of the propagation of infections, such as Ebola [13] and flu [14], in the population of different sizes. They were also applied for the modeling of COVID-19 development and regress in different cities such as Helsinki [15], New York [16], Singapore [17]. Some agent-based models are based on previously developed models for the prediction of flu pandemics, for example, NotreDame-FRED model [18], model of Ferguson's research group from Imperial College London [19] and model for simulation of the COVID-19 pandemic in Australia [20]. A number of studies utilized agent-based models for assessing the impacts of universal face mask wearing [21], digital contact tracing [22-24] and social distancing [24-26] as well as analyzing various intervention scenarios [16,19,24,27-31]. Some agent-
based models are developed for modeling COVID-19 transmission in small communities such as a university [32], a supermarket [33] and a small town [34].
The aim of this paper is the development of an agent-based model capable of simulating and predicting the progress of the COVID-19 outbreak in different regions of the Russian Federation. Another important problem to be resolved in this study is the determination of the key model parameters that provide the agreement of the simulated dynamics and actual statistics on daily new infection cases and deaths associated with COVID-19.
AGENT-BASED COVID-19 SPREADING MODEL
A general pool model, in which all the individuals (agents) may interact with each other, was employed for the fit of official statistics of COVID-19 spread. Every agent is described with a number of binary states, which values are scenario-dependent and governed by the Monte Carlo simulations: susceptible, infected, contagious, with disease manifestations, critical, recovered. The simulations feature day-scale time resolution. The model accounts for the population age structure based on the information on different disease progression in different age groups. Typical lengths of disease manifestation, progression, critical, and symptomless periods are introduced in accordance with the available data. Any agent with status "critical" may die on any day of critical period with given death probability pd.
The average number of individuals R, to which an infected agent may transmit the infection within one week in the case of no restriction measures applied, is considered as the main model parameter (spreading rate) directly related to infection transmission coefficient. Given that the probabilities to be infected from different agents are independent, the contamination probability P for a given agent interacting with the general pool in a particular day is calculated as P =
RN ■
—l-, where N is the number of infected individuals in the general pool in the current day and Nt is the total number of
agents in the considered population. The Nt number is chosen for simulations in accordance with the population of the simulated region.
The developed simulation model accounts for the efficiency of the following restrictive measures with the employment of a so-called self-isolation index, firstly introduced by Yandex (Russia) during the first wave of COVID-19 outbreak, which represents a cumulative parameter reflecting population activity based on both traffic information and activities in different internet services. The self-isolation index varies in the range between 0 and 5, and it is assumed during the simulations that its value is proportional to the percentage of agents that obey the restrictive rules and do not interact with the general pool in the current day.
The introduced rules of testing are an important part of the model, since the real data for the comparison to the results of numerical simulation are daily statistics on number of newly revealed cases and deaths. In the model, the number of daily tests for each region is either taken from official statistics or determined from the daily number of cases for the entire Russian Federation in proportion to the region population. The tunable parameter is a number of tested agents, which were in contact with an agent with positive COVID-19 test.
10 Feb 10 Apr 10Jun 10 Aug 10 Oct 10 Dec 10 Feb 10 Apr 10 Feb 10 Apr "lOJun 10 Aug 10 Oct 10 Dec 10 Feb 10 Apr
Date Date
(a) (b)
Fig. 1. Comparison of simulated scenarios and real statistical data for daily numbers of newly revealed COVID-19 (a) and lethal COVID-19 associated (b) cases in Moscow within the period of 23 March 2020 - 07 April 2021.
RESULTS
Figure 1 demonstrates the results of simulations of the epidemic progress in Moscow (Nt = 11.4 -106) derived from modelling. The spreading rate value were extracted from the simulations of the first wave performed in our previous study [35].
To provide a best-fit scenario, we manipulated with other parameters of the model, namely, the number of initial infected agents, percentage of deaths among agents in the critical state pd, and testing strategies. Each scenario was constructed by averaging five scenarios that are the closest to the real statistical data over a total of 10 realizations with the same parameters. In Fig. 1a solid green lines demonstrate calculated dynamics of daily detected cases and red solid lines demonstrate dynamics of total COVID-19 cases, while black dots indicate official statistics data, which were fitted with data for detected cases in simulations. The results indicate a larger percentage if hidden cases in the case of the second wave in comparison with the first one. The deaths statistics demonstrate hidden cases resulting in exceeding the number of daily cases in official statistics by the simulations results. This could be explained by COVID-induces deaths cases which were attributed to other causes in official statistics. The simulations were also performed for Nizhny Novgorod region and Novosibirsk region demonstrating similar dynamics. They demonstrate the prediction abilities of the model for regions with relatively small number of confirmed COVID-19 daily cases and single deaths.
CONCLUSION
In this paper we presented an agent-based model of COVID-19 epidemic spread with the employment of Monte Carlo simulation principles. The model is able to account for the age-dependent disease development, restrictive measures as well as testing system. It was validated on the statistical data for daily new cases and deaths which were reported during 1st and 2nd waves of COVID-19 pandemics from two representative regions of Russia, the Moscow city and Nizhny Novgorod region. The results of simulations show the possibilities of developed model for the prediction of disease spread with the application of restrictive measurements and different testing strategies.
ACKNOWLEDGEMENTS
The study is supported by RFBR (project no. 20-51-80004), CNPq (project no. 441016/2020-0), and NSFC (project no. 82161148012).
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