CC BY
13DOI 10.24412/CL-37135-2023-1-32-36
MODELING OF COVID-19 SPREAD: FROM CITY SCALE TO COUNTRY SCALE
MIKHAIL KIRILLIN1, ALEKSANDR KHILOV1, VALERIYA PEREKATOVA1, EKATERINA SERGEEVA1, DARIA KURAKINA1, ILYA FIKS1, NIKOLAY SAPERKIN1,2, MING TANG3; YONG ZOU3, ELBERT MACAU4, AMERICO CUNHA JR5 AND EFIM PELINOVSKY16
1 Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the Russian
Academy of Sciences, Russia Privolzhsky Research Medical University, Russia 3 School of Physics and Electronic Science, China
4 Federal University of Sao Paulo, Brazil
5 Rio de Janeiro State University, Brazil
6 National Research University - Higher School of Economics, Russia
alhil@inbox.ru
ABSTRACT
COVID-19 outbreak, which resulted in further COVID-19 pandemics, was one of the toughest worldwide challenges in recent years. It clearly showed the necessity of the development of models allowing for simulation and reasonable prediction of epidemiologic processes. Traditional approach is based on compartment models, such as SIR model, firstly introduced by Kermack and McKendrik almost 100 years ago [1]. SIR model employs the separation of population into several groups, such as susceptible (S), infected (I), and recovered (R), and further solution of differential equations describing the dynamics and interactions of these groups. Enhanced compartment models involve larger number of groups into calculations [2], distinguishing exposed (E), dead (D), in critical state (C) and others. SEIR model and its modifications were employed for fitting and predicting the spread of Ebola [3], seasonal influenza [4] and, of course, COVID-19 in a number of countries, e.g. Sweden [5], Japan [6], or large regions of countries, e.g. Italian Region of Lombardy [7]. However, compartment models are unable to account neither for random factors of epidemiologic processes nor individual behavior of persons.
We report on several developed models for the simulation of COVID-19 spread at different epidemiologic levels. We performed the analysis of the daily dynamics of newly revealed COVID-19 cases and COVID-19 associated lethal cases at macro level (whole country or big region), at meso level (megapolis, a region as a whole or a region with the realization of pendulum migration between regional center and periphery cities) and at micro level (city with the respect to its social connections). Analytical models based on logistic [8] and Gompertz [9] equations were developed for macro level. In logistic model total revealed cases N(t) is determined by equation
dN ( N \
■^"■"{'-id (1)
which gives exact solutions for N(t) and daily revealed cases I(t), respectively:
"(0 =_______(2 )
/(t) = 3T =
1 + exp[—r(t — t)] dN rNmexp[—r(t —t)]
d t { 1 + exp[ - r( t-r)]} 2 In Gompertz model the equation for total revealed cases N(t) changes to
dN t In iV
(3)
which leads to the following dynamics of N(t) and I(t), respectively:
/W
"( t) = "dexp
ir* dN i (Noi\
dN í
— = rN ( 1 —
dt V
respec
'O n©
In W,
-) (4)
exp
exp
r(t — t) r(t — t)
(5)
(
t)
JV„
(6)
We analyzed three approaches to the description of dynamics newly revealed cases I(t) and COVID-19 associated lethal cases D(t). The first and the second assume that during each wave (1, 2, ..., k) of COVID-19 spread dynamics of daily revealed cases and deaths are described independently and by standalone solutions, which allows to express full dynamics as the sum with the respect to number of waves in logistic model
rjV« ¿exp( - r£[ t-Tj])
i=l ft
{ l + exp( - r[ t-Tj)}2
( )
z(t)=L?
rZ« ¿exp( - r[ t-Tj])
{ l + exp(-rj[ t-Tj])} 2 (8)
or Gompertz model
/(t)=Zr'ln(£)exp(ln
¿=1
ft , z(t)=Zrjin(ZZ^)exp(in
[Vnf rj(t-Tj)l
V ■ Li *ooI- exp V ■ ivOO I
[Awl r( t-Tj)
exp Zoo j +
rj(t-Tj) V ■
J vooi
r.(t-Tj) ' A™
(10)
while the third approach is aimed to automodel solution for daily new lethal cases dynamics D(t) calculated from newly revealed cases I(t) (Eq. 9) by multiplication and time shifts of corresponding dynamics for separate waves:
Z(t) =^«j/j(t-Atj) (11)
It should be noted that (3) is not a solution of Gompertz equation. Developed approaches were employed for the study of COVID-19 spread in different countries and regions, including Russian Federation (Fig.1) and Nizhny Novgorod Region of Russian Federation (Fig. 2).
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Figure 1: Daily dynamics of newly revealed cases (a, c) and COVID-19 associated lethal cases (b, d, e) in Russian Federation calculated by logistic model (a, b), Gompertz model (c, d) and automodel solution for daily new deaths (e)
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Figure 2: Daily dynamics of newly revealed cases (a, c) and COVID-19 associated lethal cases (b, d, e) in Nizhny Novgorod Region calculated by logistic model (a, b), Gompertz model (c, d) and automodel solution for daily new deaths
(e)
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For the description of COVID-19 spread at meso and micro levels agent-based modeling was employed. This approach assumes that the population is a pool of agents interacting with each other. Individual behavior characteristics are determined by set of constant (e.g. age) and variable (e.g. self-isolation index) parameters. Agent based-modeling also allows accounting for random epidemic factors, restrictive measurements, vaccination etc. Agent-based modeling proved itself as a go-to tool for simulation of epidemiologic process at various levels from supermarket [10] to single city [11] and even the country [12]. We developed agent-based model for COVID-19 spread with accounting for testing system, restrictive measurements and vaccination [13, 14]. The model was also upgraded to multicentral, when studied region is assumed as a number of pools representing regional center and periphery cities with transport connections and pendulum migration between them [15]. The comparison of theoretical and simulated data with accessible official COVID-19 statistics revealed good agreement both for modelling the whole region Nizhny Novgorod Region and single cities within it, which proves the capabilities of developed models.
ACKNOWLEDGEMENTS
The study was supported by supported by RFBR (project no. 20-51-80004), CNPq (project no. 441016/2020-0), NSFC (project no. 82161148012), and partly supported in the frames of the Governmental Project of the Institute of Applied Physics RAS (Project #FFUF-2021-0014).
REFERENCES
[1] W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. of the Royal Society of London. Series A 115(772), 700-721, 1927.
[2] S. He, Y. Peng, K. Sun, SEIR modeling of the COVID-19 and its dynamics, Nonlinear Dyn. 101(3), 1667-1680, 2020.
[3] A. Bouba, K.B. Helle, K.A. Schneider, Predicting the combined effects of case isolation, safe funeral practices, and contact tracing during Ebola virus disease outbreaks, Plos One, 18(1), e0276351, 2023.
[4] V.N. Leonenko, S.V. Ivanov, Fitting the SEIR model of seasonal influenza outbreak to the incidence data for Russian cities, Russian J. Numer. Anal. Mathem. Modell., 31(5), 267-279, 2016.
[5] M. Khairulbahri, The SEIR model incorporating asymptomatic cases, behavioral measures, and lockdowns: Lesson learned from the COVID-19 flow in Sweden, Biomed. Signal Process. Control, 81, 104416, 2023.
[6] Q. Sun, T. Miyoshi, S. Richard, Analysis of COVID-19 in Japan with extended SEIR model and ensemble Kalman filter, J. Comput. Appl. Math., 419, 114772, 2023.
[7] J.M. Carcione, J.E. Santos, C. Bagaini, J. Ba, A Simulation of a COVID-19 Epidemic Based on a Deterministic SEIR Model, Front. Public Health, 8, 230, 2020.
[8] E. Pelinovsky, A. Kurkin, O. Kurkina, M. Kokoulina, A. Epifanova, Logistic equation and COVID-19, Chaos Solitons Fractals 140, 110241, 2020.
[9] E. Pelinovsky, M. Kokoulina, A. Epifanova, A. Kurkin, O. Kurkina, M. Tang, E. Macau, M. Kirillin, Gompertz model in COVID-19 spreading simulation, Chaos Solitons Fractals 154, 111699, 2022.
[10] F. Ying, N. O'Clery, Modelling COVID-19 transmission in supermarkets using an agent-based model, Plos One 16(4), e0249821, 2021.
[11] A. Shoukat, T.N. Vilches, S.M. Moghadas, P. Sah, E.C. Schneider, J. Shaff, A. Ternier, D.A. Chokshi, A.P. Galvani, Lives saved and hospitalizations averted by COVID-19 vaccination in New York City: a modeling study, Lancet Reg Health Am 5, 100085, 2022.
[12] G.N. Rykovanov, S.N. Lebedev, O.V. Zatsepin, G.D. Kaminskii, E.V. Karamov, A.A. Romanyukha, A.M. Feigin, and B.N. Chetverushkin, Agent-based simulation of the COVID-19 epidemic in Russia, Her. Russ. Acad. Sci. 92(8), 747-755, 2022.
[13] M. Kirillin, E. Sergeeva, A. Khilov, D. Kurakina, and N. Saperkin, Monte Carlo simulation of the covid-19 spread in early and peak stages in different regions of the Russian Federation using an agent-based modelling, in Saratov Fall Meeting, Chinese-Russian workshop on Biophotonics and Bioimaging-2020 1, 2020.
[14] M. Kirillin, A. Khilov, V. Perekatova, E. Sergeeva, D. Kurakina, I. Fiks, N. Saperkin, M. Tang, Y. Zou, E. Macau, E. Pelinovsky, Simulation of the first and the second waves of COVID-19 spreading in Russian Federation regions using an agent-based model, J. Biomed. Photonics Eng. 9(1), 010302, 2023.
[15] M. Kirillin, A. Khilov, V. Perekatova, E. Sergeeva, D. Kurakina, I. Fiks, N. Saperkin, M. Tang, Y. Zou, E. Macau, E. Pelinovsky, Multicentral Agent-Based Model of Four Waves of COVID-19 Spreading in Nizhny Novgorod Region of Russian Federation, J. Biomed. Photonics Eng. 9(1), 010306, 2023.
