Agent-based mathematical model of COVID-19 spread in Novosibirsk region: Identifiability, optimization and forecasting

Olga Krivorotko; Mariia Sosnovskaia; Sergey Kabanikhin

doi:10.1515/jiip-2021-0038

Article

Agent-based mathematical model of COVID-19 spread in Novosibirsk region: Identifiability, optimization and forecasting

Olga Krivorotko , Mariia Sosnovskaia and Sergey Kabanikhin

Published/Copyright: April 4, 2023

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From the journal Journal of Inverse and Ill-posed Problems Volume 31 Issue 3

Abstract

The problem of identification of unknown epidemiological parameters (contagiosity, the initial number of infected individuals, probability of being tested) of an agent-based model of COVID-19 spread in Novosibirsk region is solved and analyzed. The first stage of modeling involves data analysis based on the machine learning approach that allows one to determine correlated datasets of performed PCR tests and number of daily diagnoses and detect some features (seasonality, stationarity, data correlation) to be used for COVID-19 spread modeling. At the second stage, the unknown model parameters that depend on the date of introducing of containment measures are calibrated with the usage of additional measurements such as the number of daily diagnosed and tested people using PCR, their daily mortality rate and other statistical information about the disease. The calibration is based on minimization of the misfit function for daily diagnosed data. The OPTUNA optimization framework with tree-structured Parzen estimator and covariance matrix adaptation evolution strategy is used to minimize the misfit function. Due to ill-posedness of identification problem, the identifiability analysis is carried out to construct the regularization algorithm. At the third stage, the identified parameters of COVID-19 for Novosibirsk region and different scenarios of COVID-19 spread are analyzed in relation to introduced quarantine measures. This kind of modeling can be used to select effective anti-pandemic programs.

Keywords: COVID-19; data analysis; inverse problem; optimization; forecasting; Covasim software; regularization; identifiability; OPTUNA

MSC 2010: 65J20; 49Q12; 92C60

Funding source: Royal Society

Award Identifier / Grant number: 20-51-10003

Funding source: Council on grants of the President of the Russian Federation

Award Identifier / Grant number: MK-4994.2021.1.1

Funding source: Ministry of Science and Higher Education of the Russian Federation

Award Identifier / Grant number: 075-15-2022-281

Funding statement: This research is supported by the Russian Foundation for Basic Research, the Royal Society of London (project no. 20-51-10003) – investigation of the inverse problem for agent-based model (Sections 3, 4 and 5), by the Council for Grants of the President of the Russian Federation (project no. MK-4994.2021.1.1) – data analysis (Section 2), and by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation – short history review and analysis of numerical results.

A Comparison of optimization methods for solving the inverse problem

Analysis of the papers describing epidemiological AB models has demonstrated that no algorithm could be considered superior for identification of model parameters [12]. According to the paper “… it appears that calibrating individual-based models in epidemiological studies remains more of an art than a science.”

In our model, unknown parameters vector $\vec{q}$ $q →$ was calibrated using the OPTUNA hyperparameter optimization software [57], which is one of the latest optimizers designed to adjust hyperparameters in machine learning algorithms and neural networks. Two methods from this package (tree-structured Parzen estimator (TPE) and covariance matrix adaptation evolution strategy (CMA-ES)) were applied for comparative purposes.

A.1 Tree-structured Parzen estimator (TPE)

The tree-structured Parzen estimator (TPE) in many ways is similar to the Bayesian optimizer [38]. However, unlike the Bayesian optimizer that calculates $p (J (\vec{q}) | \vec{q})$ $p ⁢ ( J ⁢ ( q → ) | q → )$ , TPE calculates $p (\vec{q} | J (\vec{q}))$ $p ⁢ ( q → | J ⁢ ( q → ) )$ and $p (J (\vec{q}))$ $p ⁢ ( J ⁢ ( q → ) )$ to determine the parameters domain to minimize functional 𝐽 by performing Parzen window density estimation, to generate two separate distributions specifying the high and low-quality regions of the input-space respectively. For this, $l (\vec{q})$ $l ⁢ ( q → )$ and $g (\vec{q})$ $g ⁢ ( q → )$ probability distributions are introduced. Here, $l (\vec{q})$ $l ⁢ ( q → )$ is interpreted as representing the probability of a region in the input space yielding a high-quality observation, while similarly, $g (\vec{q})$ $g ⁢ ( q → )$ represents low-quality regions. A full TPE optimization procedure is described in [6].

A.2 CMA-ES

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a stochastic derivative-free numerical optimization algorithm for difficult (non-linear, non-convex, ill-conditioned) optimization problems in continuous search spaces. It belongs to the class of evolutionary algorithms and evolutionary computation. An evolutionary algorithm is broadly based on the principle of biological evolution, namely the repeated interplay of variation (via recombination and mutation) and selection: in each generation (iteration), new individuals are generated by variation, usually in a stochastic way, of the current parental individuals. Then some individuals are selected to become the parents in the next generation based on their fitness or objective function value. Like this, over the generation sequence, individuals with better and better objective function values are generated.

In an evolution strategy, new candidate solutions are sampled according to a multivariate normal distribution. Recombination amounts to selecting a new mean value for the distribution. Mutation amounts to adding a random vector, a perturbation with zero mean. Pairwise dependencies between the variables in the distribution are represented by a covariance matrix. The covariance matrix adaptation (CMA) is a method to update the covariance matrix of this distribution.

A.3 Numerical results for optimization methods

After minimization of the absolute functional using the TPE and CMA-ES algorithms, their comparative analysis was performed (Table 5). CMA-ES’s execution time was about 8 % less, but applying TPE results in 2.6 times smaller value of functional 𝐽. The values of restored parameters $\vec{q}$ $q →$ can also be found in Table 5.

Table 5

Comparative analysis of the TPE and CMA-ES algorithms to solve an inverse problem of functional 4.1 for the Novosibirsk Region. For each iteration during the identification process the ABM was launched 10 times.

Method	$\vec{q}$ $q →$			$J (\vec{q})$ $J ⁢ ( q → )$	$N_{iter}$ $N iter$	Comp. time

	$E (0)$ $E ⁢ ( 0 )$	𝛽	$\tilde{p} (X)$ $p ~ ⁢ ( X )$
TPE	110	0.02	20.2	37.1	100	12 hours
CMA-ES	75	0.012	100.5	96.8	100	11 hours

Figure 11

Iteration trends in logarithmic scale for CMA-ES (a) and TPE (b) algorithms.

(a)

CMA-ES algorithm

(b)

TPE algorithm

Figure 11 demonstrates how the functional’s values decrease during the last period for both optimization algorithms. It is noteworthy that CMA-ES reaches the maximum quite quickly, and the functional’s value does not improve starting the 40th iteration. In the case of TPE, the functional grows gradually and its maximum is 2.6 times lower than that obtained with CMA-ES. Due to these reasons, the TPE algorithm was chosen to solve the inverse problem for restoring unknown parameters in the Novosibirsk region.

B The restored COVID-19 contagiosity in Novosibirsk region from 2020-03-13 to 2021-04-24

The calibrated COVID-19 contagiosity 𝛽 in the Novosibirsk region from 2020-03-13 to 2021-04-24 for daily positive diagnosed COVID-19 people are demonstrated in Table 6. The absolute error in sense of misfit function for 𝛽 is equal to $10^{- 2}$ $10 − 2$ .

Table 6

Values of vectors ${\vec{β}}_{d}$ $β → d$ and ${\vec{β}}_{c}$ $β → c$ which were restored after solving the inverse problem for the Novosibirsk region.

Days of changes (values of ${\vec{β}}_{d}$ $β → d$ )	Changes of 𝛽 (values of ${\vec{β}}_{c}$ $β → c$ )
2020-03-13	0.0179
2020-04-02	0.0121
2020-05-02	0.0084
2020-06-09	0.0074
2020-07-23	0.0093
2020-07-27	0.0072
2020-09-04	0.0082
2020-09-17	0.0098
2020-11-29	0.0091
2020-12-17	0.0081
2021-02-07	0.0078
2021-03-27	0.0133
2021-03-31	0.0093
2021-04-24	0.0072

Acknowledgements

The authors are grateful to Dr. Cliff Kerr for support in using the Covasim package and valuable advice in setting the problem.

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Received: 2021-06-25

Revised: 2023-01-29

Accepted: 2023-01-30

Published Online: 2023-04-04

Published in Print: 2023-06-01

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DOI: doi.org/10.1515/jiip-2021-0038

Keywords for this article

COVID-19; data analysis; inverse problem; optimization; forecasting; Covasim software; regularization; identifiability; OPTUNA