Abstract
COVID-19 pandemic is severely impacting the lives of billions across the globe. Even after taking massive protective measures like nation-wide lockdowns, discontinuation of international flight services, rigorous testing etc., the infection spreading is still growing steadily, causing thousands of deaths and serious socio-economic crisis. Thus, the identification of the major factors of this infection spreading dynamics is becoming crucial to minimize impact and lifetime of COVID-19 and any future pandemic. In this work, a probabilistic cellular automata based method has been employed to model the infection dynamics for a significant number of different countries. This study proposes that for an accurate data-driven modelling of this infection spread, cellular automata provides an excellent platform, with a sequential genetic algorithm for efficiently estimating the parameters of the dynamics. To the best of our knowledge, this is the first attempt to understand and interpret COVID-19 data using optimized cellular automata, through genetic algorithm. It has been demonstrated that the proposed methodology can be flexible and robust at the same time, and can be used to model the daily active cases, total number of infected people and total death cases through systematic parameter estimation. Elaborate analyses for COVID-19 statistics of forty countries from different continents have been performed, with markedly divergent time evolution of the infection spreading because of demographic and socioeconomic factors. The substantial predictive power of this model has been established with conclusions on the key players in this pandemic dynamics.
Keywords: Epidemiological model, Probabilistic cellular automata, Genetic algorithm, Real data modelling
1. Introduction
With its outbreak in Wuhan, China, Coronavirus disease-2019 (COVID-19) has spread across the world within a few months. Due to its explosive growth and considerable rate of fatality, World Health Organization (WHO) declared COVID-19 as a pandemic and a global health emergency [1]. According to the available statistics in June, 2020, the total number of infections by SARS-CoV-2 (Severe Acute Respiratory Syndrome Coronavirus 2), the causative agent of this disease, is approaching 19 million around the world, causing around 700,000 deaths in 213 countries and territories, with no effective vaccination available in the market so far. Beyond respiratory discomforts including pneumonia, dry cough, cold and sneezing [2], [3], it has been reported to cause liver and gastrointestinal tract maladies, kidney dysfunction and heart inflammation, in cases of severe infection [4], [5], [6]. This highly infectious disease transmits from person-to-person through respiratory droplets produced by infected person. Fomite-mediated and nosocomially acquired infections are also being identified as important sources of viral diffusion [7], [8], [9]. A typical incubation time from exposure to symptoms has been reported for COVID-19, while infection transmission from asymptomatic individuals has been observed as well [10], [11], [12].
Immediately after the detection of human-to-human transmission, the government agencies of various countries started implementing several mitigation strategies to control the epidemic. The measures thus taken include social distancing, restrictions on domestic as well as international travel, cancelling social events, shutting down of public as well as commercial activities etc. which can effectively reduce the possibilities of physical human contact. Moreover, contact tracing, aggressive testing as well as hospital or home quarantine for infected individuals and suspected cases have also been executed to track and prevent further spread. However, these strategies are directly contributing to enormous economical loss. The optimum estimation of this novel disease dynamics is emerging out as a challenging problem in this context. The immense disruption caused by COVID-19, resulting into overwhelming disorder in the health, economy and lives of billions of people around the globe, has brought the necessity for accurate modelling of infectious diseases into the focus. The effect and effectiveness of this complex interplay between differing length-scales and time-scales with the applied control strategies can only be understood and predicted with the help of precisely designed quantitative models.
1.1. Models for understanding COVID-19 statistics
With a tremendous effort from researchers around the world, a spectrum of various mathematical and computational approaches is being used to understand and predict COVID-19 statistics, addressing its different perspectives. On a rudimentary sense, the studies being pursued can be segmented in two categories: (i) data science and machine learning approaches and (ii) differential equation based mathematical modelling techniques. The first group of studies trusted mostly on data mining from national/international repositories (e.g., WHO, country specific data centres etc.) or popular social media platforms to forecast the active cases and mortality data [13], [14], [15], [16], [17]. The major goal of these studies are to estimate and predict the time evolution of the disease using specific computational concepts, like Monte Carlo decision making, fuzzy rule induction, deep learning etc [18], [19], [20], [21], [22]. Some of these studies also explored impact of disease control interventions, like, travel restrictions [23], patient quarantining and isolation [24], medical facilities [25], social distancing and administrative responsibility [15] on epidemic spreading rate. Though these models are quite effective, being entirely dependent on data, the efficiency of these studies can be heavily inclined towards the data quality. As comprehensively reviewed by [26], several data-dependent models are prone to suffer from high risk of bias, which is very much probable for imprecise short time series data.
With the evidence of giving effective predictions for past pandemics [27], [28], [29], the traditional approaches of the mathematical theory of epidemiological dynamics also have driven several researchers to study COVID-19 dynamics. Theoretical modelling based approaches have been long associated to understand and predict the outbreak probabilities and seriousness of a disease, and provide key information to control the intensity [30], [31], [32], [33]. Most of the mathematical models that are being used to investigate the COVID-19 dynamics [34], [35], [36], [37] are based on variants of classical deterministic model of susceptible-infectious-recovered (SIR) that was introduced by Kermack and McKendrick [38]. Constituting a set of nonlinear ordinary differential equations (ODE), the SIR model compartmentalizes the population where susceptible subpopulation declines over time, constantly getting infected (by infectious subpopulation), and then recovered from (and gaining immunity to) the disease over time. Being powerful and computationally favourable tool to analyse epidemic, variants of this methodology are common in understanding real epidemic data [39], [40]. Though these models capture the disease transmission dynamics, being deterministic, they suffer from the assumption of homogeneous mixing, forgoing the spatial information.
For modelling real-world dynamics of a disease that spreads from close-contacts only, the tool needs to accommodate neighbourhood information. Moreover, the platform requires to take into account of stochasticity of real dynamics, spatial infection spread and inherent heterogeneity in population, which are some major limitations of the mentioned works. Thus, the identification of research gap points out in a direction of designing a methodology that addresses the above mentioned issues to understand and predict neighbourhood-dependent person-to-person probabilistic transmission of COVID-19, that should be powered with extensive computational tools for parametric optimization.
1.2. Motivation and contributions
In this study, we propose probabilistic cellular automata based dynamical model, optimized through sequential genetic algorithm for an accurate assessment of the extent of COVID-19 dynamics. The major motivation of using cellular automata (CA) is its ability in depicting extremely complex macroscopic outcomes, while being based on local interactions that trusts on the interaction of a multitude of single individuals [41], [42]. This methodology is capable of giving a direct correspondence to the physical system and also rectifies the major drawbacks of ODE models by (i) tracking individual contact processes, (ii) giving room for introducing probabilistic individual behaviour, and (iii) capturing neighbourhood as well as global spatial information. Because of these reasons, CA based approaches have been successfully used as a competent substitute method to simulate physical, biological, environmental and social contagion-like spreading [43], [44], [45], [46]. For studying past epidemics as well as interpreting COVID-19, some studies have proposed cellular automata as an alternative method [47], [48], [49], [50]. However, to capture and interpret the behaviour of real data through CA needs a large-scale parameter optimization that could be time consuming as well as sub-optimal. Thus, though being extremely flexible and powerful, CA has not been yet optimized to understand and interpret COVID-19 data for countries worldwide. To explore this, in this study, genetic algorithm (GA) has been employed, which is a well-known method for generating the optimal parameter subset through stochastic search procedures based on the principle of the survival of the fittest [51], [52], [53], [54], [55]. Cross-over and mutations, two key properties of genetic algorithm help to optimize the parameter set efficiently in limited steps. Cellular automata coupled with genetic algorithm has been used before to explore evolutionary aspects of game theoretical problems [56], but to the best of our knowledge analysing and developing understanding from real pandemic data like COVID-19 using optimized CA platform has not been attempted yet. The main contributions of this work are as follows:
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To build a CA model which is probabilistic, so that it can take into account of demographic variations, neighbourhood diversity and uncertainties of real dynamics.
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To create an easily implementable framework where optimization using GA will be done sequentially for all parameters associated with the transition rules of the CA model for real data interpretation.
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To interpret and understand COVID-19 disease transmission dynamics with an optimized CA framework, which can be extended for prediction as well.
Through this, on one hand, one can track the individual contact process through time and space; on the other hand, a self-adapting process of evolutionary strategies has been created by designing the chromosome with parametric genes and establishing fitness function that maximizes over the generations. The main limitations of the state-of-the-art algorithms and the major contributions of the proposed method are listed in Table 1 for a clear understanding. The main rationality behind this approach is that it is extremely difficult to find the optimal parameter of the complex spatial epidemiological model using random search or analytical techniques. The proposed GA based framework helps to search the parameter space more efficiently for the optimal performance of the entire algorithm.
Table 1.
Basic methodology |
Differential equation models | Data science approaches |
---|---|---|
References | [33], [34], [35], [36], [37], [39], [40] | [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] |
Limitations | a) Homogeneous Mixing b) Most models are considered as deterministic |
(a) No way to track person to person transmission. (b) No neighbourhood consideration. |
Contribution | Proposed method, (a) accommodates heterogeneity in population (b) includes stochasticity and probabilistic dynamics (c) estimates optimum epidemic dynamics parameters. (d) considers neighbourhood and demography explicitly. (e) performs robust prediction with limited data. |
The rest of this article is organized as follows: Section 2 includes the proposed concepts of epidemiological model, probabilistic cellular automata and the sequential genetic algorithm used in this work. In Section 3, the results has been elaborately discussed where the optimized CA model has been employed for simultaneously understanding as well as analysing active infections, total infections and total death caused by COVID-19 for several countries, considering the demographic and spatial population density variations. Section 4 is comprised of concluding remarks.
2. Proposed methodology
An object process diagram of the proposed method has been depicted in Fig. 1(a). The methodology starts with the infection spreads following the SEIQR epidemiological model in a random human population over a 2D grid, initialized on a country-specific basis. The parameters of the epidemiological model is continuously optimized using proposed sequential genetic algorithm to match the real country-specific infection spread data. The proposed methodology is consisted of three distinct parts
2.1. Epidemiological model
In the epidemiological model, the entire population is partitioned in five distinct parts. At the very beginning, every person was healthy but they are vulnerable to the infection. These people are denoted as susceptible (
2.2. Probabilistic cellular automata
Let
For neighbourhood criteria, modified-Moore neighbourhood or
2.2.1. Transitional probabilities
The transition probability
If a state transition
where
An empty cell does not contribute in the infection spread, and thus, self transitional probability
2.3. Parameter optimization using GA
Though PCA has potential to model the probabilistic transition of states on a spatial lattice, the main challenge to use it for modelling a real-world scenario is to find out the optimal parameters for the PCA. As the searching space for the proposed PCA model is very large, it is practically impossible to search for the optimal parameter setting manually to analyse the characteristics of the infection spread from a real data. Thus, genetic algorithm (GA) has been applied to find out the optimal parameter set given a real time-series data.
Let us assume a discrete time signal
where
For mating, two chromosomes, often referred as parents, are selected from the gene pool considering their ‘fitness’. Among two selected parents, a crossover point or a splice point is selected at
As shown by several researchers [57], the homogeneity in the gene pool increases with the generations, and as the perturbations due to mutation are typically small, the reduction of error becomes a problem after a few generations. Thus, to restrict homogeneity in the gene pool, a small number of offsprings
In our problem, the parameters
Table 2.
Notation | Description |
---|---|
Spatial lattice | |
Set of possible states on lattice | |
Set of epidemiological states | |
Total number of people at state |
|
|
|
Mapping |
|
Probability at time |
|
Transitional delay for |
|
|
Number of exposed and infected people in the |
|
Probabilities that an exposed or an infected person spreads the infection to a susceptible person when they meet |
A gene containing all the parameters of PCA method | |
Binary encoded representation of |
|
The PCA model with parameter |
|
Time series of an epidemiological state in a country | |
Time series estimate of epidemiological state from PCA | |
Estimation error of |
|
Total number of chromosome in genepool | |
Number of parents selected for mating from |
|
Fraction of |
|
Fraction of parents F that lives in the next generation |
Proposed PCA-GA has a complexity which can be approximated as
Though GA has been selected as a strategy to optimize the parameters of the proposed PCA model, it is evident that because of the generalized construction of the proposed framework, other meta-heuristic methods could also be employed to search the parameters of the spatially driven SEIQR model which is the main focus of this work. However, presence of mutation and diversification in GA help to search for better solutions as the search space is extremely large.
3. Results
To validate the effectiveness of the proposed framework, using PCA-GA, the actual statistics of COVID-19 spreads till 20th June, 2020 in different countries is used. For finalizing the data-set from available data of 213 countries, several aspects have been considered. At first, 102 countries had been dropped due to less number of reported cases (less than 1000 reported cases till 20th June 2020). Out of the remaining countries, some countries, like Iran, Greece, Paraguay etc., are removed due to data inconsistency, and finally 40 countries are randomly selected ensuring the following points:
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At least 2 countries from each continent got selected to maintain demographic diversity in our data.
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Care has been taken to maintain significant variation in population density, which we believe as a major factor contributing in disease transmission.
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It was ensured that countries from three distinct stages of COVID-19 infection are considered: (i) where the infection is significantly diminished, (ii) where the peak infection has been reached but substantial infection still persists, and (iii) where consistent growth in infection is occurring.
With these widely variant spectrum of time series data, we proceed for quantitative calibration and interpretation through the proposed methodology. All data samples are taken from the website worldometers.info.1
To point out the major contributing factors in dynamics of infection spread, for every country under consideration, three available time series, namely daily active cases, total number of infected cases and total number of deaths are accumulated. Out of these three series, the daily active cases time series is used for model formulation, and the rest are considered for model validation. It is important to mention that the population
3.1. Experimental setup
For all the simulations, PCA is initialized with a fixed lattice size of 100 × 100 with
3.2. Estimation of parameters using active cases
The daily active cases can be defined as the
where
In Fig. 2, an interesting point to notice is that the peak of the active cases are located at markedly differing time instances, and the other properties, like variance, skewness etc., of the observed distributions are also varying drastically. The fundamental differences between the fitted curves are quantified with the help of boxplot of the parameters in Fig. 3(b)–(c) by analysing basic statistical properties. The reported boxplots are specifically for the countries selected in Fig. 2. It can be noted that
Here it must be mentioned that, while performing this statistical analysis with all 40 countries, some countries were detected showing consistent outliers (not included in Fig. 3(b)–(c)) in terms of four transitional parameters:
There are also certain countries, like India, Brazil, Chile, Mexico, etc., for which the infection spreading started later than the countries like China or Italy, and the active daily cases are still growing almost exponentially. As shown in Fig. 5, PCA-GA is able to estimate the time series data for these countries where the infection is spreading rapidly. Dynamics of COVID-19 spread in these countries are of particular interest as the prediction of the peak positions in these countries might help immensely to understand the maximum socioeconomic impact of the disease at a time in that geographical location.
3.3. Validation of the proposed model
While analysing a complex dynamics like the spread of a pandemic, it is not always sufficient to model the input real data only. It is required that the optimized model should be robust and can provide meaningful interpretations without further retraining or parameter tuning for real-world applications. To validate the robustness and the effectiveness of the proposed algorithm, the optimized model is now employed for three different tasks. At first, the robustness of the optimized model is checked by estimating the total number of infected cases, followed by total number of death cases without any further training, tuning or supervision. Finally, to further validate the efficiency of the model, its performance has been evaluated for the prediction task by training the model with partitioned data and evaluating on its future predictions without any further optimization.
3.3.1. Total number of infected
The total number of infected cases
3.3.2. Total death cases
To further validate the ‘goodness’ of the estimated parameters, the parameter set
3.3.3. Prediction related to infection spread
Prediction of future events is always challenging in data modelling [62]. For the final stage of validation of the methodology, the predictive power of the model has been tested. As the impacts of this pandemic becomes far reaching as the socioeconomic contexts vary, a considerably accurate prediction about the dynamics of the infection spread can be crucial and useful in many ways. As PCA-GA successfully estimates the optimal parameter
To validate the capacity of the prediction strategy, the daily active cases of a country
3.4. Prediction for exponentially rising active cases
As the PCA-GA methodology has been elaborately validated in Section 3.3, now, in this section, it is employed for the purpose of prediction of consistently rising real epidemic data. Though the parameter estimation works well even when the minimum information about the peak position in
Fig. 9 depicts the prediction of the daily active cases using the method discussed so far. In Fig. 9, the black dotted line indicates the prediction using the optimal parameters
4. Conclusion
COVID-19 outbreak has created a massive impact all across the globe. Even after nation-wide lockdowns, extensive testing strategies and medical supports, the spread of the virus has overwhelmed several countries. Thus, it is becoming more and more important to understand the nature of the infection spread and the key parameters that are controlling the spread. In this work, we proposed a probabilistic cellular automata model to understand and depict COVID-19 spread using appropriate choice of loss functions and evolutionary optimization framework. The parameters of this cellular automata model are optimized using sequential evolutionary genetic algorithm. It has been shown that this self-adapting methodology can be highly flexible and has the power to accurately estimate time trajectories of epidemics. This model works with physically interpretable parameters, which are accessible for analysis, data collection and further experiment, and can be readily identified with ground reality. This model has been successfully employed for optimizing all these parameters simultaneously for the daily active cases, total infected cases and total deaths with extreme robustness. The performance of the model has been exhibited for a large number of countries with huge diversity in population density, continents and available healthcare infrastructures. The predictive strength of the model has also been validated extensively, and demonstrated to estimate the course of the pandemic for the countries where infection peak has not been reached yet. It is important to mention that the motivation of the work was to develop a data driven, generalized, spatial framework that can be used to estimate relevant epidemiological parameters. This methodology is so powerful and flexible that physical interpretations of the results obtained from these analyses can have a wide range implications. Once the data is properly interpreted with the proposed methodology, interesting realistic features can be identified for specific countries. For example, in a pandemic situation, easily relatable factors like population clusters, variable population density, variable health facilities at different places of a country etc, can be studied to understand and predict emergence of new hotspots which can be used to design selective area containment strategies. While we propose and establish the applicability and strength of this framework in this work, we wish address these application perspectives in a study in our upcoming research studies.
With this proposed platform, the impact of individuality on contagion process can be explicitly studied, which might be directly related to the questions like lockdown behavioural differences, influence of rumours, vaccination opinion differences etc. As the effects of more complex dynamical factors like periodic lockdown or population clusters are not considered in this present model, the prediction capability of the proposed model is not satisfactory for time series data with abrupt discontinuities in the present form. The proposed framework could be enhanced with other
CRediT authorship contribution statement
Sayantari Ghosh: Conceptualization, Methodology, Software, Validation, Writing - review & editing. Saumik Bhattacharya: Conceptualization, Methodology, Software, Validation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Footnotes
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