A review on epidemic models in sight of fractional calculus

Biomathematics has become one of the most significant areas of research as a result of interdisciplinary study. Chronic diseases sometimes referred to as non-communicable and communicable diseases, are conditions that develop over an extended period as a result of different factors like genetics, lifestyle, and environment. The most important common types of disease are cardiovascular, alcohol, cancer, and diabetes. More than three-quarters of the world's (31.4 million) deaths occur in low- and middle-income nations, which are disproportionately affected by different infections. Fractional Calculus is a prominent topic for research within the discipline of Applied Mathematics due to its usefulness in solving problems in many different branches of science, engineering, and medicine. Recent researchers have identified the importance of mathematical tools in various disease models as being very useful to study the dynamics with the help of fractional and integer calculus modeling. Due to the complexity of the underlying connections, both deterministic and stochastic epidemiological models are founded on an inadequate understanding of the infectious network. Over the past several years, the use of different fractional operators to model the problem has grown, and it is now a common way to study how epidemics spread. Recently, researchers have actively considered fractional calculus to study different diseases like COVID-19, cancer, TB, HIV, dengue fever, diabetes, cholera, pine welts, smoking and heart attacks, etc. With the help of fractional operator, we modified a mathematical model for the dynamical transmission, analysis, treatment, vaccination, and precaution leveling necessary to mitigate the negative impact of illness on society in the long run, overcoming the memory effect without defining or considering others parameters. In this review paper, we considered all the recent studies based on the fractional modeling of infectious and non-infectious diseases with different fractional operators such as Caputo, Caputo Fabrizio, ABC, and constant proportional with Caputo, etc. This review paper aims to bring all the information together by considering different fractional operators and their uses in the field of infectious disease modeling. The steps taken to accomplish the goal were developing a mathematical model, identifying the equilibrium point, figuring out the minimal reproductive number, and assessing the stability around the equilibrium point. For future direction, we consider the cancer model to study the growth cells of cancer and the impact of therapy to control infections. An equilibrium solution and an analysis of the behavior dynamics of the cell spread with treatment in the form of chemotherapy were obtained. The simulation shows that the population of cancer cells is influenced by the pace of cancer cell growth with the Caputo fractional derivative. The acquired results show how effective and precise the suggested approach is in helping to better understand how chemotherapy works. Chemotherapy medications have been found to increase immunity against particular cancer by reducing the number of tumor cells. Further, we suggested some future work directions with the help of the new hybrid fractional operator. Our innovative methodology might have significant effects on global stakeholders, policymakers, and national health systems. The current strategies for controlling outbreaks and the vaccination and prevention policies that have been implemented would benefit from a more accurate representation of the dynamics of contagious diseases, which necessitates the development of highly complex mathematical models. Microorganisms, interactions between individuals or groups, and environmental, social, economic, and demographic factors on a broader scale are all examples.

SIMULATIONS AND ANALYSIS OF COVID-19 AS A FRACTIONAL MODEL WITH DIFFERENT KERNELS

Recently, Atangana proposed new operators by combining fractional and fractal calculus. These recently proposed operators, referred to as fractal–fractional operators, have been widely used to study complex dynamics. In this paper, the COVID-19 model is considered via Atangana–Baleanu fractal-fractional operator. The Lyapunov stability for the model is derived for first and second derivative. Numerical results have developed through Lagrangian-piecewise interpolation for the different fractal–fractional operators. We develop numerical outcomes through different differential and integral fractional operators like power-law, exponential law, and Mittag-Leffler kernel. To get a better outcome of the proposed scheme, numerical simulation is made with different kernels having the memory effects with fractional parameters. [ FROM AUTHOR] Copyright of Fractals is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

On the Modeling of COVID-19 Spread via Fractional Derivative: A Stochastic Approach

: The coronavirus disease (COVID-19) pandemic has caused more harm than expected in developed and developing countries. In this work, a fractional stochastic model of COVID-19 which takes into account the random nature of the spread of disease, is formulated and analyzed. The existence and uniqueness of solutions were established using the fixed-point theory. Two different fractional operators', namely, power-law and Mittag–Leffler function, numerical schemes in the stochastic form, are utilized to obtain numerical simulations to support the theoretical results. It is observed that the fractional order derivative has effect on the dynamics of the spread of the disease. © 2023, Pleiades Publishing, Ltd.

Addendum: Predictive form of the FPM model

The article Oustaloup et al. (2021) has shown that the Fractional Power Model (FPM), A+Btm, enables well representing the cumulated data of COVID infections, thanks to a nonlinear identification technique. Beyond this identification interval, the article has also shown that the model enables predicting the future values on an unusual prediction horizon as for its range. The objective of this addendum is to explain, via an autoregressive form, why this model intrinsically benefits from such a predictivity property, the idea being to show the interest of the FPM model by highlighting its predictive specificity, inherent to non-integer integration that conditions the model. More precisely, this addendum establishes a predictive form with long memory of the FPM model. This form corresponds to an autoregressive (AR) filter of infinite order. Taking into account the whole past through an indefinite linear combination of past values, a first predictive form, said to be with long memory, results from an approach using one of the formulations of non-integer differentiation. Actually, as this first predictive form is the one of the power-law, tm, its adaptation to the FPM model, A+Btm, which generalizes the linear regression, A+Bt, is then straightforward: it leads to the predictive form of the FPM model that specifies the model in prediction. This predictive form with long memory shows that the predictivity of the FPM model is such that any predicted value takes into account the whole past, according to a weighted sum of all the past values. These values are taken into account through weighting coefficients, that, for m>−1 and a fortiori for m>0, correspond to an attenuation of the past, that the non-integer power, m, determines by itself. To confirm the specificity of the FPM model in considering the past, this model is compared with a model of another nature, also having three parameters, namely an exponential model (Liu et al. (2020);Sallahi et al. (2021)): whereas, for the FPM model, the past is taken into account globally through all past instants, for the exponential model, the past is taken into account only locally through one single past instant, the predictive form of the model having a short memory and corresponding to an AR filter of order 1. Comparative results, obtained in prediction for these two models, show the predictive interest of the FPM model.

The impact of a power law-induced memory effect on the SARS-CoV-2 transmission.

It is well established that COVID-19 incidence data follows some power law growth pattern. Therefore, it is natural to believe that the COVID-19 transmission process follows some power law. However, we found no existing model on COVID-19 with a power law effect only in the disease transmission process. Inevitably, it is not clear how this power law effect in disease transmission can influence multiple COVID-19 waves in a location. In this context, we developed a completely new COVID-19 model where a force of infection function in disease transmission follows some power law. Furthermore, different realistic epidemiological scenarios like imperfect social distancing among home-quarantined individuals, disease awareness, vaccination, treatment, and possible reinfection of the recovered population are also considered in the model. Applying some recent techniques, we showed that the proposed system converted to a COVID-19 model with fractional order disease transmission, where order of the fractional derivative ( α ) in the force of infection function represents the memory effect in disease transmission. We studied some mathematical properties of this newly formulated model and determined the basic reproduction number ( R 0 ). Furthermore, we estimated several epidemiological parameters of the newly developed fractional order model (including memory index α ) by fitting the model to the daily reported COVID-19 cases from Russia, South Africa, UK, and USA, respectively, for the time period March 01, 2020, till December 01, 2021. Variance-based Sobol's global sensitivity analysis technique is used to measure the effect of different important model parameters (including α ) on the number of COVID-19 waves in a location ( W C ). Our findings suggest that α along with the average transmission rate of the undetected (symptomatic and asymptomatic) cases in the community ( ß 1 ) are mainly influencing multiple COVID-19 waves in those four locations. Numerically, we identified the regions in the parameter space of α and ß 1 for which multiple COVID-19 waves are occurring in those four locations. Furthermore, our findings suggested that increasing memory effect in disease transmission ( α → 0) may decrease the possibility of multiple COVID-19 waves and as well as reduce the severity of disease transmission in those four locations. Based on all the results, we try to identify a few non-pharmaceutical control strategies that may reduce the risk of further SARS-CoV-2 waves in Russia, South Africa, UK, and USA, respectively.

Did cryptocurrencies exhibit log-periodic power law signature during the second wave of COVID-19?

Many studies have associated cryptocurrencies with bubbles, especially during stressed market conditions such as the recent outbreak of the second wave of COVID-19. Although the majority of studies have focused on Bitcoin, we investigate the predictability of bubble formation in the cryptocurrency market by using the log-periodic power law and we uncover some important stylized facts of this market. Our sample consists of data for a selection of 15 cryptocurrencies for the period between 1 January 2021 and 1 September 2021 which coincides with the second wave of COVID-19. We analyse 86 speculative bubbles, and we find that the cryptocurrency market has three times higher drawdown over equities during stressed market conditions. © 2022 Banca Monte dei Paschi di Siena SpA.

Impact of Network Heterogeneity on Epidemic Mitigation Strategies

We formulate an optimal control problem to find best vaccination and treatment policies to minimize the impact of an epidemic on the population. Epidemic spread on heterogeneous human contact networks is modeled using the degree based compartmental model for susceptible-infected-recovered epidemic. Our formulation allows us to study the impact of varying network heterogeneity on the mitigation strategies. Network heterogeneity is varied by using different degree distributions for the network, such as, power law, power law with exponential cut-off, and Poisson. Network heterogeneity is a proxy for social distancing measures applied on the population - as restrictions tightens, high degree hubs disappear, thus, the nature of degree distribution changes from power law to Poisson. We find that high degree nodes assume less importance in mitigating epidemics as the network heterogeneity decreases. Also, epidemics are easier to control with decrease in network heterogeneity. © 2022 IEEE.

Temporal and spatial evolution of the distribution related to the number of COVID-19 pandemic.

This work systematically conducts a data analysis based on the numbers of both cumulative and daily confirmed COVID-19 cases and deaths in a time span through April 2020 to June 2022 for over 200 countries around the world. Such research feature aims to reveal the temporal and spatial evolution of the country-level distribution observed in COVID-19 pandemic, and obtains some interesting results as follows. (1) The distributions of the numbers for cumulative confirmed cases and deaths obey power-law in early stages of COVID-19 and stretched exponential function in subsequent course. (2) The distributions of the numbers for daily confirmed cases and deaths obey power-law in early and late stages of COVID-19 and stretched exponential function in middle stages. The crossover region between power-law and stretched exponential behavior seems to depend on the evolution of "infection" event and "death" event. Such observation implies a kind of important symmetry related to the dynamics process of COVID-19 spreading. (3) The distributions of the normalized numbers for each metric show a temporal scaling behavior in 2-year period, and are well described by stretched exponential function. The observation of power-law and stretched exponential behavior in such country-level distributions suggests underlying intrinsic dynamics of a virus spreading process in human interconnected society. And thus it is important for understanding and mathematically modeling the COVID-19 pandemic.

Coupling Power Laws Offers a Powerful Modeling Approach to Certain Prediction/Estimation Problems With Quantified Uncertainty

Power laws (PLs) have been found to describe a wide variety of natural (physical, biological, astronomic, meteorological, and geological) and man-made (social, financial, and computational) phenomena over a wide range of magnitudes, although their underlying mechanisms are not always clear. In statistics, PL distribution is often found to fit data exceptionally well when the normal (Gaussian) distribution fails. Nevertheless, predicting PL phenomena is notoriously difficult because of some of its idiosyncratic properties, such as lack of well-defined average value and potentially unbounded variance. Taylor's power law (TPL) is a PL first discovered to characterize the spatial and/or temporal distribution of biological populations. It has also been extended to describe the spatiotemporal heterogeneities (distributions) of human microbiomes and other natural and artificial systems, such as fitness distribution in computational (artificial) intelligence. The PL with exponential cutoff (PLEC) is a variant of power-law function that tapers off the exponential growth of power-law function ultimately and can be particularly useful for certain predictive problems, such as biodiversity estimation and turning-point prediction for Coronavirus Diease-2019 (COVID-19) infection/fatality. Here, we propose coupling (integration) of TPL and PLEC to offer a methodology for quantifying the uncertainty in certain estimation (prediction) problems that can be modeled with PLs. The coupling takes advantage of variance prediction using TPL and asymptote estimation using PLEC and delivers CI for the asymptote. We demonstrate the integrated approach to the estimation of potential (dark) biodiversity of the American gut microbiome (AGM) and the turning point of COVID-19 fatality. We expect this integrative approach should have wide applications given duel (contesting) relationship between PL and normal statistical distributions. Compared with the worldwide COVID-19 fatality number on January 24th, 2022 (when this paper is online), the error rate of the prediction with our coupled power laws, made in the May 2021 (based on the fatality data then alone), is approximately 7% only. It also predicted that the turning (inflection) point of the worldwide COVID-19 fatality would not occur until the July of 2022, which contrasts with a recent prediction made by Murray on January 19th of 2022, who suggested that the "end of the pandemic is near " by March 2022.

The allometric propagation of COVID-19 is explained by human travel.

We analyzed the number of cumulative positive cases of COVID-19 as a function of time in countries around the World. We tracked the increase in cases from the onset of the pandemic in each region for up to 150 days. We found that in 81 out of 146 regions the trajectory was described with a power-law function for up to 30 days. We also detected scale-free properties in the majority of sub-regions in Australia, Canada, China, and the United States (US). We developed an allometric model that was capable of fitting the initial phase of the pandemic and was the best predictor for the propagation of the illness for up to 100 days. We then determined that the power-law COVID-19 exponent correlated with measurements of human mobility. The COVID-19 exponent correlated with the magnitude of air passengers per country. This correlation persisted when we analyzed the number of air passengers per US states, and even per US metropolitan areas. Furthermore, the COVID-19 exponent correlated with the number of vehicle miles traveled in the US. Together, air and vehicular travel explained 70% of the variability of the COVID-19 exponent. Taken together, our results suggest that the scale-free propagation of the virus is present at multiple geographical scales and is correlated with human mobility. We conclude that models of disease transmission should integrate scale-free dynamics as part of the modeling strategy and not only as an emergent phenomenological property.

Spatial Temporal Analysis of 40,000,000,000,000 Internet Darkspace Packets

The Internet has never been more important to our society, and understanding the behavior of the Internet is essential. The Center for Applied Internet Data Analysis (CAIDA) Telescope observes a continuous stream of packets from an unsolicited darkspace representing 1/256 of the Internet. During 2019 and 2020 over 40,000,000,000,000 unique packets were collected representing the largest ever assembled public corpus of Internet traffic. Using the combined resources of the Supercomputing Centers at UC San Diego, Lawrence Berkeley National Laboratory, and MIT, the spatial temporal structure of anonymized source-destination pairs from the CAIDA Telescope data has been analyzed with GraphBLAS hierarchical hypersparse matrices. These analyses provide unique insight on this unsolicited Internet darkspace traffic with the discovery of many previously unseen scaling relations. The data show a significant sustained increase in unsolicited traffic corresponding to the start of the COVID19 pandemic, but relatively little change in the underlying scaling relations associated with unique sources, source fan-outs, unique links, destination fan-ins, and unique destinations. This work provides a demonstration of the practical feasibility and benefit of the safe collection and analysis of significant quantities of anonymized Internet traffic.

Optimal COVID-19 Vaccination Strategies with Limited Vaccine and Delivery Capabilities

We develop a model of infection spread that takes into account the existence of a vulnerable group as well as the variability of the social relations of individuals. We develop a compartmentalized power-law model, with power-law connections between the vulnerable and the general population, considering these connections as well as the connections among the vulnerable as parameters that we vary in our tests. We use the model to study a number of vaccination strategies under two hypotheses: first, we assume a limited availability of vaccine but an infinite vaccination capacity, so all the available doses can be administered in a short time (negligible with respect to the evolution of the epidemic). Then, we assume a limited vaccination capacity, so the doses are administered in a time non-negligible with respect to the evolution of the epidemic. We develop optimal strategies for the various social parameters, where a strategy consists of (1) the fraction of vaccine that is administered to the vulnerable population and (2) the criterion that is used to administer it to the general population. In the case of a limited vaccination capacity, the fraction (1) is a function of time, and we study how to optimize it to obtain a maximal reduction in the number of fatalities. © 2021 ACM.

Power law in COVID-19 cases in China.

The novel coronavirus (COVID-19) was first identified in China in December 2019. Within a short period of time, the infectious disease has spread far and wide. This study focuses on the distribution of COVID-19 confirmed cases in China-the original epicentre of the outbreak. We show that the upper tail of COVID-19 cases in Chinese cities is well described by a power law distribution, with exponent around one in the early phases of the outbreak (when the number of cases was growing rapidly) and less than one thereafter. This finding is significant because it implies that (i) COVID-19 cases in China is heavy tailed and disperse; (ii) a few cities account for a disproportionate share of COVID-19 cases; and (iii) the distribution generally has no finite mean or variance. We find that a proportionate random growth model predicated by Gibrat's law offers a plausible explanation for the emergence of a power law in the distribution of COVID-19 cases in Chinese cities in the early phases of the outbreak.

Modeling the Waves of Covid-19.

The challenges with modeling the spread of Covid-19 are its power-type growth during the middle stages of the waves with the exponents depending on time, and that the saturation of the waves is mainly due to the protective measures and other restriction mechanisms working in the same direction. The two-phase solution we propose for modeling the total number of detected cases of Covid-19 describes the actual curves for many its waves and in many countries almost with the accuracy of physics laws. Bessel functions play the key role in our approach. The differential equations we obtain are of universal type and can be used in behavioral psychology, invasion ecology (transient processes), etc. The initial transmission rate and the intensity of the restriction mechanisms are the key parameters. This theory provides a convincing explanation of the surprising uniformity of the Covid-19 waves in many places, and can be used for forecasting the epidemic spread. For instance, the early projections for the 3rd wave in the USA appeared sufficiently exact. The Delta-waves (2021) in India, South Africa, UK, and the Netherlands are discussed at the end.

A simple electrical-circuit analogous phenomenological COVID-19 model valid for all observed pandemic phases

In physics and engineering, circuit modeling together with simple element models has been used to study concurrent physical phenomena. By using simple calculations, the underlying mechanisms that determine certain patterns can be understood. Here, we present a simple mathematical model to describe the COVID-19 pandemic time evolution. The model accounts for three phases occurring at the same pandemic wave, which are influenced by different mechanisms represented by a linear, an exponential, and a power law term, corresponding to an early stage of the contagious spread, an unconstrained spread, and a power-law increase defined by the effectiveness of the social distancing, respectively. This approach is based on parallel and series natural phenomena occurring in electrical circuits. The generality of the present tool is demonstrated using empirical data of nine countries from different continents. © 2022 Author(s).

Statistical properties of the aftershocks of stock market crashes revisited: Analysis based on the 1987 crash, financial-crisis-2008 and COVID-19 pandemic

During any unique crisis, panic sell-off leads to a massive stock market crash that may continue for more than a day, termed as mainshock. The effect of a mainshock in the form of aftershocks can be felt throughout the recovery phase of stock price. As the market remains in stress during recovery, any small perturbation leads to a relatively smaller aftershock. The duration of the recovery phase has been estimated using structural break analysis. We have carried out statistical analyses of 1987 stock market crash, 2008 financial crisis and 2020 COVID-19 pandemic considering the actual crash times of the mainshock and aftershocks. Earlier, such analyses were done considering absolute one-day return, which cannot capture a crash properly. The results show that the mainshock and aftershock in the stock market follow the Gutenberg–Richter (GR) power law. Further, we obtained higher β value for the COVID-19 crash compared to the financial-crisis-2008 from the GR law. This implies that the recovery of stock price during COVID-19 may be faster than the financial-crisis-2008. The result is consistent with the present recovery of the market from the COVID-19 pandemic. The analysis shows that the high-magnitude aftershocks are rare, and low-magnitude aftershocks are frequent during the recovery phase. The analysis also shows that the inter-occurrence times of the aftershocks follow the generalized Pareto distribution, i.e. P(τi)∝1[1+λ(q−1)τi]1(q−1), where λ and q are constants and τi is the inter-occurrence time. This analysis may help investors to restructure their portfolio during a market crash.

The correlation analysis of the daily Covid-19 new cases data series in Albania

We analyzed herein the new covid-19 daily positive cases recorded in Albania. We observed that the distribution of the daily new cases is non-stationary and usually has a power law behavior in the low incidence zone, and a bell curve for the remaining part of the incidence interval. We qualified this finding as the indicator intensive dynamics and as proof that up now, the heard immunity has not been reached. By parallelizing the preferential attachment mechanisms responsible for a power law distribution in the social graphs elsewhere, we explain the low daily incidence distribution as result of the imprudent gatherings of peoples. Additionally, the bell-shaped distribution observed for the high daily new cases is agued as outcome of the competition between illness advances and restriction measures. The distribution is acceptably smooth, meaning that the management has been accommodated appropriately. This behavior is observed also for two neighbor countries Greece and Italy respectively, but was not observed for Turkey, Serbia, and North Macedonia. Next, we used the multifractal analysis to conclude about the features related with heterogeneity of the data. We have identified the local presence self-organization behavior in some separate time intervals. Formally and empirically we have identified that the full set of the data contain two regimes finalized already, followed by a third one which started in July 2021. © 2021 Institute of Physics Publishing. All rights reserved.

Mutual relationships between SARS-CoV-2 test numbers, fatality and morbidity rates.

BACKGROUND: The number of SARS-CoV-2 tests conversely to other factors, such as age of population or comorbidities, influencing SARS-CoV-2 morbidity and fatality rates, can be increased or decreased by decision makers depending on the development of the pandemic, operational capacity, and financial restraints. The key objective of this study is to identify and describe, within the probabilistic approach, the relationships between SARS-CoV-2 test numbers and the mortality and morbidity rates. METHODS: The study is based on a statistical analysis of 1058 monthly observations relating to 107 countries, from six different continents, in an 11-month period from March 2020 to January 2021. The variable utilised can be defined as the number of tests performed in a given country in 1 month, to the number of cases reported in a prior month and morbidities and mortalities per 1 million population. The probabilities of different mortality and morbidity rates for different test numbers were determined by moving percentiles and fitted by the power law and by the three-segment piecewise-linear approximation based on Theil Sen trend lines. RESULTS: We have identified that for a given probability the dependence of mortality and morbidity rates on SARS-CoV-2 test rates follows a power law and it is well approximated by the three Theil Sen trend lines in the three test rate ranges. In all these ranges Spearman rho and Kendall tau-b rank correlation coefficients of test numbers and morbidity with fatality rates have values between - 0.5 and - 0.12 with p-values below 0.002. CONCLUSIONS: According to the ABC classification: the most important, moderately important, and relatively unimportant ranges of test numbers for managing and control have been indicated based on the value of the Theil Sen trend line slope in the three SARS-CoV-2 test rate ranges identified. Recommendations for SARS-CoV-2 testing strategy are provided.

A new logistic growth model applied to COVID-19 fatality data.

BACKGROUND: Recent work showed that the temporal growth of the novel coronavirus disease (COVID-19) follows a sub-exponential power-law scaling whenever effective control interventions are in place. Taking this into consideration, we present a new phenomenological logistic model that is well-suited for such power-law epidemic growth. METHODS: We empirically develop the logistic growth model using simple scaling arguments, known boundary conditions and a comparison with available data from four countries, Belgium, China, Denmark and Germany, where (arguably) effective containment measures were put in place during the first wave of the pandemic. A non-linear least-squares minimization algorithm is used to map the parameter space and make optimal predictions. RESULTS: Unlike other logistic growth models, our presented model is shown to consistently make accurate predictions of peak heights, peak locations and cumulative saturation values for incomplete epidemic growth curves. We further show that the power-law growth model also works reasonably well when containment and lock down strategies are not as stringent as they were during the first wave of infections in 2020. On the basis of this agreement, the model was used to forecast COVID-19 fatalities for the third wave in South Africa, which was in progress during the time of this work. CONCLUSION: We anticipate that our presented model will be useful for a similar forecasting of COVID-19 induced infections/deaths in other regions as well as other cases of infectious disease outbreaks, particularly when power-law scaling is observed.

Universal scaling of human flow remain unchanged during the COVID-19 pandemic.

To prevent the spread of the COVID-19 pandemic, governments in various countries have severely restricted the movement of people. The large amount of detailed human location data obtained from mobile phone users is useful for understanding the change of flow patterns of people under the effect of pandemic. In this paper, we observe the synchronized human flow during the COVID-19 pandemic using Global Positioning System data of about 1 million people obtained from mobile phone users. We apply the drainage basin analysis method which we introduced earlier for characterization of macroscopic human flow patterns to observe the effect of the spreading pandemic. Before the pandemic the afternoon basin size distribution has been approximated by an exponential distribution, however, the distribution of Tokyo and Sapporo, which were most affected by the first wave of COVID-19, deviated significantly from the exponential distribution. On the other hand, during the morning rush hour, the scaling law holds universally, i.e., in all cities, even though the number of moving people in the basin has decreased significantly. The fact that these scaling laws, which are closely related to the three-dimensionality structure of the city and the fractal structure of the transportation network, have not changed indicates that the macroscopic human flow features are determined mainly by the means of transport and the basic structure of cities which are invariant of the pandemic.