The authors have declared that no competing interests exist.

The spatial diffusion of epidemic disease follows distance decay law in geography and social physics, but the mathematical models of distance decay depend on concrete spatio-temporal conditions. This paper is devoted to modeling spatial diffusion patterns of COVID-19 stemming from Wuhan city to Hubei province, China. The modeling approach is to integrate analytical method and experimental method. The local gravity model is derived from allometric scaling and global gravity model, and then the parameters of the local gravity model are estimated by observational data and least squares calculation. The main results are as below. The local gravity model based on power law decay can effectively describe the diffusion patterns and process of COVID-19 in Hubei Province, and the goodness of fit of the gravity model based on negative exponential decay to the observational data is not satisfactory. Further, the goodness of fit of the model to data entirely became better and better over time, the size elasticity coefficient increases first and then decreases, and the distance attenuation exponent decreases first and then increases. Moreover, the significance of spatial autoregressive coefficient in the model is low, and the confidence level is less than 80%. The conclusions can be reached as follows. (1) The spatial diffusion of COVID-19 of Hubei bears long range effect, and the size of a city and the distance of the city to Wuhan affect the total number of confirmed cases. (2) Wuhan direct transmission is the main process in the spatial diffusion of COVID-19 in Hubei at the early stage, and the horizontal transmission between regions is not significant. (3) The effect of spatial lockdown and isolation measures taken by Chinese government against the transmission of COVID-19 is obvious. This study suggests that the role of urban gravity (size and distance) should be taken into account to prevent and control epidemic disease.

Geospatial diffusion is governed by certain scientific laws, which can be described by mathematical language. The theory and models of spatial diffusion have been initially developed in the period of quantitative revolution. Following distance decay law, the diffusion process is related to spatial interaction [

A complete dataset can help to develop new models or verify existing mathematical models. Novel coronavirus pneumonia broke out in Wuhan in January 2020, and the COVID-19 rapidly spread from Wuhan to the rest region of China with the help of Spring Festival travel rush. The space-time characteristics and mechanism behind the diffusion process are worth exploring. The new data of confirmed cases can be used to study the mathematical laws of geographical spatial diffusion. Taking Hubei Province as a study area and Wuhan as the center of spatial diffusion, we research the spatial and temporal characteristics of COVID-19 spread in Hubei Province by means of gravity models and spatial autoregressive model in this paper. The effects of city size and spatial distance on the spread of COVID-19 are investigated. Where research methods are concerned, both analytical method and experimental method are employed to make models. A series of concepts including allometric scaling, fractal dimension, and spatial autoregression are introduced into classical models. Gravity models are used to analyze the characteristics of core-periphery vertical diffusion from Wuhan, and spatial autoregression is used to investigate whether there is effect of the horizontal cross influence between different cities except for Wuhan. The goal of this paper is to reveal the geographic spatial regularity of the spread of COVID-19. In Section 2, the local gravity model is derived from the global gravity model with the help of the allometric scaling, and then the spatial autoregression term is introduced into the logarithmic linear form of the gravity model. In Section 3, the parameters of the local gravity model and the mixed gravity model comprising autoregressive term are estimated by using least squares calculations. The main calculation results and the corresponding statistical analyses are shown in this part. In Section 4, the chief related problems of the research are discussed. Finally, the discussion is concluded by summarizing the main viewpoints of this study.

The geospatial diffusion obeys the laws of distance decay. In different situations, distance decay patterns can be described by different functions [_{ij} is the origin flow, i.e., the outflow from the source, _{ji} is the received flow, i.e., the inflow to the destination, _{i} is the size of place _{j} is the size of place _{ij} is the distance from

The local gravity model can be derived and used to describe the core-periphery relationship in geographical analysis. Mathematical models fall into two categories: mechanism models and parametric models [_{ij} is the case transmission quantity associated with passenger flow _{ij},

Based on the core-periphery relationship, the place _{i} can be regarded as a constant _{0}. So Eq (

This is a typical local gravity model. The parameters of Eq (

Spatial diffusion is supposed to be a network process, including vertical spread and horizontal spread. The local gravity model discussed in subsection 2.2 mainly describes the vertical transmission, without considering the transverse diffusion. If only the diffusion of COVID-19 from Wuhan to other cities in Hubei Province is taken into account, the number of confirmed cases is directly proportional to the population size of the city, and inversely proportional to a certain power of the distance to Wuhan. Whether there is horizontal diffusion between other cities except Wuhan is ignored. In a word, the local gravity model only describes vertical diffusion effect—the relationship between Wuhan and other cities without considering horizontal diffusion effect—the connection between other cities except Wuhan. Such a method is inevitably being questioned: why the cross correlation of other cities can be ignored? In order to consider the horizontal relationship, the idea of spatial auto-regression can be introduced. Spatial auto-regression method is based on spatial autocorrelation idea. Taking natural logarithms on both sides of Eq (

Assuming that COVID-19 spread between other cities except Wuhan, we should consider the spatial self-influence which can be described by the process of spatial auto-regression. By adding autoregressive term to Eq (

The aim of this paper is at revealing the geographical mathematical regularity of spatial diffusion COVID-19. The study area includes the whole area of Hubei Province, China. The geographical elements involve the capital of Hubei Province, Wuhan, and all prefecture level cities as well as the interurban road network (

As shown in the figure, five representative dates are selected to illustrate the scale of the epidemic diffusion. The size of circles represents the scale. The gravity intensity is expressed according to the results calculated from the data of February 20, and the flow intensity is expressed through the line width. The line segments illustrate the vertical diffusion from Wuhan to other cities in Hubei. The horizontal diffusion relationship between other cities outside Wuhan are not shown. The vertical diffusion is described by local gravity model, and the horizontal diffusion is tested by spatial autoregressive analysis.

Measurement | Source | Date | Advantage | Disadvantage |
---|---|---|---|---|

National Health Commission of the People’s Republic of China | January 28 to February 23, 2020 | Cross-sectional data at multiple time points | The cut-off time of some cities is different from the majority | |

Hubei highway Atlas | Published in 2012 | Diffusion process can be well explained | Lack of complete matrix | |

Measured with GIS technology by authors | February 2020 | Complete matrix of distance | Only reflects the distance covered by private cars which may be different from bus route | |

China City Statistical Yearbook 2018 | End of 2017 | Generally reflects the scale of urban system in each region of Hubei province | The data collected two years ago are not real-time statistics |

The algorithm is the multiple linear regression based on the least square method. Gravity model is a kind of nonlinear model, which cannot be implemented in regression analysis directly. Fortunately, it is easy to transform this model into a linear relation by taking logarithms of both sides. The nonlinear fitting method can also be used to estimate the parameters of this model directly. However, the disadvantage of curve fitting may lead to a result that the overall information of the dataset cannot be utilized effectively. The reasoning lies in that the bigger data points such as the city size of Wuhan has an excessive influence on the parameters. In particular, it is worth mentioning the size distribution of Hubei’s cities follow what is called primate law rather than the rank-size law. Where urban population size is concerned, Wuhan stands head and shoulders above other cities in Hubei Province. In this case, the impact of Wuhan’s population size on the model’s parameter estimation based on curve fitting is overwhelming. After taking logarithm, the differences of city sizes lessen. It is an advisable selection to make use of multivariate linear regression analysis to estimate the parameter values of the local gravity model and the mixed gravity model with autoregressive term (

The analytical process of geographical mathematical modeling involves three basic aspects. The first is mathematical structure, the second is model parameters, and the third is statistics such as correlation coefficient and standard errors. As for the mathematical structure, different model structures are determined by different distance decay functions. Whether spatial auto-regression is considered also affects model structure. The model parameters include constant term, size exponent and distance exponent or distance coefficients. If spatial auto-regression is considered, autoregressive coefficients should be included as well. The model statistics include global statistics and local statistics. The global statistics include goodness of fit,

Let us examine the structure of the gravity model suitable for epidemic spread in Hubei at first. Model structure reflect the property at the macro level of spatial diffusion. The distance decay function of the gravity model given above follows inverse power law. In fact, the negative exponential function can be taken as the distance decay function of the gravity model as well. The mathematical properties of two decay functions are different. The exponential function bears a parameter indicating characteristic scale, while the power function has no parameter representing characteristic scale. The former implies simplicity, while the latter implies complexity [

The goodness of fit ^{2} = 0.9507,

Distance function | Autocorrelation | Partial autocorrelation | Space effect | Epidemic diffusion in Hubei Province |
---|---|---|---|---|

Tailed | Truncated | Quasi Locality: simple, limited scope, inconsistent with the first law of geography | Poor fitting effect | |

Tailed | Tailed | Long range effect: complex, unlimited scope, consistent with the first law of geography | Good fitting effect |

Secondly, the model parameters should be investigated to see the properties at the micro level of spatial diffusion. The gravity coefficient of the local gravity model contains the information of the population size of the central city, Wuhan. The value of the gravity coefficient has gradually increased, indicating that the epidemic situation of Wuhan was becoming more and more serious. The size exponent firstly decreased, and then rebounded after January 30. After February 2, the size exponent value gradually became larger than one. After February 15, it showed a downward trend again and the value became less than one after February 20. The variation implies that the impact of city size was not very prominent before February 2, and later the effect of urban population size needed to be highlighted. After February 15, the human-to-human transmission occurred less frequently because of isolation measures taken by local governments, and the scale effect declined after February 20. The distance exponent also showed a downward trend before February 1, and then gradually increased. The exponent rose sharply on February 12 (

Parameter | Property | Meaning | Variation characteristics | Geographic information |
---|---|---|---|---|

Gravitational coefficient |
Central scale factor | In proportion to the confirmed cases of COVID-19 in central cities | Showed an upward trend on the whole and dropped occasionally on February 3. | The number of people infected in the central city, Wuhan was rising rapidly. |

Size exponent |
Size exponent: secondary transmission effect | The relative share of the increased confirmed cases of COVID-19 which corresponds to the relative share of increased city size. | Decreased before February 1, then increased gradually, and decreased again after February 14. | After February 1, the size of cities has a prominent effect on the transmission of COVID-19, implying secondary transmission |

Distance exponent |
Distance scaling exponent: primary transmission effect | The relative share of the increased confirmed cases of COVID-19 which corresponds to the relative share of decreased distance from Wuhan. | Declined before February 1 and then increased gradually. | Rapid diffusion directly from Wuhan before February 1 and since then barriers to epidemic diffusion have increased, implying local isolation. |

Finally, the model statistics are analyzed for testing modeling effect. Model statistics are statistical measurements which can be used to evaluate the mathematical structure of a model or the level of confidence of parameter values. An absolutely reliable model statistic does not exist in this world. A model which cannot pass the statistical test usually has some problems. However, a model which does have some problems may also pass the statistical test. Different statistic values should be taken into account together in statistical test. From the perspective of global statistics, both the correlation coefficient square ^{2} and

In order to investigate whether there is cross transmission of COVID-19 between different regions of Hubei Province, spatial autoregressive analysis can be carried out. Eq (

Due to the adjustment of the diagnosis standard of COVID-19, the statistical caliber has varied several times. The most significant change took place on February 12, 2020. Statistically speaking, the growth rate of the total number of confirmed cases on February 12 is an abnormal value. From February 19 to 26, the statistical caliber seemed to be changeable again for the value is abnormal to some extent. Here are two trend lines based on two sets of data with different statistical calibers. The first logistic curve is more consistent with the growth rate, and the second fractional logistic curve can reflect the capacity of confirmed cases. The bifurcation point of these two curves is around February 4 and 5. The growth rate of COVID-19 reached its peak on these two days.

The goodness of fit ^{2} = 0.9517, _{adj}^{2} is comparable. The former is _{adj}^{2} = 0.9436, and the latter is _{adj}^{2} = 0.9404.

The insignificance of spatial autoregressive coefficient indicated that the spatial autocorrelation of epidemic transmission in Hubei Province was not significant. The experiment shows that the spatial autocorrelation is indeed insignificant in Hubei Province (Limited to the space of the paper, the relevant issues will be discussed separately). After the outbreak of the epidemic, local governments may take necessary lockdown and isolation measures [

The form of a mathematical model reflects the macro structure of a system, while the parameters of the model reflects the characteristics of the micro element correlation or interactions of the system. Accordingly, the statistics (e.g., ^{2}, ^{2},

The gravity models are based on distance decay law which can be expressed by a certain type of impedance function. In geography, the impedance functions are also termed distance decay functions, which are not limited to negative exponential function and inverse power function. General typology of distance decay functions include power function, exponential function, square root exponential function, normal function, lognormal function, gamma function, and so on [

Type | Name | Function | Entropy maximization | Characteristic scale |
---|---|---|---|---|

Linear Model: | _{0}− |
Yes | ||

Normal | _{0}exp(−^{2}) |
Yes | ||

Exponential | _{0}exp(− |
No | ||

Square root exponential | _{0}exp(−^{1/2}) |
No | ||

Logarithmic | _{0}− |
No | ||

Power law | _{1}^{−α} |
Yes | ||

Lognormal | _{1}exp(−^{2}) |
No | ||

_{1}^{−α} |

_{0}, _{1},

The characteristics of certain temporal process could be reflected from the pattern of spatial diffusion, for the spatial pattern and the temporal process depend on each other. The growth of confirmed cases of an epidemic usually appears as an S-shaped curve [

Stage | Three stage division | Four stage division | ||
---|---|---|---|---|

Spatial diffusion | Temporal growth | Spatial diffusion | Temporal growth | |

Primary stage | Initial stage | Primary stage | Initial stage | |

Diffusion stage | Celerity stage | Spatial diffusion stage | Acceleration stage | |

Diffusion stage | Celerity stage | Hierarchical diffusion stage | Deceleration stage | |

Shrinking stage | Terminal stage | Shrinking stage | Terminal stage |

In geographical analysis, it’s natural to use gravity model to study the pattern and process of diffusion of epidemic. However, the objective of this paper lies chiefly at the gravity scaling analysis based on allometric relation, gravity model and spatial autoregressive analysis. In a sense, this paper is devoted to developing a local gravity model to describe the spatial diffusing processes based on core-periphery patterns. The main task of sciences is to make models [

The main academic contribution of this paper is to develop the spatial diffusion model based on local gravity. Based on the allometric scaling relation, the local gravity model about the passenger flow is transformed into the local gravity model about the case transmission quantity. A series of COVID-19 papers published have been published, showing many interesting research results. However, in terms of research objectives, results and nature, this paper is quite different from the published papers. The application of mathematical tools in scientific research has two main functions: one is to sort out observational data, and the other is make theoretical models. In the previous works about COVID-19, mathematics was mainly employed to process data, while this paper is to develop theoretical model of spatial diffusion. The shortcomings of this study are as follows. The first is the limitations of data. The urban population size is obtained from the statistical data collected two years ago, and the total number of confirmed cases may not be completely accurate. The second is the limitations of the research area. This paper takes the administrative boundary of Hubei Province as the boundary of study area, but the spatial transmission of epidemic cannot be prevented by the administrative boundary. The third is the limitation of time. Since log linear regression analysis is used in this paper, the values must be larger than 0. So the dates before January 28 were not included in the time series analysis of parameters due to the incomplete datasets and the difficulties in comparing. Despite all these deficiencies, the general trend of the COVID-19 is very clear. The day-by-day comparison showed that the modeling results based on epidemic data not only reflected certain spatial regularities, but also revealed significant geospatial information. Further study could consider breaking through the administrative boundary and analyzing epidemic transmission from Wuhan in a larger research area.

As a basic mathematical method of geospatial analysis, gravity model is useful in researching the temporal and spatial characteristics of epidemic transmission. The local gravity model for spatial diffusion was derived from the global gravity model and the allometric scaling relation between passenger flow and case transmission quantity. Which differs from the traditional local gravity model which is suitable for passenger flow. Then the newly derived local gravity model was employed to analyze the spatial diffusion from Wuhan to other cities in Hubei, and the spatial auto-regression based on the gravity model was used to investigate whether there is interaction between all prefecture-level cites in Hubei Province. The analysis results reflect the quality of the models. This study demonstrated that the spatial diffusion process of COVID-19 in Hubei Province of China was dominated by gravitational rule. Through the comparison of model structure, the analysis of the model parameter change and the variation of model statistics, chief conclusions about spatial diffusion of COVID-19 in Hubei Province can be drawn as follows.

This can be judged by the distance decay functions. According to the matching analysis of the local gravity model with observation data, the goodness of fit of the gravity model based on power-law distance decay is better than that of the gravity model based on negative exponential distance decay. This indicates that the diffusion mechanism of COVID-19 is complex and the spread is not limited within a clear boundary. That is, in theory, COVID-19 can spread from Wuhan to an infinitely distant place. The reality is that COVID-19 once spread almost all over China from the central city, Wuhan. This conclusion may be superfluous, but it is of some significance. It proves that the epidemic diffusion is not localized, and the gravity model based on negative exponential function cannot well describe the spatio-temporal evolution of epidemic effectively.

This can be judged by the mathematical structure of the local gravity model. The total number of confirmed cases of COVID-19 is directly proportional to a certain power of population size, and inversely proportional to a certain power of distance. Before February 1, the impact of distance from Wuhan was prominent, which indicated that the epidemic situation in Hubei Province was mainly caused by direct transmission from Wuhan, namely, spatial diffusion process. After February 1, 2020, the effect of city size became increasingly prominent while the distance decay exponent gradually increased. This indicated that the speed of direct transmission from Wuhan to other places had been controlled to a certain extent due to the isolation measures taken by Chinese government. The effect of the cardinal number of infected people in each region is manifested, and the secondary diffusion, namely, hierarchical diffusion mode, have emerged since February 1, 2020.

This can be judged by the results of spatial auto-regression analysis. The spatial autoregressive term is introduced into the gravity model, but the autoregressive coefficient is not significant. In general, the significance of autoregressive coefficient has gradually increased, but the confidence level failed to reached 80%. The result shows that due to the timely isolation measures, the interaction of COVID-19 spread between different regions in Hubei has not played an important role. This also demonstrates that the isolation measures taken by the government of China are effective although some people regarded “lockdown” as a clumsy method. Geospatial isolation is a necessary means to prevent and control the diffusion of COVID-19 when vaccines and efficacious drugs against the virus have not been developed yet.

The main results of multiple linear regression based on Eq (

(DOCX)

The main results of multiple linear regression based on Eq (

(DOCX)

This contains the original data and calculation process for the gravitational scaling analysis on spatial diffusion of COVID-19 in Hubei Province, China. The data are follows: (1) Population sizes of all the cities in Hubei Province. (2) Spatial distance vectors and matrix. (3) Cumulative number of novel coronavirus pneumonia confirmed at different dates.

(XLSX)

I would like to thank the two anonymous reviewers and Dr. Jing Sun whose interesting and constructive comments were very helpful in improving the quality of this paper.

PONE-D-21-07713

Gravitational Scaling Analysis on Spatial Diffusion of COVID-19 in Hubei Province, China

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Reviewer #1: It is a good paper with a clear presentation of the model, its implementation in modeling the pandemic, and a thorough interpretation of the results.

My comments are minor and discretionary:

1. The distance decay functions employed by the study include power and exponential functions. I understand that the two are the most popular ones used in the literature. The literature also suggests additional functions such as square-root exponential and log-normal (Taylor, 1983, Distance decay in spatial interactions. In Concepts and Techniques in Modern Geography) and Gaussian (Shi et al. 2012, Annals of AAG 102, 1125-1134). Future work may expand the selection set.

2. On interpretation/discussion of the result, I am not surprised by the strong effect of Wuhan on peripheral cities and no much between peripheral cities themselves, which is consistent with the spatial interaction pattern in passenger flows (even before the pandemic) in Hubei. In other words, if actual passenger flows could be obtained in the region (which is not easy in China), one could predict/explain the pandemic by that observed data without modeling the interaction.

3. I like the information presented in Table 2 and Figure 2, which are considered the main results of the paper. However, Table 2 seems to be redundant as Figure 2 tells the same info with better visual effect.

4. The lockdown in late Jan in the region completely brought population movement to a stop. I'd suspect that the temporal lag in the diagnosis cases in various cities reflects the incubation period.

Reviewer #2: This research proposes an integrated mathematical model combining gravity model with spatial autoregression model to investigate the spatial diffusion of COVID-19 in Hubei province, China. Generally, this manuscript provides us sufficient information about model selection and establishment, data source, and statistical test. The effectiveness of the proposed method was tested empirically and statistically. The analytical results are of certain enlightening value for understanding the spatial diffusion process of COVID-19, especially in terms of the effects of population size and spatial distance. However, to be accepted for publication, the presentation still need improvement. Detailed comments are as follows:

The mathematical model proposed in this study is suitable for certain conditions. Specifically, the gravity model describes the one-directional transmission of COVID-19 from Wuhan to other cities, and the spatial autoregression model considers horizontal diffusion between cities (except Wuhan) within Hubei province. More details about the epidemic prevention and control measures (i.e., spatial isolation of cities and the corresponding time spots of the implementation) should be provided to illustrate that the practical situation in Hubei is consistent with applicable conditions of the model.

Reviewer #3: Comments to the Author

The authors used gravitational scaling analysis to study the spatial transmission of COVID-19 in Hubei, China. I provided my feedbacks based on my first reading and my experiences.

Abstract:

The manuscript needs to explore the results more deeply. I don’t think the main result in the Abstract “The local gravity model based on power law decay can effectively describe the diffusion patterns and process of COVID-19 in Hubei Province, and the goodness of fit of the gravity model based on negative exponential decay to the observational data is not satisfactory” is informative, or useful. The authors only described the model function, not the COVID-19 transmission. Most readers are not interested in the goodness of fit or confidence level, which should be within the acceptable level, as default.

Conclusion part in the abstract is more important, which should replace the current results part. Yet, the authors need to clarify some information here: what does long-range effect mean, please explain? “the size of a city and the distance …affect the total number of confirmed cases”, positively affect or negatively affect, or just affect? What are direct and horizontal transmission?

The authors need definitely rewrite the abstract. Please refer to some journals’ guidance, like landscape ecology, whose abstract is structured.

Main text:

Since 2020, there have been a series of COVID-19 papers published, many of which focus on spatial transmission, like epidemiology study. The authors need to make a brief review/comparison of these publications, since the authors’ stance is from the urban geography perspective, your strengthen?

Section 3.1 should not place in the Results part.

Table 2. I cannot directly find useful information from this long table. Suggestion, plot some columns?

The authors’ work focuses on Wuhan, the first city attacked by the COVID-19 as reported. It would be useful to make a comparison study, with other provinces (sub-nation unit) in China or other countries using the model.

OVERALL

The paper provides an important perspective in studying COVID-19 transmission, which is useful and has potential implications in policy making. I hope that my feedback comments are useful to help improve it in this regard.

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Gravitational Scaling Analysis on Spatial Diffusion of COVID-19 in Hubei Province, China

PONE-D-21-07713R1

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PONE-D-21-07713R1

Gravitational Scaling Analysis on Spatial Diffusion of COVID-19 in Hubei Province, China

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