Spatiotemporal Transmission Model to Simulate an Interregional Epidemic Spreading

Infectious disease spread is a spatiotemporal process with significant regional differences that can be affected by multiple factors, such as human mobility and manner of contact. From a geographical perspective, the simulation and analysis of an epidemic can promote an understanding of the contagion mechanism and lead to an accurate prediction of its future trends. The existing methods fail to consider the mutual feedback mechanism of heterogeneities between the interregional population interaction and the regional transmission conditions (e.g., contact probability and the effective reproduction number). This disadvantage oversimplifies the transmission process and reduces the accuracy of the simulation results. To fill this gap, a general model considering the spatiotemporal characteristics is proposed, which includes compartment modeling of population categories, flow interaction modeling of population movements, and spatial spread modeling of an infectious disease. Furthermore, the correctness of a theoretical hypothesis for modeling and prediction accuracy of this model was tested with a synthetic data set and a real-world COVID-19 data set in China, respectively. The theoretical contribution of this article was to verify that the interplay of multiple types of geospatial heterogeneities dramatically influences the spatial spread of infectious disease. This model provides an effective method for solving infectious disease simulation problems involving dynamic, complex spatiotemporal processes of geographical elements, such as optimization of lockdown strategies, analyses of the medical resource carrying capacity, and risk assessment of herd immunity from the perspective of geography. Key Words: geospatial heterogeneities, health geography, interregional population interaction, spatiotemporal analysis, transmission modeling.


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stablishing models that correctly simulate the spread of infectious diseases is an effective method to understand and recognize the spreading process.The spread of infectious diseases usually depends on the behavior of individuals, the flow of people, and the heterogeneity of different spatial environments.These factors endow the results of spread with considerable spatial characteristics (Zhu, Li, and Hou 2022).Therefore, the analysis of the spread of infectious diseases from a spatial perspective, especially the establishment of a model of infectious disease transmission from a spatial perspective, is a common task for researchers in multiple disciplines, such as behavioral geography, health geography, and geographic information systems (Xi et al. 2023).Facing raging COVID-19 and unpredictable infectious diseases in the future, establishing effective spatiotemporal infectious disease models has become an important research topic for geographers.
The process of spatial transmission of infectious diseases can be influenced by both interregional population movements (e.g., the inflow of latents created by infected persons who have left the city for other cities) and differences in the conditions under which viruses are transmitted within different regions (e.g., different control measures are usually taken in different cities, which results in different efficiencies of transmission of infectious diseases in different cities).Theoretically, the former reflects the theory of spatial interaction (Roy and Thill 2003;H. Zhang et al. 2022), in which the process of epidemic evolution at the city scale is linked through population movements.The latter reflects the theory of spatial heterogeneity (Anselin 1989), in which differences in population density, intensity of crowd travel, and other elements cause spatial differences in epidemiological parameters such as effective reproduction numbers and transmission rates in different cities (J.Huang and Kwan 2022).
In the face of an outbreak of a highly transmissible infectious disease, governments need to have access to complete real-time data on the infections to formulate effective prevention strategies and assess the actual loss caused by the epidemic.Simulations used to obtain relevant data and optimize strategies are an important tool to address infectious diseases in modern society.Combined with the preceding theoretical background analysis, the spatiotemporal transmission model of infectious diseases needs to first consider the heterogeneity of population movement in each region and the regional heterogeneity of transmission conditions.
Because the spatial transmission of infectious diseases is a complex spatiotemporal system, the population flow process will affect the epidemiological parameters within each region, and the changes in the epidemiological parameters will change the proportion and number of latents and infections in the region.It can conversely influence the subsequent population flow process.Therefore, this article proposes a hypothesis, that the model can simulate the spatiotemporal transmission process of infectious diseases more reasonably by constructing a heterogeneity mutual feedback mechanism conforming to spatial law based on considering different types of heterogeneity.
There are a few models that typically consider the heterogeneity of population movement or the spatial heterogeneity of transmission conditions.Existing models, however, do not consider the mutual feedback between the aforementioned interregional population movement heterogeneity and the regional heterogeneity of transmission conditions driving the former variation, which leads to the neglect of the interplay of spatial heterogeneity in the whole process, substantially simplifying the complex process of spatial transmission of infectious diseases and reducing the prediction accuracy of the spatial distribution of infections.
The main purpose of this article is to propose a general method for the simulation and prediction of the interregional spread of infectious diseases from the perspective of complex spatial dynamic systems.The mutual feedback between heterogeneities is realized through a combination of interregional movement of various classified populations and intraregional transmission of infectious diseases subsequent to the movement.First, the conceptual framework of spatiotemporal infectious disease spread is established based on the general rules of COVID-19 transmission as an example.Then, the specific implementation process and algorithm of the model are introduced through rigorous stylized equations.Finally, a synthetic data set and real-world data set are provided to evaluate the correctness of our theoretical hypothesis and accuracy of the model via controlled experiments.

Literature Review
At present, numerous models simulating infectious diseases are available, and the method system is relatively complete.Scholars have classified these models from different perspectives (Riley 2007;Bian 2013).This article divides the existing models into classic simulation models and spatial simulation models according to whether spatial logic is introduced.Classic simulation models provide the most basic method for analyzing the prevention and transmission mechanism of infectious diseases.Classic models, however, have difficulty simulating the process of infectious diseases spread in geospatial space because they do not incorporate the simulation process in a more realistic geospatial scenario (Cantrell, Cosner, and Ruan 2009).
This section focuses on spatial simulation models and introduces the development trend of these models.The regional scale of the analysis affects infectious disease transmission process modeling.Spatial simulation models are further divided into three categories-microscopic models, mesoscopic models, and macroscopic models-according to the differences in the modeling of human mobile behavior processes determined by the scale differences in the scale of spatial units.

Spatial Transmission Model of Infectious Disease from a Micro Perspective
Infectious disease simulation models from a microscopic perspective are used to study infectious disease transmission phenomena occurring in specific small areas.These models have been developed along with system simulation techniques and pedestrian dynamics, and are used in risk assessments in locations with high pedestrian flows such as university campuses, cruise ships, and airplane cabins (C.-Y.Li and Yin 2023).These models focus on individual movement behavior (e.g., step and duration of individual movement), the way individuals interact with each other (e.g., social force models), and the mechanisms of virus transmission in space (direct, droplet, and aerosol transmission), mostly using individualbased models (IBMs) or agent-based models (ABMs; Harweg, Bachmann, and Weichert 2023).These models enable the study of both the processes by which transmission of infectious diseases occurs in individuals and the influence of differences in individual behavior and spatial facilities on the transmission of infectious diseases (Honey-Ros es et al. 2021).
Microscopic models usually require modeling how individuals move and how viruses are transmitted.They are able to capture the state of individuals in the space of a place, where the individuals are located, and when infection occurs.These models often involve very precise location calculations during process simulation, and thus the operation efficiency will be reduced when the number of individuals and simulation duration increase (Gosc e, Barton, and Johansson 2014).The models are less generalizable and are currently used for only some small indoor and outdoor sites because of the substantial variation in function and spatial layout of different sites (C.-Y.Li and Yin 2023).Moreover, these models are not sufficient to model the transmission process of infectious diseases on a large regional scale or even across regions.
After assessing the special transmission characteristics of COVID-19 (C.-Y. Li and Yin 2023), the researchers designed new simulation methods for public spaces ranging from college campus and academic buildings, and studied elements such as the effectiveness of nonpharmacological interventions at different levels (Romero, Stone, and Ford 2020;C.-Y. Li and Yin 2023).

Spatial Transmission Model of Infectious Disease from the Meso Perspective
When an infectious disease analysis is performed within a larger area with very high heterogeneity (e.g., neighborhoods, cities), it is considered a spatial simulation model of infectious diseases at the mesoscopic scale.Mesoscale simulations of infectious disease transmission processes emphasize the influence of individual or group mobility across sites, the population density within sites, and the heterogeneity of site environments on the transmission process.Meanwhile, they typically ignore the effects of individual behavioral interaction processes in microscale models (Bian et al. 2012;Q. Huang et al. 2022).Depending on whether the basic human unit used for modeling is a population or an individual, models can be classified as population-based or individual-based models.
Population-based models generally consider a single site or a spatial grid where spatial heterogeneity exists as the basic spatial cell and model the movement of the population between cells, thereby simulating the spread of infectious diseases (Lai et al. 2015).The size of population movements is usually determined based on movement data and gravity models (Zhou et al. 2022).In IBMs, agents are mainly used to portray the characteristics of human individuals, and the transmission process of infectious diseases among individuals is simulated via the interaction between agents.Two modeling approaches have been developed: (1) agents exist and move in the spatial cells, and the contagion process occurs within the spatial cell (Q.Huang et al. 2022;Zhou et al. 2022); and (2) the flow of places and intersite agents as a whole is considered a network, and the transmission process of infectious diseases is modeled based on the analytical paradigm of spatially complex networks (Mao 2014;Zhong and Bian 2016).
The cellular automata model is also used in both models (Holmes 1997).Each cell species can contain one or more individuals, and the influence between cells can consider only neighboring regions or is customized to set the number of movable steps (Sirakoulis, Karafyllidis, and Thanailakis 2000;Bian 2013).Mesoscopic models are suitable for analyzing the patterns of infections occurring in different locations within the city, thus assisting in the development of accurate prevention and control strategies (Q.Huang et al. 2022;Zhou et al. 2022).These models usually oversimplify movement processes by only considering starting and ending points, making it difficult to obtain detailed information about infection processes.Such models are also difficult to extend directly to cross-city modeling of infectious diseases (Zhou et al. 2022).Recent studies indicate that the incidence of COVID-19 in urban areas is correlated with the sociospatial structure (Galacho-Jim enez et al. 2022).Innovative models like metapopulation network models can incorporate more detailed variables to simulate infectious disease transmission in cities (Calvetti et al. 2020).

Spatial Transmission Model of Infectious Disease from a Macro Perspective
The macro infectious disease model is used to dynamically simulate the process of infectious disease spread across regions, which involves the interaction process among large spatial units such as countries, regions, and cities.This scenario usually ignores individual behavior in the space unit and regards the space unit as a whole.These models consider the interaction between space units and the population flow (Sattenspiel and Dietz 1995;Colizza and Vespignani 2008).These models can be divided into nonspatial interaction process models and spatial interaction process models according to whether the population flow among the geospatial units is calculated in the modeling process.
Nonspatial interaction process models consider the spatial heterogeneity of the spread of infectious diseases in each spatial unit, as well as the influence of the interaction between spatial units.The methods for considering the spatiality in the modeling include (1) introducing parameters with spatial heterogeneity (Colizza and Vespignani 2008), (2) incorporating population mobility as a parameter (Meyer and Held 2017), and (3) introducing the spatial interaction strength directly into the model as a parameter (Grenfell, Bjørnstad, and Kappey 2001).The simulation results from these models might be more reasonable because the spatial differences are considered.The spatial interaction process model usually establishes the movement process of individuals or populations in the model based on mobile data obtained over a larger spatial range (e.g., global flight data and domestic mobile data).They couple the results of the movement to the infectious process to calculate the spread of the infectious disease in each region (Hou et al. 2021).
Individual-based spatial interaction process models simulate infectious disease transmission when individuals move between countries or cities, and have been used to analyze disease transmission under protective conditions (Chang et al. 2020).These models consider individual properties and movement between regions, producing detailed and accurate simulation results.Moreover, these models have been used to predict the transmission path and are used in many research scenarios (Chinazzi et al. 2020;T. Zhang and Li 2021).Because the data for simulating individual movement are difficult to obtain and the census data for setting individual attributes lag behind, simulation accuracy is affected (Riley 2007;Fair, Zachreson, and Prokopenko 2019).
Population-based spatial interaction process models construct networks by setting different regional units as nodes and population movement among regions as edges.The heterogeneity (social structure and population dynamics) of each spatial unit in the model is typically considered such that the process of infection occurs differently within each region (Sattenspiel and Dietz 1995;Colizza and Vespignani 2008).Population movement data are introduced to calculate the transmission process and changes in the total population or number of infections.These models are widely used due to their high operational efficiency and ability to model infectious disease transmission across multiple regional units (Yin Ling et al. 2021).Most existing models ignore the interaction effect between the infection process within the spatial units and the flow of people among spatial units.This approach leads to neglect of the mutual feedback and causal relationships of variables between regions during the entire transmission process.Thus, the models lack system integrity.Population mobility and population density can promote the spatial spread of COVID-19 in a large regional scope, and researchers have developed a number of models to assess the impact of mobility restrictions and spatial heterogeneity on outbreaks (T.Zhang and Li 2021;Xi et al. 2023).

Method Conceptual Framework of the Spatiotemporal Transmission Model for Infectious Disease
In this article, we propose a macro spatiotemporal infectious disease model from the perspective of mutual feedback between heterogeneities, which can simulate the movement of populations and virus carriers among regions and the intraregional spread of infectious diseases.This model has two assumptions: (1) the same person moves from one region to another in one day but not to a third region, and (2) the individual does not get infected on trains, planes, or in other vehicles during the journey from one area to another.The regions studied refer mainly to countries and cities rather than areas within cities or finer scales.Based on these assumptions and mutual feedback mechanism, the whole model is divided into three main steps: compartment classification of the population (compartment modeling), population flows among regions (flow interaction modeling), and spread of the infectious disease within a region (spatial spreading modeling) based on these steps.The model is named the spatiotemporal transmission model of infectious disease (STTM).The conceptual framework of the STTM is shown in Figure 1.
First, in the compartment modeling, the population is classified into different groups according to whether the individuals have been infected and their infection status.The idea is similar to the compartments used to classify goods.Thus, the complex interaction and infection relationship between individuals is transformed into the interaction and infection relationship within and between groups.
In the next step of flow interaction modeling, the interregional population movements are transformed into interregional movements of different groups.This forms a complex population flow network that makes data analysis and statistics challenging.On the one hand, methods must be developed to clearly identify the number and destination of each group flow in each region.On the other hand, how do the interregional population flow and intraregional transmission that have occurred affect the upcoming flow processes?The construction of flow rules and implementation of algorithms must be based on actual conditions to determine the number of inflowing and outflowing individuals in each group in each region in the same period.This process provides a statistical basis for further analysis.
The final step is the spatial spreading modeling, which is a model of the infectious disease transmission process based on the population classified according to mechanisms and methods for modeling infectious disease dynamics.Finally, with an interval of days, the numbers of different groups in each region can be updated, and the intraregional spread of the infectious disease and the infection simulation of the population in each region can be obtained each day.The preceding paragraphs describe the main implementation logic of the overall STTM.
From a macro perspective, the transmission process within each region in the STTM is affected by regional heterogeneity of transmission conditions (proportion and number of different groups, contact probability), the interregional population movement heterogeneity, and the mutual feedback of these heterogeneities.In the flow interaction model, the total inflow and outflow of the population among all regions forms a dynamically balanced system.The population flow can change the proportion and number of each group within each region, which will affect the contact probability in spatial spreading model, causing a change in the transmission rate when the transmission process occurs within a region.These processes change the groups in every region and will affect the proportion and number of each group in the floating population in the subsequent flow interaction model.We have provided a detailed logical introduction to the construction of a mutual feedback mechanism in a macro infectious disease model.Using China as an example, the mutual feedback process occurs among more than 300 cities.Any city will be affected by more than 300 cities, and all these cities will conversely be affected by this city.The advantages of this model are obvious, compared with the nonspatial model or the model without the mutual feedback mechanism.

Compartment Modeling of Population Categories.
The compartment concept used in the traditional infectious disease dynamics model can quantitatively analyze the number of infections according to the characteristics of infectious diseases.The first step of the STTM is to build a crowd compartment, namely, compartment modeling.This step makes the model applicable to the transmission scenario of infectious diseases.First, three conditions are required for the spread of an infectious disease: the source of the infection, susceptible individuals, and a transmission route.Compartment modeling for a city (Figure 2A) is able to quantify these three conditions.Specifically, individuals (infected or latent) at different pathological stages and uninfected individuals (persons with antibodies and healthy persons without antibodies) coexist.
If the modeling unit is the individual, the low efficiency caused by the complex behavior of a large number of individuals and the privacy problems brought by the acquisition of individual information will be confronted in the simulation process (Riley 2007).To avoid these problems, in this model, the population is classified into groups to establish a compartment from a macro perspective.The interactive behavior of individuals can be transformed into group behavior, which can not only reduce the difficulty of data acquisition and analysis, but also establish the correct transformation relationship of the different types of groups.For COVID-19, three main transmission modes have been identified among individuals: droplet transmission, contact transmission, and aerosol transmission (World Health Organization-China 2020).When susceptible people interact with carriers, the susceptible people are likely to be infected.The probability of infection for the susceptible group is similar to their probability of individual infection when interacting with groups having an infectious capability (World Health Organization-China 2020).
The rules of virus transmission between any classified group are determined in this model.Drawing from the consistency of behavioral interaction between individuals and groups, the compartment model in this article divides the population into four main categories, S, E, I, and R, as shown in Figure 2B.S denotes a susceptible person who is not infected but does not have immunity, E denotes an exposed person who has an infectious ability but does not show clinical symptoms, I denotes an infected person who has the ability to infect and shows clinical symptoms, and R denotes a removed person who has died or obtained immunity after infection.Through this classification, the difficulty of tracking all individuals to obtain information is avoided, and information such as the number of people with a particular infection status can easily be obtained to simulate the spread of an infectious disease from a macro perspective, as shown in Figure 2C.Individual movements occur among regions according to individual needs and objective conditions.Figure 3A shows that population movements occur among city clusters.Population movements are defined as a variety of short-term, repetitive, or periodic movements of population flows among regions.First, without interregional population movement, an infectious disease characterized by human-to-human transmission will not spread to a healthy area, and an epidemic does not occur.Second, the speed of population movement will also affect the spread of infectious diseases.A faster rate of population movement means that people interact with each other at a higher frequency; thus, the risk of infection is increased (Wan and Wan 2022).Moreover, different regions have their own epidemic spread characteristics and spread ceilings.Such geographical heterogeneities are neglected in the classic compartment models.In conclusion, it is necessary to consider the influence of population movement in the model.
As shown in Figure 3, according to the characteristics of the group itself and the anti-infection measures, S, E, I, and R are further divided into two categories: those who can move between cities and those who cannot.As S and E cannot be visually rated as being at-risk, they belong to groups that can move between cities.In the context of the early stages of the spread of COVID-19 in China, infected people with any acute respiratory illness were often identified as suspected patients.Although these patients are not diagnosed, they still have limited mobility (World Health Organization 2020).I and R are quarantined according to the prevention and control policy.Thus, I and R are set as immovable populations (World Health Organization-China 2020).
Based on this classification, intercity population movements are divided into three situations (Figure 3B).As shown in Figure 3B, pathogens are present in cities C 1 and C 2 ; S and E move from city C 1 to city C 2 and from city C 2 to city C 1 .As shown in Figure 3B, city C 3 does not have pathogens; S and E in city C 4 will move into city C 3, and only S in city C 3 moves into city C 4 .As shown in Figure 3B, no pathogens are present, and only a flow of S is observed between the two cities.Thus, a conceptual model of flow in the course of the spread of the infectious disease can be established.

Spatial Spreading Modeling of Infectious Disease.
The process of simulating the spatial spread of infectious disease is based on compartment modeling and flow interaction modeling.First, the model divides the daily spread of the infectious disease into two parts: the flow process and the infection process.The flow process changes the population categories within a single geographic unit and the population distribution between regions.The infection process changes only the population categories within a region and does not affect the differences in spatial distribution.
Based on the preceding discussion, the traditional compartment model is used to simulate changes in the numbers of different population groups, and the data calculated using this model are the basis for the subsequent calculation.Using these multiple iterative analyses, the quantitative differences and changes in the spatial distribution of the population categories in the target area are simulated in a certain period.
As shown in Figure 4C, S is converted to E through the effects of both E and I. Furthermore, E will be aggravated by their own illness into I, and I will similarly be converted to R. The dead individuals in R are inherently immobile, and the recovered individuals in R and I are affected by the prevention measures (World Health Organization-China 2020), resulting in limited mobility; thus, I and R are not considered mobile.The control measure in Figure 4A shows the quarantine measures for I and R.
In the daily calculation of the course of infection, population movements have a "first impact" on the number of SEIR groups in each region; then, the spreading process within the region has a "second impact" on these groups.In fact, infections still exist during the population movement process, but they do not affect the spatial distribution of the infection outcomes.Thus, the second impact mainly causes the mutual transformation of different groups within a region, leading to a gradual decrease in the number of healthy people and an increase in the number of infections, as shown in Figure 4B.In this infection pattern, changes in quantity are related to only the population base after the population movements in the same day, regardless of the infection that occurs during the process of population movement.Pie charts in Figure 4A and 4B visually represent changes in different groups.
In contrast to the traditional SEIR model, the transformation foundation of this model is based on flow data; namely, the method for analyzing population flow in a geographic analysis is used to simulate the spread of infectious disease.The distance, degree of connectivity, and influence between regions affect population flow and are reflected through flow data.The process of population flow, together with geospatial heterogeneity factors such as population vitality, affects the process of transmission within each region.

Algorithm Description and Implementation
In this study, the implementation of the STTM consists of three main steps from a macro perspective: (1) different types of population flow algorithms between regions (macroscale; e.g., regions, cities, countries, etc.); (2) the model of virus transmission within the region described; and (3) the iterative calculation performed for a period using the population flow algorithm and virus transmission algorithm, with the iterative process.

Initial Conditions.
The initial population category data within each region are the basis of the calculation using the model, and the cases provided use cities at the regional scale.The city cluster in Figure 5A that consists of several cities is expressed as City ¼ fA, B, C, D, ::::::g, and its corresponding population set is G ¼ fF A , F B , F C , F D , ::::::g: Figure 5 shows only four cities in the city cluster (A, B, C, and D) as the study object.The city instance set is City 0 ¼ {A, B, C, D}, and its corresponding population set is G 0 ¼ fF A , F B , F C , F D g: Here, these areas are visualized in a bar chart of their spatial location and the various categories of their populations.
During the transmission of different diseases, the population categories will have different classifications.In this case, the city population categories are classified as the susceptible (S), exposed (E), infected (I), and removed (R), where S and E are movable and I and R are immovable.Each city forms a set of categories of the city population according to the designated classification rules, wherein Areas with a boundary are units of population movement.Each region has a corresponding inflow (F in ) and outflow (F out ).In Figure 6, the population movements among four cities are shown as an example.
As shown in the previous population movement settings, I and R in this model do not have the ability to flow.Thus, interregional population movements are attributed to two factors, the mobility of S and E, namely, the overall mobile population set fF ij g ¼ {fS ij g, fE ij g}.fS ij g and fE ij g denote the movable population sets of S and E, respectively.S and E are movable in the process of population movements among cities, which in this article are called the inflow set of S {S in }, the outflow set of S {S out }, the inflow set of E {E in }, and the outflow set of E {E out }. Figure 6  According to the flow rules of this model, the formula based on the law of population movements of the compartment model is summarized in Equations 1 to 13 (i and j in the formula refer to any city involved in population movements with no fixed restrictions).
Regarding formulas of population movements, Equations 1 and 2 are used to obtain the migration rates of S (u S ) and E (u E ), referring to the ratio of S and E in the overall mobile population.The migration rate is determined by the population categories in each city, with different migration rates, where fu S g and fu E g are the sets of migration rates for S and E in the city cluster, respectively.
Equations 3 and 4 obtain the numbers of S and E (equal to the product of the migration rate and the total number of migrants) in the process of the population movements from city i to city j.The number of migrants varies from city to city, and fS ij g and fE ij g represent the mobile population set of S and E in all cities, respectively.
Equations 5, 6, and 7 describe the numbers in the overall population, S, and E that flow into city i during the process of city i population flow to city j, respectively.{F in }, {S in }, and {E in } are the inflow sets of the overall population, S, and E, respectively, to each city.
Equations 8, 9, and 10 describe the numbers of the overall population, S, and E that flow out of city i during the process of city i population flow to city j, respectively.{F out }, {S out }, and {E out } are the outflow sets of the overall population, S, and E, respectively, from each city.
Equations 11, 12, and 13 describe the population numbers of the overall population, S, and E in city i after the population movements, respectively, namely, the newly formed set of population categories in each city after movement.
After the numbers of population movements among city clusters are calculated, an origin-destination (OD) matrix of the various groups of populations is generated (Figure 7E).Because the overall population flow OD matrix is obtained from known data, after the initial population categories within each city are determined, the corresponding migration rates of S and E of each city are calculated.The population flow OD matrix of S and E among cities is obtained as the initial data from the improved SEIR model in the city that is subsequently used by the model, which can be calculated iteratively.An example of the process used to determine population movement among the four cities is shown in Figure 7D, 7E, and 7F.
Spread of an Infectious Disease within the Region.The infectious diseases considered in this study have an incubation period, so the SEIR model should be used in the epidemic dynamics model.Due to the existence of geospatial heterogeneity, the traditional SEIR model cannot be used in research on the interregional transmission process of infectious diseases and in simulating the process of virus proliferation within subregions after a spatial interaction.Combined with the theoretical analysis of the spatial spreading modeling, the spatiotemporal-SEIR (ST-SEIR model) of the interregional flowing state variable is constructed and used for each subregion.All subregions within the spatial range are calculated using the ST-SEIR model, and the number of different groups within each subregion must be recounted to record the results for the transmission of infectious disease during the current transmission cycle and prepare for the next transmission cycle.Next, the symbols and parameters (Table 1) and formulas used in the calculation are described.
The ST-SEIR model represents the transformation relationship of different groups.Its conceptual diagram presented in Figure 8A shows that the transformation relationship occurs in the four different Equation 14 represents the number of S within a subregion after a transmission cycle, with the first item indicating the number of S within the subregion after the population flows across the region.The second item is the total number of S infected by the I and becoming E within the subregion.The third item is the total number of S being infected by E and becoming E within the subregion.The second and third items are infection items, which result in newly infected individuals who must be removed from S. The number of I, R, and total population in the subregion before the process of spatial spreading The number of S and E in the subregion before the process of flow interaction S tþ1 E tþ1 I tþ1 R tþ1 The number of S, E, I, and R after a transmission cycle b 1 The probability of transition from S to E after S contacts I b 2 The probability of transition from S to E after S contacts E r t ð Þ The  Equation 15 represents the number of E within a subregion after a transmission cycle, with the first item indicating the number of E inside the subregion after the interregional population flows.The second and third items are infection items, namely, the same as those in Equation 14, which indicates that new I exists in the form of E in the E population within the subregion.The fourth item indicates that a portion of E within the subregion is converted into E during the transmission cycle and must be removed from E.
Equation 16 represents the number of I being infected within a subregion after a transmission cycle, with the first item representing the total number of people being infected within the subregion at the end of the previous transmission cycle.The second and third items represent the number of people who were converted from E to I in the transmission cycle within the subregion and the number of people who were converted from I to R, respectively.
Equation 17 indicates the number of R who died or recovered within a subregion after a transmission cycle, with the first item representing the total number of R in the subregion at the end of the previous transmission cycle and the second representing the number of people who transitioned from I to R.
City x is shown in Figure 8B as an example.The ST-SEIR model is used to calculate infection methods within the city.City x has experienced a process of population movement with other cities on day t, resulting in the numbers of various groups of people classified as S 0 t , E 0 t , I t , and R t : Based on these data, Equations 14, 15, 16, and 17 are used to calculate the infection results for city x on day t, which are S tþ1 , E tþ1 , I tþ1 , and R tþ1 , respectively.Then, we apply this calculation to the four cities A, B, C, and D mentioned in Figure 7 and calculate the results for infection in these cities during any of the transmission cycles, as shown in Figure 8C.

Case Study Synthetic Data Set
Data and Parameter Description.
The synthetic data set is used to simulate the transmission process of infectious diseases in city clusters.The objectives of the experiments in this section are (1) to prove the correctness of the theoretical hypothesis presented in this article, and (2) to demonstrate the effectiveness of the STTM.The main focus of the experiments here is to calculate the number of daily infections in each city and the number of flowing latents among cities using STTM.
Figure 9 shows the sixteen cities constructed in the simulation, as well as their population vitality, population categories, population mobility data, and necessary infectious disease parameters.The synthetic data set contains population, intercity population mobility data, and infectious disease parameters for each city, corresponding to the population categories data set (D R ), the population mobility data set (D F ), and the infectious disease parameter data set (D S ), respectively.
As shown in Figure 9B, the population categories data set (D R ) contains six fields: city ID (City ID), total population (Pop_sum), total number of susceptible individuals (Pop_susceptible), total number of individuals with a latent infection (Pop_exposed), total number of infected individuals (Pop_infected), and total number of dead or recovered individuals (Pop_removed).The total population (Pop_sum) represents the population size of each city.Here, the minimum and maximum population sizes are set to 5 million and 10 million, respectively.We assumed that city 2 is the first city to be infected with one initial latent individual (E).
For the population mobility data set (D F ), as shown in Figure 9C, the flow rate of the population between each city is presented in a population flow matrix, where rows and columns represent the departure city and the destination city, and each element in the matrix represents the number of people traveling from one city to another.The sum of each row in the matrix represents the total outflows per day for a certain city, and the sum of each column represents the total inflow per day in a certain city.The maximum and minimum sizes of the total inflows and outflows are set to 15,000 and 7,500, respectively.The daily population movements among cities are shown in the population mobility matrices in Figure 9C.The first group is used before the number of infections reaches the threshold, indicating that no external force is involved in the infectious disease, and the daily contact rate of larger and smaller cities is set to 4 and 3, respectively.The second group is used after the number of infected individuals reaches the threshold, indicating that external forces are involved in the spread of the infectious disease.As a result, the transmissibility of infectious diseases is reduced by a factor of three to four, leading to the end of the infectious process.
In this experiment, sixteen cities were initially free of infections until the first latent of an infectious disease (E) appeared in city 2. At this point the infectious disease transmission process begins.This E carrying the virus spreads it from city 2 to other cities through population movements, and the infection will continue to occur in and spread to other cities.After the number of infections in all cities reaches a threshold of 120,000, the transmissibility of the infectious disease is reduced by human interventions.Simulating the spatiotemporal transmission process of infectious diseases under human intervention conditions, the process at that spatial scale is considered to be finished when the number of new infections within each city continues to drop to 100.
Combined with the theoretical analysis, the number of infections within each city, which affects the rate of infection growth, is dramatically influenced by heterogeneity of interregional population movement, the heterogeneity of regional transmission conditions, and the mutual feedback between these two heterogeneities.

Results.
Figure 10 illustrates the whole process of the spatial spread of infectious diseases.According to the calculation, the total number of infected persons in all cities reached the threshold for increasing control measures on day 120 and the threshold for the end of transmission on day 194.Observing the infection curves, there were significant differences in the curves of different cities by the end of the epidemic, which shows the variability of the infection process in different cities.Additionally, the infection results can be roughly divided into three groups.The infection curve of all cities fluctuates different degrees, which are caused by the combined effect of the population movement process among cities and the transmission process of infectious diseases occurring within cities and the mutual feedback between them on the transmission process within each city.
The results in Figure 10 indicate that different heterogeneities and their mutual feedback significantly influence the spatial transmission process of infectious diseases.Specifically, Figure 11B, for example, demonstrates the infection process from day 98 to day 115.The order of the number of city infections changed during this period, and the highlighted part in Figure 11C shows the city IDs where the order changed.There are three main reasons for this change.First, the transmission conditions within each city are heterogeneous, as illustrated in Figure 10.Second, the process of population flow differs across cities.The OD flows in Figure 11E show the number of latents for all flows up to the day 98 cutoff, and the differences in latent flows also affect the intracity transmission process.Third, the fluctuations of each infection curve mentioned in the previous paragraph show that the interaction between different heterogeneities also affects the transmission process.
In summary, the STTM, which considers multiple types of geospatial heterogeneities and mutual feedback mechanisms between heterogeneities, can more reasonably model the spatiotemporal transmission process of infectious diseases.

Real-World Data Set
Study Area, Data, and Parameter Description.
A real-world data set is used to simulate the spatiotemporal transmission process of the COVID-19 epidemic in mainland China between 1 January 2020 and 22 January 2020.The aim is to use the STTM to simulate the spatiotemporal transmission process of the COVID-19 epidemic in the real world and compare it with the results of the study on the medical history of SARS-CoV-2 infections, thereby reflecting the practical value of the model.
The main goal of the experiment is to establish 104 infections in China according to a study (Guan et al. 2020) of the medical history of SARS-CoV-2 infectors to simulate the process of interregional flow of infectious diseases using population flow data and calculate the daily infection status of each prefecture-level city in the Chinese mainland from 1 January 2020 to 22 January 2020, in combination with the parameters of the early infectious diseases caused by SARS-CoV-2.The reason for choosing this research period is that the global awareness of SARS-CoV-2 was low during this period, and the observation data required by the STTM, such as population movements, daily exposure rate, and infectious disease parameters, had not yet been affected by policies or people's attitudes toward epidemic prevention.After lockdown measures were launched in Wuhan, China, and the strictest emergency response regulations were launched in all provinces on 23 January 2020, people's travel and their awareness of COVID-19 changed.Hence, the observed data were affected by human factors.
The real-world data set contains three different data sets (Table 2): the China population categories dataset (D RC ), which includes the population of the 354 cities of the Chinese mainland and the number of latent (E) or infected (I) individuals who already existed on 1 January 2020; the China population mobility data set (D RC ), which includes the vast majority of daily population flow data among most prefecture-level cities of the Chinese mainland from 1 January 2020 to 22 January 2020; and the infectious disease parameter data set (D FC ).These three data sets have the same form as the three data sets used in previous section.The total population of each prefecture-level city in the D RC comes from National Bureau of of China (China Statistical Yearbook 2020).The judgment of whether each prefecture-level city contains latent persons or confirmed infections is based on notifications from the National Health Commission of China (World Health Organization-China 2020) and an investigation of the medical history of patients infected with SARS-CoV-2 (Guan et al. 2020).According to the standard of whether person was infected before 10 January 2020, thirteen provinces are regarded as the areas with latent or diagnosed infected people.In these thirteen provinces, 104 latent (E) or infected (I) individuals were identified according to the literature (J.Zhang, Litvinova, Liang, et al. 2020).Among them, Wuhan city had a total of eighty-seven latent or infected persons (World Health Organization-China 2020), and the remaining seventeen latent or infected persons were distributed in the prefecture-level cities using population as a weight.The distribution results are shown in Figure 12A.
The population mobility data among cities in D FC were calculated using the hypernetwork model algorithm based on Baidu Migration Big Data (Xiu et al. 2021).The author of these data has proven that the accuracy of the population flow data among cities obtained by calculating the R 2 measure exceeds 90 percent.Figure 12B shows the population movement among every pair of cities in China in January 2020 in the form of a population flow matrix.
b 1 and b 2 in D SC describe the transmissibility of SARS-CoV-2 in a natural environment.The values of these two parameters (Table 3) were calculated from the mean value of the basic reproductive number (R 0 ) compiled in the literature (Chen et al. 2020) and derived using the method in the literature (Youssef and Scoglio 2011).r and c are the mean values selected according to the literature (Yang et al. 2020).r varies by city and with time (J.Zhang, Litvinova, Liang, et al. 2020).The method of obtaining these data is described in the Supplemental Material.Figure 13 shows the results of the largescale spread of COVID-19 on 10, 17, and 22 January 2020 caused interregional transmission and intraregional population contact with infected individuals under the condition of existing initial infections in some areas.From the perspective of spatial spreading, Wuhan, a city with a large number of initial infections, had the fastest rate of new infections and always had the most infections; the number of infections in other cities in Hubei Province, cities around Hubei Province (the province to which Wuhan belongs), Beijing (the capital of China, in northern China), Shanghai (the largest city on the east coast of China), and other cities also increased rapidly.Other cities were affected to varying degrees and gradually experienced a certain scale of infection.From the perspective of the process of infectious diseases, the total number of infections in many of the cities at different stages was not considerable at.Over time, however, the number of cities with more infections increased, transitioning from a power-law distribution on 10 January 2020 to a skewed distribution on 22 January 2020.
The simulated infection curve shown in Figure 14 reveals that from 1 January 2020, infections and the total number of newly infected people also increased daily.By 22 January 2020, the number of simulated infections in China reached 16,850, and the number of simulated infections in Wuhan reached 7,617.This result is within the confidence interval (CI) of the total number of infections in China and Wuhan,16,829 (95 percent CI [3,797,30,271]) and 13,118 (95 percent CI [3,797,30,271]), respectively, during the period from 10 January 2020 to 23 January 2020, obtained by optimal fitting in a paper published in Science (R. Li et al. 2020).Compared with existing At the beginning of the outbreak in China, the daily confirmed cases did not represent the number of new infections (World Health Organization-China 2020).A large number of infections were not diagnosed.The new real-world cases shown in Figure 14 represent the number of reports ten days after the simulation results.The reason for selecting these data as a reference is that the peak patient incidence is concentrated between 24 and 28 January 2020, whereas the peak number of confirmed cases is concentrated on 5 February 2020, approximately ten days later (J.Zhang, Litvinova, Liang, et al. 2020).These data are still limited by the detection capability and diagnostic criteria, and they cannot not fully count the number of infections (J.Zhang, Litvinova, Wang, et al. 2020).Simulation results show a general overestimation due to the low understanding of all sectors of society, the limited detection ability, and the standards for case diagnosis at the beginning of the COVID-19 outbreak, resulting in the interval between infection and diagnosis of infected persons reaching several weeks.This issue resulted in a lag in the data.The simulation results are compared with the existing studies mentioned in the previous paragraph to study the accuracy of the model.
The results obtained using the STTM to simulate the early process of the SARS-CoV-2 spread in China are supported by theoretical research and are reasonable compared with the empirical research results from patients with COVID-19 in China.This outcome proves the accuracy of the STTM in simulating the actual epidemic transmission process.

Discussion Theoretical Contributions
The main advantage of the proposed model in this article for infectious diseases with large regional scope, large population size, and interregional transmission is that it considers the mutual feedback between the heterogeneity of interregional population movements (the number of inflow and outflow populations and the proportion of susceptible and immune persons among them, etc.) and the spatial heterogeneity of intraregional transmission conditions (the proportion of susceptible and immune persons within a region, daily contact number, etc.).This enforces the spatial connection and further improves the accuracy of the spatiotemporal transmission simulation of infectious diseases.
The contribution of this article is to design a modeling approach for infectious diseases from the perspective of complex spatiotemporal dynamic systems, and the following mechanisms are introduced in the model: (1) the influence of population on epidemic transmission, as emphasized in spatial interaction theory; (2) the influence of spatial differences on spreading rates, as emphasized in spatial heterogeneity theory; and (3) mutual feedback processes between spatial heterogeneities in different regions based on population movement, as emphasized in systems theory.The rationality of considering these theories for modeling infectious diseases is further proven based on controlled experiments.

Application Prospect the Model
This model has two potential applications.First, researchers can directly use this model as a tool to simulate the spatial transmission process of infectious diseases and obtain data on the spatiotemporal distribution of the number of infections.The model provides source code and software that allows users to set up their own spatial areas for the occurrence of infectious diseases and adjust the initial conditions for spatial transmission of infectious diseases to conduct comparative experiments and scenario-specific experiments.
Second, the model is universal and the researcher can easily expand on it.The model contains the main modeling processes such as the compartment modeling of population categories, the population flow interaction module, and the spatial spreading of infectious diseases module.Each step is expressed by means of programming, which is easy to calculate and extend.For example, to study the spatial differences in population immunity levels, an immunized population can be introduced in the compartment modeling module; to study the effects of travel restriction measures, new movement rules can be introduced in the population flow interaction module.

Limitations and Future Directions
This model has two defects.First, it does not consider the heterogeneity of individuals, and the probability of the infection process and movement across regions experienced by individuals in this article is the same.This causes the model to ignore the randomness and heterogeneity of individual behavior, leading to a lack of confidence in the results.Second, the variation in transmission parameters in the model currently only considers the effect of changes in the number and proportion of each type of population.In the whole life cycle of infectious disease transmission, many other factors could influence the change in transmission parameters, such as the epidemic scale of infectious disease, personal protection awareness, and epidemic prevention measures from health institutions.
These two shortcomings can be addressed by (1) introducing individual rules of discrete and stochastic behavior, and (2) introducing transmission parameters that can be adjusted adaptively according to the general behavior patterns of the population in response to infectious diseases.

Conclusion
Spatial interactions and spatial heterogeneity are prevalent in geographic phenomena and are important in the spatiotemporal transmission of infectious diseases.In this article, we propose a hypothesis that the mutual feedback between the heterogeneity of population movements and the heterogeneity of transmission conditions significantly affects the prediction accuracy of the applied model.Based on this hypothesis, we construct an infectious disease simulation model that considers this mechanism.We prove the correctness of the theoretical hypothesis presented in this article by observing the variation in the ranking of the number of infections by region due to mutual feedback of heterogeneity based on the simulation results obtained with a synthetic data set.The validity of the model is assessed by comparing the simulation results of the real-world data set with the confirmed cases data.
This method is applicable to the simulation modeling of the spatial transmission of infectious diseases significantly influenced by interregional population mobility and regional differences in transmission conditions.When this method is used in other application scenarios, the spatial distribution of the infections at different times can be effectively simulated and predicted by setting initial conditions such as the interregional population flow, the number of susceptible individuals and infections in each region, and the transmission parameters for a given infectious disease.The model itself has the capability to be extended, and if other infectious disease mechanisms or population flow patterns need to be considered, the corresponding module in the method can be directly extended to solve specific problems.

Figure 1 .
Figure 1.Conceptual framework of the spatiotemporal transmission model of infectious disease (STTM).

Figure 2 .
Figure 2. The compartment conceptual model is based on the transportation of goods in the real world to different groups in terms of the infectious disease status.(A) People in a city in the real world.(B) A warehouse with the population classified.(C) Population categories in city A.

Figure 3 .
Figure 3. Conceptual model of people's flow interaction among regions.(A) Population flow of different groups among cities.(B) Flow rules for different groups among cities.

Figure 4 .
Figure 4. Conceptual model of the spread of infectious diseases within a region.(A) Spread of infectious diseases at t 0 : (B) Spread of infectious diseases at t 0 þ T: (C) Spreading rules.
and F D ¼ S D , E D , I D , R D f g are the city population category instance sets and the components of the population set G 0 , which corresponds to the city instance set C 0 : The statistical table in Figure 5B consists of the statistics for S, E, I, and R within four cities and is a visual representation of G 0 : Population Flow among Regions.
shows a diagram visualizing interregional population movements, where blue arrows represent the flow of S, orange arrows represent the flow of E, and black arrows represent the flow of the overall population.The flows of S and E are integral parts of the flow of the overall population.

Figure 5 .
Figure 5. Data from the original population categories.(A) City cluster and the composition of different groups.(B) Statistical table of population categories.

Figure 6 .
Figure 6.Sets of population flow among regions.

Figure 7 .
Figure 7. Process of population flow among regions.
number of contacts per person per day in a subregion r Probability of the transition of E to I c Probability of the transition of I to R Note: S ¼ susceptible; E ¼ exposed; I ¼ infected; R ¼ removed.

Figure 8 .
Figure 8. Infection process model of disease within the city.

For
the infectious disease parameter data set (D S ), as shown in Figure two sets of infectious disease dynamics parameters are used in this experiment.

Figure 9 .
Figure 9. Synthetic data set for sixteen cities. (A) Sixteen cities. (B) Regional population category data.(C) Flow data.(D) Infectious disease parameter.

Figure 10 .
Figure 10.Infection curve of cities and the different heterogeneities of each city.

Figure 11 .
Figure 11.Infection curve and map of the spatiotemporal spread of infectious diseases in the experimental sample area.

Figure 12 .
Figure 12.Infection process model of disease within a city.(A) The distributions of E and I in the population category data set.(B) Flow rate of the population.

Figure 13 .
Figure 13.Map showing the simulated infection results in China.The line graph represents the number of cities in different stages of infection on different dates.

Figure 14 .
Figure 14.Infection curve and the number of new cases: Real observation and simulation experiments.

Table 1 .
Symbols and parameters used in the spatiotemporal-SEIR (ST-SEIR) model

Table 2 .
Categories and descriptions of real-world data set