A symbiosis between cellular automata and dynamic weighted multigraph with application on virus spread modeling

Abstract

The pattern of coronavirus spread at different geographical scales verifies that travel or shipment by air, sea or road are potential to transmit viruses from one location to somewhere far away in a very short time. Simulation and analysis of such a situation requires the development of models that support long distance transmission of viruses. Cellular Automata (CA) are a family of spatiotemporal computational models frequently employed in analysis of biomedical systems. A CA consists of a topological combination of units called cells as well as a transition function that propagates the configuration of cells locally and step by step. In this paper, we first present some patterns that show the local interaction between CA cells is not sufficient for virus spread modeling, especially at large spatial scales. Then, we generalize the concept of CA by providing a symbiosis between the neighborhood relationship of cells and the transmission channels represented by a dynamic weighted multigraph. Furthermore, we characterize the capabilities of the proposed modeling tool in simulation of the virus spread, and estimating the risk control during the movement restrictions and related health protocols. Finally, we simulate the coronavirus outbreak in the five study areas including three states and two countries. Our experiments using the proposed model verify that the proposed model is capable of formulating different ways of virus transmission, including long-distance transmission, and supports high-precision simulation of the pandemic.

1. Introduction

The large amounts of data and information generated due to the spread and longevity of viruses and infectious diseases provide an opportunity for scientists to extract the knowledge needed for making decisions in similar situations [1], [2], [3], [4], [5], [6], [7], [8] as well as control measure issues [9], [10], [11], [12]. Modeling devices, as a result of knowledge representation, offer several advantages by their representational, computational and analytical potentials. Models can be developed, tested and adapted for different applications, especially when data on events that occurred over a wide range of time and space are available.

Cellular Automata (CA) [13] are a family of spatiotemporal mathematical and computational tools employed in a variety of modeling and simulation contexts such as epidemic, growth, flow, propagation, evacuation and transport dynamics [14], [15], [16], [17], [18], [19], [20], [21], [22]. The successful applications of CA shows its high potential for simulating and analyzing how changes spread locally over time [23].

A CA consists of a collection of cells arranged in a multi-dimensional space and a dynamical rule (transition function) which updates their configuration synchronously. The transition function calculates the state of each cell based on the state of its neighbors where, these simple local interactions and computations between cells, form a complex global behavior [24], [25], [26]. The cells usually represent real spaces in various sizes ranging from atomic structure of matter to urban texture components or even larger [27], [28]. In physical space modeling / simulation, the migration of agents or the spread of phenomena among the cells causes change in their state through a number of discrete time steps [19], [21]. Therefore, the dynamics of CA depend on the set of states, the geometry of cells, the definition of neighborhood relationship, and the state calculation method of transition function.

Cells are the basic blocks of CA that are located in a topological arrangement. A cell has a number of configuration parameters that change over time. Each configuration is called a state, and at each time step, it is calculated according to state of the neighboring cells. In a CA, the set of neighbors of each cell is determined based on the topological relationships of the cells and the parameter of the neighboring radius. Figure 1 shows tree sample CA with different topologies. Here, the cell numbered by 0 is considered as the base cell and other cells are numbered with their neighborhood radius from it.

Fig. 1.

Three sample CA and neighbor cells around a base cell indicted by 0.

Various geometries and distributions are possible for cells, each of them affects the interaction between cells [23], [13]. Interaction between cells occurs passively through the transition function, in such a way that updates state of all cells in parallel. Figure 2 illustrates the general structure of a transition function in classic definition of CA. Here, $S_{i,t}$ corresponds to the state of cell $i$ in time step $t$, $D_{j,i}$ represents the distance of cell $i$ from cell $j$, and $r$ is the neighboring radius. Accordingly, the transition function calculates the next state of cell $i$ based on the current state of itself and its neighbors within a maximum radius of $r$.

Fig. 2.

Inputs and output of a transition function for cell $i$ in a classic CA.

In the classic sense, to define cell neighborhood, the focus of attention is on the absolute location of cells and a neighborhood radius that the higher the value, the more cells can interact with each other [23]. This model does not support relational conceptions of space as well as remote transmission of agents or data without intermediaries. On the contrary, by focusing on the interaction between cells through transmission pathways, the adjacency relationship can be represented using graphs or vector layers (in Geospatial Information Science), and even the neighborhood radius can be considered on these structures [29], [30], [31], [32], [33]. However, providing a communication network between cells in the form of a 0/1 graph allows the modeling of direct communication between two non-adjacent cells, it is not capable of modeling the intensity of communication between cells. For example, in applications such as virus spread modeling that population areas are considered as CA cells, the amount of traffic between them cannot be modeled.

Modeling the dynamic of heterogeneous spaces using CA leads to defining the weights of neighborhood to reflect the varying impacts of nearby cells [34], [35], [36]. The weighted adjacency presentation can increase the modeling power of CA, especially in cases neighboring cells do not necessarily interact and influence each other similarly. In Fig. 3 , the transition function of a heterogeneous CA is presented. Here, $W_{j,i}$ corresponds to the weight of communication between cells $j$ and $i$. The weighted relationship of cells provides modeling the heterogeneous transport of virus-transmitting agents, but it is not enough for manipulating the changes of health or political protocols.

Fig. 3.

Inputs and output of a transition function for cell $i$ in a heterogeneous CA.

Another feature that make CA more flexible is the ability to adjust the connectivity strength for cells during simulation [36], [37], [38]. This feature is very useful for modeling and analysis of control measures such as blocking the entry of imported cases by mandatory quarantine upon arrival, an entry ban for nonresidents and requiring a negative test result from arrivals, passenger monitoring and other protective protocols applied dynamically during the outbreak of COVID-19 [39], [40], [41], [42], [43], [44], [45]. Figure 4 , presents the transition function of a heterogeneous CA that is capable of dynamic adjusting the weights. Here, $W_{j,i,t}$ indicates the weight of communication between cells $j$ and $i$ in time step $t$. Although this model includes powerful features for modeling the dynamics of systems and propagating changes in them, it is not sufficiently abstract to support the interaction of CA cells through different communication layers. In other words, for cases where several types of neighborhoods need to be defined between two cells, the model should be more capable.

Fig. 4.

Inputs and output of a transition function for cell $i$ in a heterogeneous CA with dynamic connectivity.

The aim of this research is study of spatiotemporal pattern of new coronavirus spread at different geographical scales to discover pathways that have accelerated the pandemic [46] of COVID-19, and use the resulting knowledge to develop a generalized CA that support required features for virus spread modeling and simulation. In this direction, we focus on the sequence of first case reported in the subdivisions of study regions and extract the patterns corresponding to the virus transmission channels. Then, we develop the proposed CA by employing a multigraph with dynamic weights as the interaction model of CA cells. As well, we show the impact of controls and constraints related to health protocols are supported. In addition, we compare the results of the simulation of the corona virus outbreak in the five study areas with the actual data and show the high capability of the proposed model in predicting the spread.

2. COVID-19 spread pattern

We investigate th pattern of coronavirus spread at different levels of political divisions of the world. At each spatial scale, we extract the desired patterns according to the first case reported timeline of the political divisions of the study area. We also consider the researches that realize the potential virus transmission pathways as well as the impact of monitoring, controls and restrictions for mitigating virus spread, using field studies or laboratory activities [40], [45], [47], [48], [49].

Figure 5 presents a timeline of the first confirmed cases of COVID-19 on the worldwide scale during December 2019 and January 2020. This timeline is extracted from the information presented in Wold Health Organization (WHO) coronavirus disease dashboard [50]. On December 2019, the first case was observed in China, and then on January 13, 2020, the first case outside of China was reported in Thailand. The detection trend of COVID-19 in other countries is also shown on the chart and indicates that in January, the rate of virus spread in the world was very high. Now, considering a long-distance virus transmission pattern in a short time, On January 20, the first case was reported in North America (United States of America), on January 24, the first case was reported in Europe (France), on January 25, the first case was reported in Australia, and on January 29, the first case was reported in Middle East (United Arab Emirates). The countries included in this pattern are colored in red. Also, the short time of formation of this pattern on a global scale indicates the spread of the virus through the airlines.

Fig. 5.

A timeline of COVID-19 developments during December 2019 and January 2020. The countries marked in red, form a long-distance virus spread pattern. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

We study the pattern of COVID-19 spread in the USA because of its vastness, the large number of infected people, wide extent of virus spread, and the availability of the required information through online dashboards of health departments of different states. Figure 6 illustrates the state by state development of COVID-19 in USA where, the default color of states is white, the yellow mean the first case reported on the same day and the red mean the case reported in the previous days. According to maps Fig. 6a to d, which run from January 21, to February 1, the pattern of development through neighboring states has not occurred; especially on January 24, (Illinois marked in yellow) and February 1, (Massachusetts marked in yellow) when there is a great physical distance from the previously infected states. Infected cases were also reported in Florida and Texas on March 1, and 4, respectively, while no cases were reported in neighboring states. On the other hand, the first reported case in the state of Nevada is on March 5, while the two neighboring states, California and Arizona, have been infected with the corona virus since January 26, and also in the other neighboring state, Oregon, since February 28, the first Infection was reported. As another example, focusing on West Virginia, the last state that report the first case of COVID-19 (Fig. 6); the first neighboring states were infected on March 5, (Fig. 6), the other two neighboring states on March 6, (Fig. 6), and by March 10, (Fig. 6) all neighboring states were infected. In addition, the state of New Mexico reported the first case on March 11, while in neighboring Arizona, the first case was reported about six weeks ago on January 26. The information leads us to the idea that at a country scale, the proximity of provinces has less effect on the spread of the virus than communication lines such as railroads and airlines. Intuitively, this idea seems plausible because train stations and airports are crowded, and in each transfer, a significant number of people on the train or aircraft are in contact with each other.

Fig. 6.

The evolution map of COVID-19 first case report in USA during January, February and March 2020. The map is simplified and all states and areas are not included. Areas are colored in white (default), yellow (first case report) or red (some cases are reported in previous days). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

We generated a state level virus spread map based on data extracted from Indiana COVID-19 dashboard [51]. This dashboard is updated daily with information reported to Indiana State Department of Health (ISDH) in the last 24 h. Figure 7 shows the spread of coronavirus in the state of Indiana, until March 12. As shown in Fig. 7a, the first case of COVID-19 reported in Marion County on March 6. It is the most populous county in Indiana and its administrative center is Indianapolis, the state capital and largest city. According to the maps from March 8, to 12, two different distribution patterns can be seen in the state. The first pattern is related to the spread in the counties adjacent to Marion. Since the state capital is located in this county and has a crowded connections with other cities, this pattern of virus spread seems natural. The second pattern is related to non-adjacent counties, where at the time of reporting the first case in each county, no case of COVID-19 was reported in any of the neighboring counties. These patterns show that population flow communications are more important than neighborhood communications in modeling the spread of the virus at the state level and by county division.

Fig. 7.

Coronavirus spread in the state of Indiana, USA during March 2020.

The patterns of coronavirus spread at different macroscopic scales such as worldwide, countrywide and state level suggest that important factors in addition to the geographical neighborhood of the regions are effective in transmission of coronavirus. Since COVID-19 is a highly transmittable and pathogenic viral infection [52], travel or shipment by air, sea or road are potential to transmit the virus from one location to somewhere far away in a very short time. Therefore, in the next section, we will consider the transmission pathways as well as how to control their impact on the spread of viruses in the development of the proposed model.

3. Symbiosis between cellular automata and dynamic weighted multigraph

In this section, we introduce a Dynamic Weighted Multigraph-based Cellular Automaton (DWMCA) that is adapted for modeling the propagation of changes by different type of transmission channels with dynamic capacity.

Definition 3.1

A weighted graph is an ordered pair $G=(V,E)$, where:

• •$V$ is a set of vertices called nodes.
• •$E:V\times V\to [0,1]$ is called the set of weighted edges.

Definition 3.2

A dynamic weighted multigraph is defined as $G=(V,Y,I,D)$, where:

• •$V$ is the set of nodes.
• •$Y$ is the set of weighted layers, such that for each layer $E\in Y$ the ordered pair $G=(V,E)$ is a weighted graph. Also, by $\left|Y\right|$ we indicate the number of layers.
• •$I:Y\to [0,1]$ is called the impact factor of layers.
• •$D$ is the graph dynamic and in each time $t\in \mathbb{Z}$, updates $Y_t$ and $I_t$ to $Y_{t+1}$ and $I_{t+1}$, respectively.

Definition 3.3

Let $G=(L,Y,I,D)$ be a dynamic weighted multigraph. A dynamic weighted multigraph-based CA is a 7-tuple $DCA=(L,S,c,c_0,G,f,f_a)$, where:

• -$L$ is a lattice of cells.
• -$S$ is a set of states.
• -$c:L\times \mathbb{Z}\to S$ indicates the configuration of CA (the states of all cells) in time $t\in \mathbb{Z}$. In other words, $c_t$ returns the state of all cells int time $t$.
• -$c_0$ is the initial configuration of CA.
• -$G$ represents the dynamic weighted multi-layer neighborhood on $L$.
• -$f:L\times 2^E\times 2^c\to S$ for some $E\in Y$, is the transition function.
• -$f_a:2^S\to S$ is the aggregation function.

As well, by transition $T:c\times f\times f_a\to c$ we mean the simultaneously application of the transition function $f$ and aggregation function $f_a$ to all cells and calculating $c_{t+1}$ from $c_t$ for some $t\in \mathbb{Z}$. We note that this paper does not deal with the CA with memory [53], where calculating new configuration relies on not only current configuration but also past configurations.

Figure 8 illustrates the general structure of a transition on cell $i$ in a $k$-layer DWMCA, where $k=\left|Y\right|$. Here, $D_{j,i,t}^{\left(r\right)}$ and $W_{j,i,t}^{\left(r\right)}$ are the distance and the weight of the connection between cells $j$ and $i$ through layer $r$ at time $t$, respectively. The transition function generates a tuple of $k$ states ${\widehat{S}}_{i,t+1}=〈S_{i,t+1}^{\left(1\right)},S_{i,t+1}^{\left(2\right)},\dots ,S_{i,t+1}^{\left(k\right)}〉$, where $S_{i,t+1}^{\left(r\right)}$ is the next state of cell $i$ calculated through layer $r\in \{1,\dots ,k\}$. The aggregation function calculates the next state $S_{i,t+1}$ from the tuple ${\widehat{S}}_{i,t+1}$.

Fig. 8.

Inputs and output of a transition for cell $C_i$ in DWMCA.

In fact, DWMCA is the resulted of symbiosis between CA and DWM, so that the DWM represents cell neighborood relation, and the transitin function is adapted accordingly. Therefore, in DWMCA, the cell neighborhood graph is defined by DWM which results to the following properties rather than CA:

• -Each DWMCA represents a set of neighborhood relations (called layers), each of which is modeled by a dynamic weighted graph.
• -The neighborhood relations are dynamic and may change over time.
• -Each layer has an impact factor that may change over time.
• -Compared to CA, the definition of transition function is extended to a pair of transition function and aggregation function to support manipulating multiple neighborhood relations.

Now, we define a DWMCA model for simulating covid-19 outbreak. Of course, our goal is not determining the degree of infection for areas, and we consider the first case reported. In this model, the geographic regions such as countries or provinces are considered as set of cells $L$. Also, we assume that dynamic weighted multigraph $G=(L,Y,I,D)$ is defined on different pathways (such as airlines, roads, railways and waterways) between these areas. Therefore, to define the desired automaton, we specify the set of states, the transition function and the aggregation function.

• •The set of states:$S=[0,1]$.
• •
  The transition function: To define the transition function, we consider the neighborhood radius to be 1. Therefore, the next state of each cell is calculated based on the current state of itself and the cells directly related to it (in different layers). Also, we consider a nonnegative growing factor $g\in \mathcal{R}$ for the dynamics of each cell. Hence, the transition function for each cell $l\in L$ at time $t\in \mathcal{Z}$ is defined as follows:

  $$\forall E\in Y_t:f_E\left(l\right)=1\wedge \left(\right(c_t\left(l\right)\times g)\vee \underset{l^{\prime }\in L}{\bigvee }{c}_t\left(l^{\prime }\right)\times E(l^{\prime },l))$$ (1)
• •
  The aggregation function: At first, the set of most effective layers for the cell at time $t$ is defined by:

  $$R_t\left(l\right)=\underset{E\in Y_t}{\bigvee }{I}_t\left(E\right)\times f_E\left(l\right)$$ (2)

  $$Y_t^{\prime }\left(l\right)=\{E\in Y_t|I_t\left(E\right)\times f_E\left(l\right)=R_t\left(l\right)\}$$ (3)

  Then, the amount of propagation data received through the most effective layers is calculated using the aggregation function as follows:

  $$f_a\left(l\right)=\underset{E\in Y_t^{\prime }}{\bigvee }{f}_E\left(l\right)$$ (4)

This model has three free parameters $c_0$ (the initial configuration), $g$ (the growing factor) and $G$ (the DWM corresponding to neighboring relationship) that can be adjusted according to the application. Also, an important issue in simulation with this automaton is the interpretation of states. For example, a threshold can be used to divide states into two classes infected and non-infected. It should be noted that in the proposed model, we have considered the state of each cell to mean the degree of risk in that cell, and different procedures can be used to analyze $c_t$ (the set of states in time step $t$). The parameter $g$ can also be considered smaller for situations where control measures within cells are greater. To model neighborhood relationships by DWM, the weight of layers can be adjusted to model the degree of risk of that layer. For example, in some areas (such as the USA) most travel is by air, and in other areas (such as Iran) the volume of road and rail travel is significant. Also, the degree of boundary control measures for blocking the entry of imported cases can be manipulated by updating the weight of the DWM edges. In the following, we investigate some of the behavioral features of this automaton.

Proposition 3.4

Suppose$g>0$and in the time step$t$, cell$l^{\prime }$is in state$c_t\left(l^{\prime }\right)>0$. Then, every neighbor$l$of cell$l^{\prime }$will be in state$c_{t+1}\left(l\right)>0$in time step$t+1$.

Proof

According to Eq. (1), existence of expression ${\bigvee }_{l^{\prime }\in L}{c}_t\left(l^{\prime }\right)\times E(l^{\prime },l)$ guaranties that if $E(l^{\prime },l)>0$ and $c_t\left(l^{\prime }\right)>0$, then $f_E\left(l\right)>0$. Thus, Eq. (2) gives the result that $R_t\left(l\right)>0$. Therefore, according to Eq. (3), the set of effective layers $Y_t^{\prime }\left(l\right)$ is not empty and contains layers through which the transition function calculates non-zero states. Hence, Eq. (4) results in $f_a\left(l\right)>0$. □

Proposition 3.5

Suppose$g>1$and in the time step$t$, cell$l$is in state$c_t\left(l\right)>0$. Then, there exists a time step$t+\Delta$such that cell$l$will eventually be in state$c_{t^{\prime }}\left(l\right)=1$for all$t^{\prime }\ge t+\Delta$.

Proof

According to Eq. (1), existence of expression $c_t\left(l\right)\times g$ guaranties that $f_E\left(l\right)\ge c_t\left(l\right)\times g^{\Delta }$ in time step $t+\Delta$. Hence, $f_E\left(l\right)$ is ascending over time for $g>1$. So, it can be simply proved that with increasing time, the limit of $f_E\left(l\right)$ is 1 and as a result, the value of the aggregation function will also be 1. □

Corollary 3.6

Suppose$g>1$and in the time step$t$, cell$l$is in state$c_t\left(l\right)>0$. Then, there exists a time step$t+\Delta$such that every neighbor$l^{\prime }$of cell$l$will in state$c_{t^{\prime }}\left(l^{\prime }\right)=1$for all$t^{\prime }\ge t+\Delta$.

Proof

This is a direct result of Propositions 3.4 and 3.5. □

Therefore, in this model, two important factors in the contamination of a cell (reaching the risk level of the contamination threshold) are border control measures (weight of edges as well as impact of communication layers) and also internal control measures (parameter $g$).

4. Experimental results

We consider three states Colorado, Indiana and Utah in the United States, as well as two countries United States and Iran as the study areas and generate evolution maps of spreading coronavirus and compare them with the maps derived from Colorado [54], Indiana [51], Pourghasemi et al. [55], USA [56]. Table 1 presents the specification of our study areas where, the first column is assigned to the area ID, the second column specifies the name of the area, the third column specifies the area type, the fourth column indicates the CA cell granulation level for area modeling, and the fifth column indicates the number of CA cells corresponding to that area.

Table 1.

Specifications of the study areas and the corresponding models.

$\text{ID}$	$\text{Area}\text{Name}$	$\text{Area}\text{Type}$	$\text{Cell}\text{Type}$	$\text{Cells}$
$CO$	$Colorado$	$State$	$County$	64
$IN$	$Indiana$	$State$	$County$	92
$UT$	$Utah$	$State$	$County$	29
$USA$	$USA$	$Country$	$State$	49
$IR$	$Iran$	$Country$	$Province$	31

We implemented DWMCA by C++ programming language and defined a threshold parameter ${\tau }_{inf}=0.6$ to generate virus development maps. Then, for each cell $l$ with $c_t\left(l\right)\ge {\tau }_{inf}$, we concluded that it is infected. This value has been obtained experimentally and is fixed in all our experiments. Also, to define the weighted multigraph in the simulation, the transportation and related layers such as airlines, railroads, Eisenhower Interstate System (EIS) and other National Highway System (NHS) of roads are extracted from On [57], Open [58]. We presented each layer as well as adjacency graph of cells (counties, states or provinces) in the form of a separated adjacency matrix (layer) of the weighted multigraph. During the simulation, we did not change the weight of the edges, but slightly increased the impact factor of the layers except for the airline layer. Table 2 presents the initial impact factor of the layers airlines, railroad, EIS, NHS and adjacency graph of cells for the study areas.

Table 2.

The Default Impact Factor of Layers for the study areas.

$\text{ID}$	$\text{Airline}$	$\text{Railroad}$	$\text{EIS}$	$\text{NS}$	$\text{Adj}$
$CO$	0.9	0.5	0.4	0.3	0.1
$IN$	0.9	0.5	0.4	0.3	0.1
$UT$	0.9	0.5	0.4	0.3	0.1
$USA$	0.9	0.6	0.4	0.3	0.1
$IR$	0.5	0.7	-	0.7	0.2

In addition, we used the population of cells [51], [56] to calculate the weight of the edges between them. Here, Eq. (5) calculates the weight of each existing edge between cells $i$ and $j$ in layer $E$.

$$E(i,j)=\frac{populatio{n}_i}{populatio{n}_i+populatio{n}_j}$$ (5)

Also, the growing factor is set to $g=1.1$, and for each cell $i$, we used two variables $d$ and $a$ to keep the values needed for calculating the state. The value of $d$ changes by transmission of data from other cells, but $a$ is a boolean variable with value $true$ only for the counties that have airports and flights outside the state. The function $c$ is defined by Eq. (6).

$$c_t\left(l\right)=\{\begin{array}{cc}{d}_l+0.01\hfill & \text{if}a=true,\hfill \\ d_l\hfill & \text{otherwise}.\hfill \end{array}$$ (6)

To analyze the simulation results of each of the study areas, we have prepared virus development maps in which the maps for each day include a pair of maps in which white means uninfected areas, orange means infected spots based on real data and blue means contaminated points based on simulation information. We have also prepared graphs and statistical tables corresponding to these maps. Table 3 explains the symbols used in this section. For example, $True+$ indicates the number of cells that are reported infected in both the simulation and the actual data. As well, $False+$ indicates the number of cells reported infected in the simulation but not in the actual data. The same definitions can be extended to $True-$ and $False-$.

Table 3.

The explanation of the symbols used in the tables and graphs in this section.

$\text{Symbol}$	$\text{Title}$	$\text{Simulation}\text{Report}$	$\text{Real}\text{Data}\text{Report}$
$True+$	$truepositive$	$infected$	$infected$
$False+$	$falsepositive$	$infected$	$notinfected$
$True-$	$truenegative$	$notinfected$	$notinfected$
$False-$	$falsenegative$	$notinfected$	$infected$

To measure the accuracy of the simulation results, we use the $\%True$, which is calculated according to Eq. (7).

$$\%True=\frac{(True+)+(True-)}{numberofcells}\times 100$$ (7)

Each chart also has two axes; The vertical axis corresponds to the number of cells and the horizontal axis corresponds to the sequence of coronavirus spread evolution maps, which are sorted by date in ascending order.

4.1. Colorado

Figure 9 provides a comparison between six simulated maps and the corresponding maps obtained from real data reported in [54], [56]. The default state of the cells is not infected, which is marked in white. The orange cells are areas where COVID-19 infection is reported, and the blue cells are areas infected based on the simulation results. It is notable that the initial map and the probability of infection in the simulation was prepared based on the information on the number of cell flights out of the state. In Fig. 9a, one cell (Denver) is declared correctly infected, but another cell (Adams) is mistakenly declared infected. Thus, the simulation on 64 cells detected one true positive and one false positive infection case. Here, the number of true negative cases is 61 and there is one false negative case (Chaffee). These statistics are shown in Fig. 10 in column $a$, where the accuracy of this map is 97%. In addition, the county of Adams reported the first case of infection on March 9, [56] which is only three days away from the simulation result.

Fig. 9.

The simulation results of daily coronavirus spread in the state of Colorado during March and April 2020. White: not infected, Orange: infection reported in [6,50], Blue: infection reported by simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10.

The statistical results of coronavirus spread simulation for the state of Colorado during March and April 2020.

After one week, according to Fig. 9, 19 infected cells are reported by the simulation where the result includes 13 true positive and 6 false positive cases. Also, the number of true negative and false negative cases are 44 and 1, respectively. Also, according to column $f$ in Fig. 10 the accuracy of this map is 89%.

Figure 9 presents the virus spread and its corresponding simulation map at the beginning of the third week. Here, 29 infected cells are reported by the simulation where it includes 22 true positive and 7 false positive cases, and the number of true negative and false negative cases are 32 and 2, respectively. Also, based on column $j$ in Fig. 10 the accuracy of this map is 86%. Furthermore, based on columns $p$ and $v$ in Fig. 10, the accuracy of the simulation results for the next two weeks is 80% and 83%, respectively.

Our experimental results using the proposed model to simulate the spread of coronavirus in the state of Colorado confirm that the accuracy obtained in this state is at least 80%. The simulation maps have also reported all state counties infected since April 2, while about 19% of states have not yet been infected. One of the reasons for this error is the lack of transmission restrictions data in the simulation.

4.2. Indiana

Figure 11 provides a comparison between 30 simulated maps of Indiana and the corresponding maps obtained from real data reported in [51]. Based on the maps, the pattern of coronavirus spread in the state of Indiana is close in both our simulations and the reported data.

Fig. 11.

The simulation results of daily coronavirus spread in the state of Indiana during March and April 2020. White: not infected, Orange: infection reported in [50], Blue: infection reported by simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In Fig. 11a, one cell (Marion) is declared correctly infected, but another cell (Adams) is mistakenly declared infected. Thus, the simulation on 92 cells detected one true positive and one false positive infection case. Here, the number of true negative cases is 90 and false negative is 0. As well, the county of Adams reported the first case of infection on March 9, [51] which is only three days away from the simulation result (Fig. 11).

Figure 12 includes the statistical data derived from virus spread evolution maps of Fig. 11. The data verifies that the minimum accuracy of the simulation results is 79%. Of course, the minimum accuracy occurred on April 25, and two days later, on April 27, the accuracy reached 88% and increased in the following days.

Fig. 12.

The statistical results of coronavirus spread simulation for the state of Indiana during March and April 2020.

4.3. Utah

The first reported case in Utah was on March 7, [56]. As shown in Fig. 13 a, the first infected county is Utah in simulation but Salt Lake in [56]. However, they are neighbors and in later maps the spread patterns are close in the simulation and the actual data. Also, the simulation results in the next map, which is related to March 9, show the Salt Lake as an infected county.

Fig. 13.

The simulation results of daily coronavirus spread in the state of Utah during March and April 2020. White: not infected, Orange: infection reported in [50], Blue: infection reported by simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

According to Fig. 13c, d, g and i, the simulation based on the proposed model successfully identified two long-distance transmission patterns. The first pattern belongs to Washington where, simulation reported the first case on March 12, and [56] reported this county is infected on March 15. The second pattern is about Grand County where detected by simulation on March 18, and based on [56] this pattern occurred on March 27.

Figure 14 presents comparison statistics between the simulation results and [56] data based on evolution maps illustrated in Fig. 13. Here, the minimum accuracy of the simulation results (according to row $\%True$) is at least 79%, which verifies that the spread patterns in the simulation are close to the real patterns.

Fig. 14.

The statistical results of coronavirus spread simulation for the state of Utah during March and April 2020.

4.4. United states

To simulate the spread of coronavirus across USA, we defined cells at the state level (with a slight tolerance) so that the model has 49 cells. Also, there are two considerations about data and simulation:

• •Maritime traffic is not included in the simulation, but when defining the neighborhood graph for cells, all the western states that have a blue border are considered neighbors, and all the eastern states that have a blue border are considered neighbors.
• •We extracted the traffic information of the international flights of the states from Bureau [59] and then increased the probability of infection for the states that have the most international flights. The infection status of the origin of flights is ignored here.

According to Fig. 15 , the first case of coronavirus was reported on January 21, 2020 in Washington state, but simulation maps show three other states. In fact, the initial maps, which are based on the risk probability calculated from international flight traffic of states, have different initial values. However, from January 26, (Fig. 15) the COVID-19 outbreak maps (simulation / actual data) become more similar which indicates that the error of the calculated probabilities is negligible. Also, the development maps such as Fig. 15c, d, e, g and j show that DWMCA detects the spread of the virus in areas whose neighbors are not yet infected because it is capable of modeling and simulating the long-distance transmission of viruses.

Fig. 15.

The simulated maps of coronavirus spread in USA during January and March 2020. White: not infected, Orange: infection based on [50], Blue: infection reported by simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The statistical data in Fig. 16 , shows that the accuracy of the simulation maps is more than 70%. Of course, the lowest accuracy is belong to the map illustrated in Fig. 15, which is 71%. Also, the number of its false positive cells is greater than the number of its true positive cells, but in the next few days, a significant number of false positive cells have become true positive. Therefore, considering that the incubation period of COVID-19 is more than one week, we can say that the actual accuracy of the map in Fig. 15 is more than 71%.

Fig. 16.

The statistical results of coronavirus spread simulation for USA during January and March 2020.

4.5. Iran

To generate simulation maps of COVID-19 outbreak in Iran, its provinces are considered as cells. Unlike countries such as USA, where a large volume of travels are done by air, in Iran most travels are done through railroads as well as intercity highways. Therefore, the weight of the airline layer was less and the weight of railroads and highways was considered more, compared to the four previously studied areas.

In our experiments, we consider the highest probability of infection for the Iranian capital, Tehran because the most active international airport in the country is located there. However, the first case in Iran was reported on February 19, 2020 in Qom province, which is a neighbor of Tehran and the largest international airport in the country is located between them. Of course, two days later, on February 21, the first case was reported in Tehran. In Fig. 17 , the virus outbreak maps for both the simulation case and [55] have approximately the similar patterns, relating to railroads and intercity highways. In fact, the country’s busiest routes are from Tehran to the recreational provinces in the north, the religious province in the northeast of the country and the Tehran highway to the south (Bushehr province, where the country’s largest port is located).

Fig. 17.

The simulated maps of daily coronavirus spread in Iran from February 19, 2020 to March 03, 2020. White: not infected, Orange: infection reported in [38], Blue: infection reported by simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

According to Pourghasemi et al. [55], the outbreak of COVID-19 occurred in all the provinces over a period of two weeks, which makes it difficult to measure the accuracy of the simulation. Fig. 18 contains diagrams and tables of statistical data related to simulation maps. The data show that the simulation accuracy on February 24, and 25 is 77% and 71%, respectively, but in other maps the accuracy is more than 80%.

Fig. 18.

The statistical results of coronavirus spread simulation for Iran during February and March 2020.

5. Conclusion

We studied the spread of COVID-19 at different spatial scales and extracted some patterns from the first cases reported in different places. Based on the extracted knowledge, we considered patterns of local as well as remote transmission of infectious disease / viruses to develop the proposed model. In this direction, we provided a symbiosis between the concept of neighborhood in cellular automata and dynamic weighted multigraph to develop a model called DWMCA. The model is capable of simulating the outbreak of infectious diseases under various transmission restrictions. Also it provides multi-layer modeling of communication channels and configuring them dynamically. Finally, we implemented a simple version of DWMCA by C++ to simulate the spread of COVID-19 in five study areas including three states Colorado, Indiana and Utah in USA as well as two countries USA and Iran. Although we did not take into account some data such as travel bans and the effects of health protocols in this simulation, the virus development maps obtained are highly consistent with the maps derived from actual data and reached accuracy above 77% at the state-level and above 70% at the country-level. Furthermore, comparing the simulation results with real data verifies that DWMCA is capable of successfully simulating virus transmission over long distances.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.