Epidemic monitoring in real-time based on dynamic grid search and Monte Carlo numerical simulation algorithm

Monte Carlo algorithm

The Monte Carlo algorithm is a statistical simulation method that relies on the principles of probability and statistics. It is a numerical calculation approach that associates a given problem with a specific probability model. Through extensive sampling or statistical simulation using a large set of random or pseudo-random numbers, the algorithm obtains an approximate solution for the problem at hand (Shao, 2012; Zhu, 2014).

The spread of infectious diseases is a stochastic process, where the transmission occurs as a signal from an information source to multiple receiving sources. These receiving sources act as new information sources, potentially transmitting the disease further. However, the decision of whether and how these receiving sources propagate the information again involves probability events. In handling such probability events, Monte Carlo simulation proves to be highly advantageous (Chen et al., 2022).

(1) A probabilistic process can be described as a series of events or actions influenced by random factors or uncertain outcomes. In the context of the Monte Carlo algorithm, it involves analyzing and simulating this probabilistic process in a randomized manner. For problems that inherently possess randomness, the algorithm accurately captures and models the probabilistic nature of the process. However, for deterministic problems that lack inherent randomness, a constructed artificial probability process is necessary. In this case, certain parameters of the artificial probability process are designed to represent the solutions to the desired problem. Essentially, a problem that lacks inherent randomness is transformed into a problem with probabilistic properties through the construction of an artificial probability process.

(2) To implement sampling from a known probability distribution, one of the fundamental techniques used in the Monte Carlo method, random variables with specific probability distributions are generated based on the constructed probability model. This is the reason why the Monte Carlo method is often referred to as random sampling. Among the various probability distributions, the uniform distribution on the interval (0, 1) holds significant importance. Random numbers obtained from this distribution serve as subsamples of the uniform distribution and can be generated using mathematical recurrence formulas on a computer. Different methods exist for random sampling from known distributions, and unlike uniform sampling from (0, 1), these methods rely on generating random sequences, which essentially means generating random numbers. Thus, the generation of random numbers serves as the fundamental tool for Monte Carlo simulations.

Dynamic grid algorithm

In this article, an epidemic real-time monitoring plan is proposed, considering the characteristics of virus infection. The plan involves dividing the population based on different living areas, considering the spread of the epidemic within each designated area (Su et al., 2020).

In the event of an undetected carrier of the virus emerging within the population, a uniform and random sampling process is implemented across all partitioned areas to identify infected patients. Currently, domestic nucleic acid testing technology is well-developed, with an average turnaround time of six hours for test results. If positive patients are detected in the test results obtained on the same day, the specific block where these positive patients are located, as well as its surrounding area, will be subject to a lockdown measure (Zhang et al., 2021; Zhong et al., 2020).

To mitigate the risk of infection spread, as the surrounding blocks have close proximity to the area where positive patients are identified, it is crucial to implement isolation measures simultaneously in both the block with positive patients and the neighboring blocks. Additionally, on the second day, nucleic acid testing is conducted on individuals residing in the surrounding blocks to monitor and detect potential infections among the population in those areas. This proactive approach aims to prevent the further transmission of the epidemic beyond the initial block and promptly identify and respond to any potential outbreaks in the surrounding vicinity.

In the scenario where not all infected individuals can be detected through nucleic acid testing on the first day, it is important to consider the possibility of undetected infected individuals continuing to transmit the virus to people in their vicinity. To simplify the problem, it is assumed that an undetected infected person can only infect individuals in the eight adjacent partitioned areas. In subsequent days, the testing strategy follows a fixed-point approach, where the same uniform random sampling test points used on the first day are utilized. This means that the testing locations remain consistent throughout the monitoring period. The aforementioned steps are repeated for long-term testing until there are no confirmed cases for 14 consecutive days. This indicates that the epidemic in the region is under control. Different symbols or representations are used to denote detection points, infection points, and quarantine areas, as illustrated in Fig. 1.

For the convenience of analysis, the regional population is divided into an n ×n grid consisting of square areas with n rows and n columns. Each block area within the grid is represented by a two-dimensional vector (x, y), indicating its position in the x row and y column. To optimize the utilization of medical resources, a random sampling test is conducted on a subset of these n² blocks. The size of the sampling area is determined as [20%, n²], where [x] denotes the largest integer less than or equal to x. Subsequently, the specific block area to be tested is identified, and its coordinates are recorded, commonly referred as (x₁, y₁), (x₂, y₂), ... , (x_[20%n²], y_[20%n²]). Equation (1) is defined to record the position of the detection point on the first day. (1)${\displaystyle S_1=\left\{\left(x_1,y_1\right),\left(x_2,y_2\right),\dots ,\left(x_{\left[20\text{\%}{n}^2\right]},y_{\left[20\text{\%}{n}^2\right]}\right)\right\}.}$ Equation (2) displays the location coordinates of the surrounding block areas if the infected person is detected on day T, and the coordinates of this detection point are known. (2)${\displaystyle \left\{\begin{array}{c}\left(x_i-1,y_i\right)\hfill \\ \left(x_i+1,y_i\right)\hfill \\ \left(x_i,y_i-1\right)\hfill \\ \left(x_i,y_i+1\right)\hfill \\ \left(x_i-1,y_i+1\right)\hfill \\ \left(x_i+1,y_i-1\right)\hfill \\ \left(x_i-1,y_i-1\right)\hfill \\ \left(x_i+1,y_i+1\right)\hfill \end{array}\right.}$

Equation (3) represents the location set of the isolated area on day T. (3)${\displaystyle \begin{array}{c}{F}_t=\left\{\left(x_i-1,y_j\right),\left(x_i+1,y_j\right),\right.\hfill \\ \left(x_i+1,y_j-1\right),\left(x_i,y_j-1\right),\hfill \\ \left(x_i+1,y_j-1\right),\left(x_i,y_j\right),\hfill \\ \left(x_i-1,y_j+1\right),\left(x_i,y_j+1\right),\hfill \\ \left.\left(x_i+1,y_j+1\right)\right\}\hfill \end{array}}$

Equation (4) demonstrates the set relationship between the location set A_t, which represents the block area where the infectious virus is detected on day T. (4)${\displaystyle \begin{array}{c}{A}_t\subset S_t\cup F_t\hfill \\ S_{t+1}=\left(S_t\cup F_t\right)-A_t\hfill \end{array}}$

Based on the analysis of the set relationship, it is assumed that the epidemic is under control on day T. This means that on the day (t-14), all infected block areas are detected, and the adjacent block areas are isolated. Consequently, the sets An, Fn, and Sn are all stable. The mathematical relationship is expressed as A_n = A_t−14, F_n = F_t−14, S_n = S_t−14 (n ≥ t). Hence, the determination of the steady state can serve as a means to assess and control the epidemic situation in the region.

The iteration mode of variables in each dynamic grid search is shown in Equation (5). (5)${\displaystyle \left\{\begin{array}{c}{A}_n=A_{n-1}+{\text{new}}_{A_n}\hfill \\ F_n=F_{n-1}+{\text{new}}_{F_n}\hfill \\ S_n=r\text{and}+F_n\hfill \\ d_n=\sum _{j=1}^{\text{length}\left(F_n\right)\text{length}\left(F_n\right)}\sum _{k=1}\left\{\begin{array}{c}{\left[F_n\left(j,1\right)-F_n\left(k,1\right)\right]}^2+\hfill \\ {\left[F_n\left(j,2\right)-F_n\left(k,2\right)\right]}^2\hfill \end{array}\right\}\hfill \\ \text{source}=F_n\left[\text{min}\left(d_n\right)\right]\hfill \end{array}\right.}$

Here, new_An represents the newly added infection points, new_Fn represents the newly added isolation points on the nth day, and length(F_n) represents the F_n number of elements.

Once the spread of the epidemic is contained in this phase d_n, Equation (5) calculates the sum of Euclidean distances between the nth infection point and all infection points. The smallest d_n corresponding corner marker N is then selected as an approximation for the coordinates of the initial infected person F_n[(x_n, y_n)].

Model assumptions

To simplify and streamline the problem-solving process, the model is built upon the following assumptions:

1. There are 100 majors in a college, and the number of people in each major is 500.

2. There are 2,000 communities in the city, and the number of people in each community is 3,000.

3. The nucleic acid test results are highly accurate, with no occurrences of false negatives or false positives.

4. The nucleic acid test results are conducted and released on the same day.

5. There is a 100% infection rate from infected points to surrounding points.

6. The quarantined population is immune to both infection and transmission.

7. All individuals will willingly adhere to the measures of the monitoring program and actively cooperate.

8. There are no sudden increases in the transmission of the epidemic, aligning with the principle of contiguous contagion.

9. There are eight neighboring individuals to an infected point.

10. Each cycle of epidemic transmission can be contained within a specific timeframe.

11. The medical reserve resources are sufficient.

Establishment and solution of epidemic monitoring model in colleges and universities

For colleges and universities with a population of 50,000, a division of the student body into major-based groups is implemented due to increased interactions among individuals of the same major. Assuming there are 100 majors in the school, with each major consisting of 500 individuals, the major areas are simplified as 10 ×10 square blocks. Each major is considered an infection point with specific horizontal and vertical coordinates. Within this model, an infected major can only transmit the infection to the surrounding eight adjacent majors. Additionally, a fixed-point nucleic acid test is conducted uniformly and randomly on 20% of the 10 ×10 block area (referred to as mixed sample tests), while each individual within a specific major undergoes a single test (known as single sample tests) to identify any infected individuals. By employing the established model, we can approximate the location of the first infected person within a specific major and determine the duration in days for the current round of the epidemic to spread among different major areas.

Figure 2 illustrates the flowchart for college monitoring. Once the approximate location of the major of the first infected person is determined, the individuals within that major are further divided into 24 ×24 areas for analysis, considering each person as an independent infection point. By employing a similar approach as professional traceability, we approximate the coordinates of the first infected person within the major and predict the duration of epidemic transmission among individual members in this particular round.

Once the ongoing round of the epidemic in colleges and universities is effectively managed, we can gather mixed sample data on the current round of transmission detection among majors, as well as single sample test data on transmission detection between individuals. This data enables us to obtain the approximate coordinates of the first infected person’s major and the approximate coordinates of the first infected individual. Moreover, it is important to note that the virus spreads synchronously between different majors and within the major. Therefore, by taking the maximum value of the transmission time between majors and individuals in this round of the epidemic, we can approximate the overall duration of this particular round.

In order to validate the model’s accuracy, this article assumes the availability of a known infection point. The model proposed in this study is then employed to address this scenario. The objective is to trace the approximate geographical coordinates of both the major and individual origins of the initial infected person. An error analysis is conducted by comparing these coordinates with the provided hypothetical infection point. Additionally, by considering the maximum time taken to trace the approximate coordinates of the major and the individual back to the source of the first infection, we can approximate the duration of this current epidemic cycle. This process can be described as a dynamic grid search.

During each iteration of the dynamic grid search, a random location coordinate is assigned to the initial infected person, denoted as the “Center.” The initial nucleic acid test point, referred to as the “Rand,” is uniformly and randomly determined within the global area. By utilizing the model above, we can predict the approximate location coordinate of the first infected person, denoted as the “Source,” after the conclusion of this epidemic cycle.

This article utilizes the concept of the Monte Carlo algorithm to address the limitations and unpredictability associated with a single solution. By iteratively running the dynamic grid solution process multiple times, the average value of multiple predictions is calculated and considered as the final prediction outcome. This approach aims to overcome contingencies and enhance result stability.

Initially, the infection point is positioned at Center (5, 5) of the major area. Following five simulations, the average duration for epidemic control across different major areas is determined to be 16.5 days, while the estimated number of mixed mining reaches 59.14. The last traceability coordinate calculated is Source (5, 5.14). The relative error for the horizontal and vertical coordinates of the initial simulation point, Center (5, 5), is reported to be (0%, 2.8571%).

For the position of the initial infected person in the area block Center (5, 5), we take Center (12, 12). After running 100 simulations, the average duration for epidemic control within the profession is found to be 18.01 days, and the estimated number of single sample tests is 446.64. The final traceability result indicates the source as (12, 12.09). The relative error for the horizontal and vertical coordinates is reported to be (0%, 0.75%). These results demonstrate that the relative error of the major’s horizontal and vertical coordinates obtained through source tracing does not exceed 3%, while the relative error for the initial infected person’s coordinates derived from the source tracing remains under 1%. Consequently, the model’s accuracy is validated.

In summary, the final solution result reveals that the duration of the epidemic in colleges and universities for this round is 18.01 days. On average, there have been 59.14 instances of mixed mining and 30016.64 instances of single mining. Figure 3 presents a visual representation of two simulation results during the dynamic grid search process.

Establishment and solution of the urban epidemic monitoring model

In a city with a population of 6 million, we make the assumption that there are 2,000 neighborhoods, with each neighborhood consisting of approximately 3,000 people. To facilitate analysis and simulation, we further divide the community into 46 ×46 areas. Each of these areas is then subdivided into smaller 56 ×56 regions for consideration. According to the epidemic transmission rule, infected individuals within a given area can only transmit the infection to people residing in the surrounding areas.

In the first step, we randomly select 20% of the 46 ×46 block areas for fixed-point nucleic acid testing, which is referred to as mixed sample tests. Then we conduct single sample tests on all individuals within a selected community area to identify any infected individuals within the community. By employing the established model, we can approximate the location of the community where the first infected person is situated and determine the duration of the epidemic for the respective community areas.

Once we have obtained the approximate location, the individuals within that community are further divided into 56 ×56 areas for analysis. In this case, each person is treated as an independent infection point. Similar to community tracing, we employ the same methodology to estimate the approximate coordinates of the first infected individual and predict the duration of epidemic transmission among the individual members. The monitoring process is illustrated in Fig. 2, which depicts the flowchart of the monitoring procedure. Once the current round of urban epidemic is brought under control, we can gather mixed sample test data for detecting transmission between communities and single sample test data for detecting transmission between individuals. Additionally, we can obtain the approximate coordinates of the community where the first infected person is located and the approximate coordinates of the individual who is the first to be infected. It is important to note that the spread of the virus occurs synchronously between community areas and within communities. Therefore, by considering the maximum value of the transmission time between communities and individuals during this round of the epidemic, we can approximate the overall duration of this round of the epidemic.

The specific solution process is the same as the university model. The flow chart can be seen in Fig. 4.

To overcome the limitations of a single solution, this article incorporates the Monte Carlo algorithm. The dynamic grid solution process is iterated multiple times, utilizing the concept of a loop. By doing so, multiple predictions are generated. To enhance result stability and reduce contingency, the final prediction result is obtained by calculating the average value of these multiple predictions.

The initial infection point is located at Center (23, 23) of the community block area. Through 1,000 simulations, the average duration of epidemic control between community areas is determined to be approximately 18.488 days. The estimated number of mixed sample tests is calculated to be 1727.324. The last traceability coordinate obtained is Source (22.9885, 22.9925). The relative error for the horizontal and vertical coordinates of the initial simulation point is reported to be (0.05%, 0.032609%).

In the case of the initial infection within the population of the community block, the center point is adjusted to (28, 28) for analysis. After conducting 1,000 simulations, the average duration for epidemic control within the community is determined to be approximately 18.285 days. The expected number of single sample tests is calculated as 2419.455. The final traceability result indicates the source as (28.015, 28.011), with a relative error of (0.053571%, 0.039286%) for the horizontal and vertical coordinates. These results demonstrate that the relative error for the horizontal and vertical coordinates of the community where the first infected person is located does not exceed 0.05%, while the relative error for the horizontal and vertical coordinates of the first infected individual obtained from the source tracing remains below 0.06%. Hence, the accuracy of the model is verified. In summary, the final solution for this round of the epidemic in the city is as follows: the average duration is determined to be 18.488 days. Throughout the epidemic, there has been an average of 1727.324 instances of mixed mining and 5.18 million instances of single mining. Figure 5 provides a visual representation of two simulation results during the dynamic grid search process.