An infectious disease threshold searching method based on a high-order infectious disease model

Abstract

The present application belongs to the technical field of infectious disease prevention and treatment, and specifically relates to an infectious disease threshold searching method based on a high-order infectious disease model. The present application comprises the following steps: constructing a high-order SEIS infectious disease model; searching for a Lyapunov function by using a quadratic neural network and an ICNN neural network algorithm, so as to judge the global dynamic behavior of the infectious disease system, and to perform dynamic stability evaluation and threshold searching according to the global dynamic behavior. Compared with the prior art and model, the present application is closer to the actual transmission law of infectious diseases, and can strictly prove the dynamic asymptotic behavior of an epidemic by means of machine learning in mathematics.

Description

Technical Field

[0001] This invention belongs to the field of infectious disease prevention and control technology, specifically relating to an infectious disease threshold search method based on a high-order infectious disease model. Background Technology

[0002] The concept of complex networks is used to characterize the structure of natural systems and is widely applied to understanding dynamic processes. Many natural and human phenomena can be explained by constructing complex networks of interactions among people. Furthermore, the transmission process of viruses is increasingly attracting the attention of researchers. Therefore, network-based mathematical models play a crucial role in understanding the dynamics of infectious diseases. Due to the characteristics of network structures, most network-based epidemic transmission models are described as follows: nodes in the network are assigned different states (e.g., susceptible or infected), and the epidemic spreads based on the contact behavior between two nodes, simulated using edges.

[0003] Among existing infectious disease models, the SIRS model is the most widely used. Through stability analysis, it can reveal local dynamics near the equilibrium point and make a simple prediction of the epidemic's trajectory. However, such models are often incomplete. They do not consider the multifaceted transmission behavior when multiple people are involved, nor do they take into account the impact of dynamic phenomena during the incubation period of infectious diseases. Furthermore, their judgments on epidemic behavior are relatively narrow. Summary of the Invention

[0004] The purpose of this invention is to provide a method for searching infectious disease thresholds based on a high-order infectious disease model, so as to provide suggestions for infectious disease prevention and control decisions.

[0005] The present invention provides a method for searching infectious disease thresholds based on a high-order infectious disease model, comprising constructing a high-order SEIS infectious disease model; using the ICNN neural network algorithm to search for the Lyapunov function (this search method has been mathematically proven to strictly satisfy all properties of the Lyapunov function), thereby determining the global dynamic behavior of the infectious disease system, and performing dynamic stability assessment and threshold search accordingly; the specific steps are as follows:

[0006] S1: Construction of a high-order SEIRS infectious disease model

[0007] Based on the real-world scenarios of viral transmission in modern biomedical research, this invention proposes the following higher-order susceptibility-exposure-infection-recovery infectious disease model, also known as the higher-order SEIRS model:

[0008] (1)

[0009] The meanings of each parameter are as follows:

[0010] Effective contact rate The conversion rate from latent nodes to infected nodes. Recovery rate The rate of transformation from a recovery point to a susceptible point; The number of nodes in a high-order network. The order of a high-order network. The probability that node i is in a susceptible period. The probability that node i is in the latent period. The probability that node i is in the disease stage. The probability that node i is in the recovery period. The first-order propagation efficiency between node i and node j. : 'Simple The efficiency of transmission between them : The weight of the singleton.

[0011] In most cases, we don't need particularly high-order infectious disease models. Generally, N=2 is sufficient to simulate the vast majority of situations, and the second-order SEIRS model in this case takes the form:

[0012] (2)

[0013] in, This represents the first-order average propagation efficiency. This represents the second-order average propagation efficiency, while This represents the weight of the two-dimensional simplex.

[0014] S2: Using the model constructed above, perform numerical simulations to find three possible dynamic equilibrium points. , as well as And their coordinates are respectively denoted as ;in, This refers to the disease-free equilibrium point, which definitely exists. This refers to the equilibrium point where a small percentage of the population is affected by the disease. This refers to the equilibrium point where a large population becomes ill; if only two equilibrium points are found, then these two equilibrium points are... , If only one equilibrium point is found, then that equilibrium point is... .

[0015] S3: Calculate the key values ​​for the epidemic dynamics:

[0016] Under the parameters of the above model, the following key values ​​were calculated: Threshold for local stability , Does it exist threshold? , Does it exist threshold? Equilibrium point The second, first, and zeroth order terms of the characteristic polynomial of the Hessian matrix , , ;Will Considered as about The quadratic function, and its corresponding second-order, first-order, and zero-order terms. , , ; and the two roots of the quadratic function. and :

[0017] ,

[0018] , (3)

[0019] ,

[0020] (4)

[0021] (5)

[0022] (6).

[0023] S4: Preliminary assessment of the existence of the system's equilibrium point and local stability.

[0024] The existence and stability of the epidemic equilibrium point can be judged based on the table below:

[0025] .

[0026] in

[0027] (7)

[0028] This means that the characteristic polynomial mentioned in S4 is always greater than 0 in its domain; Represents the equilibrium point The infection rate at that time The settlement method is given in equation (4).

[0029] S5: Build a neural network and design the loss function

[0030] Furthermore, this invention utilizes cutting-edge neural network algorithms to determine the global stability of the epidemic equilibrium point; these algorithms have been rigorously proven mathematically to be feasible. This invention employs quadratic neural networks and ICNN neural networks respectively to complete this determination process:

[0031] (a) Constructing a quadratic form function using a feedforward multilayer neural network:

[0032] make It is a neural network with tanh as the activation function, and If it is a parameter vector, then the undetermined Lyapunov function in it is of the form:

[0033] (8)

[0034] (II): Constructing the V function using ICNN:

[0035] ICNN stands for Input Convex Neural Network, and its specific form is as follows:

[0036] (9)

[0037] in, and All are real-valued weights. The weights are positive, and F is the indicator function. It is an adjustable hyperparameter, g(•), which is an activation function, typically tanh. For another activation function with the following form:

[0038] (10)

[0039] After building the neural network, a loss function needs to be designed. The loss function is defined as:

[0040] (11)

[0041] In practice, its specific form should be:

[0042] (12)

[0043] in, It is an adjustable positive hyperparameter. From the following set Sampling in:

[0044] , From a set The number of samples in the sample.

[0045] S6: Use neural networks to evaluate system stability and find the threshold.

[0046] right After sufficient subdivision, the Lyapunov function is found using the neural network built in step S5. At this point, for There will be two propagation thresholds and The specific judgment is as follows:

[0047] when At that time, there was a question about Lyapunov function, i.e. The overall situation is stable, which means the pandemic will eventually end;

[0048] when At that time, there was a question about Lyapunov function, i.e. The overall situation is stable, which means that the epidemic will eventually break out on a large scale.

[0049] when At that time, there is no Lyapunov function, which means that the epidemic will exhibit a fluctuating state.

[0050] Compared with the prior art, the present invention has the following advantages:

[0051] (1) Compared with traditional infectious disease models, this invention considers the influence of the incubation period on the overall transmission dynamics. Furthermore, by utilizing the concept of a simplex in topology, it considers the mutual influence of multiple nodes at a higher dimension, rather than only considering the mutual contact between pairs of nodes as in previous models. At the same time, by introducing weights of order, it can more flexibly adjust the relationship between unilateral and simplex influences, thereby helping to simulate the dynamics of infectious diseases more accurately.

[0052] (2) This invention first uses mathematical dynamics analysis tools to accurately analyze the existence of equilibrium points and local stability, thus laying the groundwork for subsequent global stability evaluation using neural networks;

[0053] (3) This invention is the first to utilize cutting-edge quadratic neural networks and ICNN neural network artificial intelligence algorithms to evaluate the global stability of the equilibrium point. This evaluation result has been proven to be mathematically rigorous, and therefore has high accuracy and robustness. At the same time, by finding the threshold of epidemic transmission, we can make more effective judgments on the development prospects of the epidemic, which is conducive to the government and society in formulating corresponding epidemic prevention policies. Attached Figure Description

[0054] Figure 1This is a schematic diagram of the infectious disease threshold search method based on a high-order infectious disease model of the present invention.

[0055] Figure 2 These are examples of social networks. Among them, (a) is a random network, (b) is a small-world network, and (c) is a scale-free network.

[0056] Figure 3 It is for different The infection probabilities obtained by numerical simulations of different social networks, with values ​​corresponding to these probabilities, change over a long period of time. The relationship between them. Among them, (a) corresponds to a random network, (b) corresponds to a scale-free network, and (c) corresponds to a small-world network.

[0057] Figure 4 Is the loss function and The relationship.

[0058] Figure 5 This is an example of a Lyapunov function found through a neural network. Among them, Detailed Implementation

[0059] The present invention will be further described below with reference to the embodiments and accompanying drawings. Obviously, the embodiments described are only some, not all, of the embodiments of the present invention.

[0060] All other embodiments obtained by those skilled in the art without inventive effort are within the scope of protection of this invention.

[0061] Example 1:

[0062] The infectious disease threshold search method based on a high-order infectious disease model provided by this invention includes the construction of the high-order infectious disease model and the determination of equilibrium point stability, etc. Figure 1 As shown, the specific steps are as follows:

[0063] S1: Construct a high-order SEIRS model as follows:

[0064]

[0065] Its second-order SEIRS model is:

[0066] .

[0067] S2: Using the above-mentioned higher-order SEIRS model, numerical simulations were performed to find three possible dynamic equilibrium points. , as well as And denote their coordinates as . .

[0068] S3: Calculate the key values ​​as follows:

[0069] ,

[0070] ,

[0071] ,

[0072]

[0073]

[0074] .

[0075] S4: Preliminary assessment of the existence of the system's equilibrium point and local stability:

[0076] .

[0077] in:

[0078] (7).

[0079] S5: To further evaluate the global stability of the dynamics of the high-order infectious disease model, a quadratic neural network was constructed:

[0080]

[0081] Or ICNN neural network

[0082]

[0083]

[0084] And design the loss function:

[0085] .

[0086] S6: Use social neural networks to find the Lyapunov function, evaluate the global stability of the system, and find the corresponding propagation threshold.

[0087] There are three types of classic social networks: (a) random networks [1] (b) is a small-world network [2] (c) is a scale-free network. [3] Its form is as follows: Figure 2 As shown.

[0088] First select Figure 2Numerical simulations were performed on three classic social network generation methods: random networks, small-world networks, and scale-free networks. In the random network, the connection probability of each edge is 0.5. In the small-world network, the ratio of clustering and average path length between this network and an equivalent random network of the same degree is 3. In the scale-free network, the degree distribution γ value is 2.5.

[0089] Then select the following parameters:

[0090] 1. First-order average propagation efficiency ;

[0091] 2. Second-order average propagation efficiency ;

[0092] 3. Recovery rate ;

[0093] 4. The conversion rate from latent nodes to infected nodes ;

[0094] 5. The rate of transformation from a recovery point to a susceptible point ;

[0095] 6. The initial value is that there is a 0.1 probability of being infected, and the rest are susceptible.

[0096] Weights for different second-order simplexes Simulations were conducted on three different types of networks, and the asymptotic behavior of their infection probabilities was compared with... The relationship between them is as follows Figure 3 As shown. Figure 3 This indicates that for different types of networks, The increase in [something] will promote a large-scale outbreak of the epidemic. Furthermore, it can be seen that for different networks and [other factors]... All have thresholds ,when The pandemic will eventually disappear, and when... The larger, The smaller the threshold, the more severe the epidemic. This completes the simulation process of the higher-order SEIRS model, but we still cannot obtain another threshold. The value of is not determined because even after a long period of time, the infection probability at a certain point in time cannot reflect whether the epidemic is in a major outbreak or in a state of fluctuation. At this time, it is necessary to rely on the power of neural networks.

[0097] Regarding the evaluation of stability using neural networks, two methods were used for different... Find the Lyapunov function, whose loss function is similar to... Relationship such as Figure 4 As shown. Figure 4 This indicates that, regardless of Whether the value is small or large, the final value of the loss function is 0, which means that the constructed function satisfies the necessary conditions for the Lyapunov function. Therefore, the epidemic dynamics have global stability at this time. At the median value, the loss function increases sharply, indicating that the dynamics of the epidemic are inconsistent under different states, thus suggesting that the epidemic is in an oscillating phase. and Does a Lyapunov function exist, that is, as... Figure 4 As shown, this is the case where the loss function is 0. From... Figure 5 It can be known that, and They are as well as The Lyapunov function. This is because when When the value of the function constructed through the neural network is closer to the epidemic-free point (i.e., the zero point), it becomes smaller. This indicates that the zero point is the zero of the Lyapunov function. It is a globally stable point. Conversely, when At that time, the closer the value of the function constructed through the neural network is to The smaller the point value, therefore It is a globally stable point. Ultimately, this shows that... Furthermore, the neural network can successfully determine the global stability of the system's equilibrium point.

[0098] References

[0099] [1] Erdős P, Rényi A. On random graphs I[J]. Publicationesmathematicae, 1959, 6(1): 290-297;

[0100] [2] Watts DJ, Strogatz S H. Collective dynamics of 'small-world'networks[J]. nature, 1998, 393(6684): 440-442;

[0101] [3] Barabási AL, Albert R. Emergence of scaling in random networks[J]. science, 1999, 286(5439): 509-512.

Claims

1. A method for searching infectious disease thresholds based on a high-order infectious disease model, characterized in that, This includes constructing a high-order SEIS infectious disease model; using the ICNN neural network algorithm to search for Lyapunov functions to determine the global dynamic behavior of the infectious disease system, and based on this, performing dynamic stability assessment and threshold search; the specific steps are as follows: S1: Construction of a high-order SEIRS infectious disease model Based on the real-world scenarios of viral transmission in modern biomedical research, a higher-order susceptibility-exposure-infection-recovery infectious disease model, namely the higher-order SEIRS model, is constructed: （1） The meanings of each parameter are as follows: Effective contact rate The conversion rate from latent nodes to infected nodes. Recovery rate The rate of transformation from a recovery point to a susceptible point; The number of nodes in a high-order network. The order of a high-order network. The probability that node i is in a susceptible period. The probability that node i is in the latent period. The probability that node i is in the disease stage. The probability that node i is in the recovery period. The first-order propagation efficiency between node i and node j. : 'Simple The efficiency of transmission between them : The weight of the singleton; When N=2, it is a second-order SEIRS model, which has the following form: （2） in, This represents the first-order average propagation efficiency. This represents the second-order average propagation efficiency. Represents the weight of the two-dimensional simplex; S2: Using the model constructed in S1, numerical simulations are performed to find three possible dynamic equilibrium points: , , And record their coordinates as ;in, This refers to the disease-free equilibrium point, which definitely exists. This refers to the equilibrium point where a small percentage of the population is affected by the disease. This refers to the equilibrium point where a large population becomes ill; if only two equilibrium points are found, then these two equilibrium points are... , If only one equilibrium point is found, then that equilibrium point is... ; S3: Calculate the key values ​​for the epidemic dynamics: Threshold for local stability , Does it exist threshold? , Does it exist threshold? Equilibrium point The second, first, and zeroth order terms of the characteristic polynomial of the Hessian matrix , , ;Will Considered as about The quadratic function, and its corresponding second-order, first-order, and zero-order terms. , , ; and the two roots of the quadratic function. and : ， ， （3） ， （4） （5） （6） S4: Preliminary assessment of the existence of the system's equilibrium point and local stability. The existence and stability of the epidemic equilibrium point are judged based on the table below: ； in: （7） S5: Build a neural network and design the loss function (a) Constructing a quadratic form function using a feedforward multilayer neural network: make It is a neural network with tanh as the activation function, and If it is a parameter vector, then the undetermined Lyapunov function in it is of the form: （8） (II): Constructing the V function using ICNN: ICNN, or Input Convex Neural Network, takes the following form: （9） in, and All are real-valued weights. The weights are positive, and F is the indicator function. It is an adjustable hyperparameter, g(•), which is an activation function. For another activation function with the following form: （10） After building the neural network, a loss function needs to be designed; the loss function is defined as: （11） In practice, its specific form is as follows: （12） in, It is an adjustable positive hyperparameter. From the following set Sampling in: , From a set The number of samples in the sample; S6: Use neural networks to evaluate system stability and find the threshold. right After sufficient subdivision, the Lyapunov function is found using the neural network built in S5. At this point, for There will be two propagation thresholds and The specific judgment is as follows: when At that time, there was a question about Lyapunov function, i.e. The overall situation is stable, which means the pandemic will eventually end; when At that time, there was a question about Lyapunov function, i.e. The overall situation is stable, which means that the epidemic will eventually break out on a large scale. when At that time, there is no Lyapunov function, which means that the epidemic will exhibit a fluctuating state.

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