A Method for Estimating Infectious Disease Parameters Based on Integrated Kalman Filtering and Social Networks

Abstract

This invention belongs to the field of infectious disease prevention and control technology, specifically a method for estimating infectious disease parameters based on ensemble Kalman filtering and social networks. The method includes: constructing a regional social network to characterize the transmission path of an infectious disease; obtaining the number of observations showing the percentage of infected and removed individuals during the spread of the epidemic in that region; establishing an infectious disease model between nodes on the social network based on the SIR infectious disease model and the social network; obtaining the estimated average infection probability between nodes, the average removal rate of nodes, and the corrected percentages of infected and removed individuals at the current time through ensemble Kalman filtering; correcting the node states of the model at the current time based on the corrected percentages of infected and removed individuals from the ensemble Kalman filtering; and obtaining the estimated infectious disease parameters after the ensemble Kalman filtering converges. This invention can effectively estimate the transmission parameters of an epidemic in relatively closed areas and improves the accuracy of the estimation by utilizing social network information.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of infectious disease prevention and treatment, and particularly relates to an infectious disease parameter estimation method. BACKGROUND

[0002] Infectious diseases are a serious challenge to human survival and health. Modeling and predicting infectious diseases are of great significance to control and eliminate the occurrence and prevalence of infectious diseases and protect people's health. The most commonly used infectious disease model is collectively referred to as the compartment model, which divides the population into several compartments for study through certain division, for example, the SIR model is commonly used to describe the disease transmission process of diseases such as measles and chickenpox, which can be cured and the infected persons permanently retain the immunity to the disease. Kalman filtering is an effective method for estimating infectious disease parameters. However, the current infectious disease parameter estimation method based on integrated Kalman filtering ignores the importance of social networks and can only estimate the macro infection rate and recovery rate of the population, but cannot accurately estimate the parameters of discrete nodes at a fine granularity. The method estimates the infection probability between nodes that exist in contact and the recovery probability of the node itself by using social networks, which plays a better guiding role in infectious disease prevention and treatment. SUMMARY

[0003] The purpose of the present application is to provide an infectious disease parameter estimation method based on integrated Kalman filtering and social networks in view of the shortcomings or deficiencies of the infectious disease parameter estimation method.

[0004] The infectious disease parameter estimation method based on integrated Kalman filtering and social networks provided by the present application comprises obtaining the social network in the region, observing the infection rate and the removal (recovery) rate of the infectious disease, and estimating the infectious disease parameters by using the integrated Kalman filtering algorithm. The specific steps are as follows:

[0005] Step 1: Construct the social network in the region to represent the transmission path of the infectious disease; for example, the social connection relationship between students in a closed school;

[0006] Step 2: Obtain the observation of the infection rate and the removal rate in the epidemic transmission process of the region;

[0007] Step 3: Based on the SIR infectious disease model and the social network, establish the infectious disease model between nodes on the social network;

[0008] Step 4: Obtain the average infection probability β between nodes, the average removal rate γ of the node and the corrected infection rate and removal rate at the current time by using the integrated Kalman filtering;

[0009] Step 5: According to the corrected infection rate and removal rate after the integrated Kalman filtering, correct the node state of the model at the current time every time window with a length of L;

[0010] After the integrated Kalman filter converges in step 6, the estimated parameters of the infectious disease are obtained.

[0011] The detailed instructions for each step are as follows.

[0012] Step 1: Construct a regional social network to characterize the transmission paths of infectious diseases;

[0013] In social networks, nodes represent people in a region, and undirected edges represent connections between nodes (people). That is, the two nodes (people) connected by an edge have a social relationship and can be infected by each other.

[0014] by Figure 1 For example, 20 nodes represent 20 people in a region, and the edges represent potential transmission paths due to social relationships. In the following steps, taking M nodes as an example, denoted as node1, node2, ..., node... M .

[0015] Step 2: Obtain the observed data on the percentage of infected and removed people during the spread of the epidemic in the region. This data can be obtained from statistics during the epidemic or from publicly available data from relevant departments.

[0016] Step 3: Based on the SIR infectious disease model and social networks, establish an infectious disease model between nodes on the social network.

[0017] Suppose there is a social network G with M nodes, each node having three states: s - susceptible, i - infected, and r - removed. A removed node becomes immune and cannot be reinfected. Two parameters, β and γ, represent the probability of infection and the probability of removal, respectively. The following rules are followed in modeling the transmission process:

[0018] Step 3.1. Find the set of susceptible nodes adjacent to the infected node;

[0019] Step 3.2. Traverse the set of susceptible nodes, take a random number, and if the random number is less than β, then infect the neighboring node, that is, change the state of the neighboring node from s to i;

[0020] Step 3.3. After the traversal is complete, take a random number. If the random number is less than γ, the infected node becomes the remover node, that is, the state changes from i to r.

[0021] Step 4. Obtain the estimated average infection probability β and average removal rate γ between nodes at the current time using ensemble Kalman filtering, along with the corrected percentages of infected and removed individuals; estimate parameters β and γ using ensemble Kalman filtering. The specific steps are as follows:

[0022] Step 4.1. Select the number N of particles (i.e. the number of sample models in the ensemble Kalman filter), for example 50, 100, 200, and maintain a separate social network infection model for each particle;

[0023] Step 4.2. Establish the social network relationship of M nodes according to the social network obtained in step 1, and assume that the initial infection rate satisfies the normal distribution Reasonably set the mean i0 and variance P of the initial infection rate i Sample the initial infection node rate of each particle of N particles, and on each particle, set the initial infection node according to the initial infection node rate; set the initial removal node rate of all particles to 0; that is, on each particle, part of the sampled node state is s-infected, and the remaining node state is r-susceptible;

[0024] Step 4.3. Assume that the initial infection probability satisfies the normal distribution Sample the initial infection probability β of each particle 0，1 , …, β 0，N ; Similarly, sample the initial removal probability γ of each particle 0,1 , …, γ 0，N ; Here, β0, P β , γ0, P γ are set based on priori, and the method does not require high accuracy of the initial value, so the method is simple to operate;

[0025] Step 4.4. Traverse all particles k = 1, …, N, and know the state of the nodes in the social network in the kth particle at t-1 and the current estimate of the model parameters β t，k and γ t，k of the kth particle, execute the infection process shown in step 3 for each particle's social network; for the kth particle, obtain its infection rate at time t and removal rate

[0026] Step 4.5. Denote the state predicted by the kth particle model at time t as:

[0027]

[0028] where φ(·) is an arbitrary invertible function that maps the domain from [0, 1] to [-∞, +∞], for example φ(x) = tan((x-0.5)π) / 300; calculate the average predicted state of the algorithm:

[0029]

[0030] Step 4.6. Let the observation predicted by the model for the k-th particle at time t be denoted as . Linear observations were used. The algorithm's average predicted observations are:

[0031]

[0032] Step 4.7. Obtain the percentage of infected individuals in the actual epidemic dataset at time t. t and the percentage of people removed R t Let the observation y t =[I t ,R t Assuming the observed noise follows a multivariate normal distribution with mean 0 and variance R, the variance H of the observed noise is adaptively set based on the observed data. t H t [0,0]=(0.04I t ) 2 H t [1,1]=(0.04R t ) 2 Furthermore, upper and lower bounds are set for the diagonal elements, and the elements at other positions in the covariance matrix are set to 0;

[0033] Step 4.8. Correct the predicted state according to the following steps:

[0034]

[0035]

[0036]

[0037]

[0038]

[0039] in, This refers to the algorithm's estimate of the percentage of actual infected people in the region at time t. Estimated percentage of actual removals Average infection probability and average removal probability Simultaneously, the algorithm outputs the corrected state for each particle: Particle k corrects the percentage of infected people at time t Correct the number of people removed Corrected infection probability Correct removal probability

[0040] Step 5. Modify the state of the revised Kalman filter algorithm to the infection model of particles, divided into modification of parameters and modification of nodes.

[0041] Step 5.1 Modification of parameters. At each step, the revised infection probability and the revised removal probability of the particles are modified to their infection model, that is, for the prediction process of particle k at time t+1 (step 4.4), the revised infection probability and removal probability of step 4.8 are used.

[0042] Step 5.2 Modification of nodes. Define a time window L, and every L steps, modify the revised actual infection proportion and the revised actual removal proportion of all particles to their infection model; one of the innovations of this method is how to reflect the revised macroscopic infection proportion and removal proportion based on the social network infection model to the microscopic network node state of the particles. Specifically, record the time when each node state changes from s to i and the time when it changes from i to r; the revision of the state mainly has the following 4 forms:

[0043] (1) s→i: Traverse the currently infected nodes, and form a susceptible set S neighbor from the neighbor nodes of the state s, and select the nodes to infect with equal probability. Update the state advance time of the newly infected nodes to the current t+0.5, which is used to distinguish the newly infected nodes in the prediction phase;

[0044] (2) i→r: Sort the currently infected nodes in the order of infection time, remove the first infected first, and remove the nodes infected at the same time with equal probability. Update the state advance time of the removed nodes to r to the current t+0.5, which is used to distinguish the removed nodes in the prediction phase;

[0045] (3) r→i: Sort the currently removed nodes in the order of removal time, and the last removed first back to the infected state, and the nodes removed at the same time back with equal probability. The state advance time of the back nodes to r is recorded as empty;

[0046] (4) i→s: Sort the currently infected nodes in the order of infection time, and the last infected first back to the susceptible state, and the nodes infected at the same time back with equal probability. The state advance time of the back nodes to i is recorded as empty.

[0047] Since the result of a single step of the system has randomness, a parameter time window L is introduced, that is, the state revision of the system is performed according to the following method:

[0048] Update the parameters of the system at each step, that is, the parameters β and γ that need to be assimilated;

[0049] Update other states of the system, i.e. the fraction of infected and removed, per L step.

[0050] The specific modification of the specific step is:

[0051] Step 1: if the current fraction of removed is greater than the value after the integrated Kalman filter correction, execute r→i; if less, execute i→r;

[0052] Step 2: if the current fraction of infected is greater than the value after the integrated Kalman filter correction, execute i→s; if less, execute s→i.

[0053] Step 6. After the integrated Kalman filter converges, the estimation of the parameters of the infectious disease is obtained. It can be further used for prediction of other regions or time of the infectious disease and research on the performance of the infectious disease.

[0054] The main features and advantages of the present application are:

[0055] (1) The infectious disease model of discrete node infection on a social network is modeled, and the parameters of the model are estimated under the condition of only observing the overall fraction of infected and removed;

[0056] (2) A node state correction method based on infectious disease prior and state transition time is proposed;

[0057] (3) An adaptive observation noise method based on observation suitable for parameter estimation of the infectious disease model of discrete node infection on a social network is proposed;

[0058] (4) The method can be extended to parameter estimation of other infectious disease models such as SIS, SEIR, SEIS based on social networks.

[0059] The present application does not require the state of each person in the region to be known, but only the overall infection situation to be observed. The present application can be effectively used to estimate the epidemic transmission parameters in a relatively closed region, and the accuracy of the estimation can be improved by using social network information. BRIEF DESCRIPTION OF DRAWINGS

[0060] Figure 1 It is a schematic diagram of a social network.

[0061] Figure 2 It is a graph of the number of three kinds of people (s-susceptible, i-infected, r-removed) in the data set.

[0062] Figure 3 It is a schematic diagram of the parameter estimation results of the method of the present application. Among them, (a) is the β parameter, and (b) is the γ parameter. DETAILED DESCRIPTION

[0063] The present application will be further described below through a simulation example combined with the drawings.

[0064] The infectious disease process of a closed area with M = 5000 people is simulated, assuming that its social network is a Watts Strogatz (WS) small-world graph with an average degree of 5 and a reconnection rate of 0.1. When generating the actual epidemic data set, the initial number of infected people at time t = 0 is 0.002, that is, 10 people are infected, and the initial number of removed people is 0, that is, 0 people are removed. The infection rate β = 0.005 and the removal rate γ = 0.002 are set, that is, at each time, the probability that an infected person infects any one of its susceptible neighbors is 0.005, and the probability that an infected person becomes a removed person is 0.002. The simulation is performed for 3000 rounds, and the data set is generated as shown in Figure 2 . Figure 2 The number of three kinds of people in the data set is shown, where the horizontal axis represents time and the vertical axis represents the number of people. s (that is, the blue curve) represents the number of susceptible people, i represents the number of infected people, and r represents the number of removed people. As time goes on, the number of susceptible people gradually decreases, the number of removed people gradually increases, and the number of infected people first increases and then decreases. In the experiment, only the number of infected people and the number of removed people in the area shown in the figure can be observed, and the state of each person cannot be observed.

[0065] For integrated Kalman filtering, the initial particle infection rate parameter i0 = 0.01, P i = 1e-4, the initial particle removal parameter β0 = 0.01, P β = 1e-4. The number of particles N = 200, and the time window L = 5. The estimation results of the infection rate β and the removal rate γ of the present method are shown in Figure 3 . Among them, Figure 3 (a) shows the prediction of the value of the parameter β, where the horizontal axis is time and the vertical axis is the predicted value of β. ‘beta gt’ (gt is the abbreviation of ground truth) is the true value of β, and ‘beta pre’ (pre is the abbreviation of prediction) is the predicted value of β by the algorithm. Similarly, Figure 3 (b) shows the prediction of the value of the parameter γ, where ‘gamma pre’ is the predicted value of the algorithm, and ‘gamma gt’ is the true value. As can be seen from the figure, at the initial time, that is, t = 0, the random initialization of the predicted value of β is poor, and after t = 2500, the predicted value of β approaches the true value, and the effect is good.

[0066] It can be seen that the present method can accurately estimate the parameters of the infectious disease on the social network.

Claims

1. An infectious disease parameter estimation method based on integrated Kalman filtering and social network, characterized in that, The method comprises the following steps of: acquiring a social network in a region, observing an infection number ratio and a removal number ratio of an infectious disease, and estimating parameters of the infectious disease by using an integrated Kalman filtering algorithm. Step 1, constructing a social network in the region to represent a transmission path of the infectious disease; In the social network, the nodes represent the population in the region, and the undirected edges represent the connection between the points, that is, the two nodes connected by the edge have a social relationship and can be infected by each other; let the number of nodes be M, denoted as ; Step 2, acquiring observation data of the infection number ratio and the removal number ratio in the process of epidemic transmission in the region, the data being obtained from statistical data in the process of the epidemic or public data provided by a relevant department; Step 3, establishing an infectious disease model between nodes in the social network based on an SIR infectious disease model and the social network; A social network of M nodes is denoted as G, and there are three states for each node: - susceptible, - infected, - removed; The removed nodes will have immunity and will not be infected again, two parameters and represent the infection probability and the removal probability, respectively; in modeling the infection process, the following rules are followed: Step 3.1, finding adjacent neighbors of an infected node as a susceptible node set; Step 3.

2. Traverse the susceptible node set, take a random number, if the random number is less than , then infect the neighbor node, i.e. the neighbor node state is changed from to ; Step 3.

3. After the end of the iteration, take a random number, if the random number is less than , then the infected node turns into a removed node, i.e. the state changes from to ; Step 4. Obtain the average infection probability between nodes at the current time by integrated Kalman filtering , the average removal rate of nodes , and the proportion of infected people and the proportion of removed people after correction; estimate the parameters and by integrated Kalman filtering; the specific steps are: Step 4.1, selecting a particle number N, and maintaining a separate social network infectious model for each particle; Step 4.

2. Establish the social network relationship of M nodes according to the social network obtained in step 1, assuming that the initial infection proportion satisfies the normal distribution , the mean and variance of the initial infection proportion are reasonably set and , the initial infection node proportion of particles is obtained by sampling, and on each particle, the initial infection node is set according to the initial infection node proportion; the initial removal node proportion of all particles is set to 0, that is, on each particle, the state of the partially sampled node is ; Step 4.

3. Assume the initial infection probability follows a normal distribution , sample the initial infection probability of each particle ; Similarly, assume that the initial removal probability satisfies the normal distribution , sample the initial removal probability of each particle ; here, based on prior settings; Step 4.

4. Traverse all particles Known Moment The state of the nodes in the social network and the current estimate of the model parameters in the th particle And , the infection process shown in step 3 is performed separately for the social network of each particle; for the th particle, the proportion of the number of infected people at the moment And the proportion of the number of removed people ; Step 4.

5. Note the time instant The state predicted by the particle model is: , （1） where, To map the definition domain from to any invertible function, compute the average predicted state: ， （2） Step 4.

6. Note that the time of the particle model prediction is , the linear observation is , and the average predicted observation is ， （3） Step 4.

7. Obtain the actual epidemic data set at time t , and the removal rate , let the observation , assume that the observation noise obeys the multivariate normal distribution with mean and variance , set the variance of the observation noise according to the observed data , and set the upper and lower bounds of the diagonal elements, and the rest of the elements of the covariance matrix ; Step 4.8, modifying the predicted state according to the following steps: ， （4） where, ; , is the estimate of the actual infection proportion of the population in the region at time , is the estimate of the actual removal proportion of the population in the region at time , is the average infection probability ; meanwhile, the algorithm outputs the corrected state of each particle: : the corrected infection proportion of particle k at time t , the corrected removal proportion of particle k at time t , the corrected infection probability of particle k at time t , the corrected removal probability of particle k at time t ;​​ Step 5, modifying the state modified by the integrated Kalman filtering algorithm to the infectious model of the particle, and dividing the modification into modification of parameters and modification of nodes; Step 5.1 Modification of parameters; the modified infection probability of a particle is used in each step and the modified removal probability The modification is applied to the infection model of the particle, i.e. for the prediction process of particle k at time t + 1, as in step 4.4, the modified infection probability and removal probability of step 4.8 are used; Step 5.2, modification of nodes; defining a time window L, and modifying the actual infection number ratio and the actual removal number ratio of all particles modified at every L steps to the infectious disease model thereof; Step 6, obtaining estimation of the parameters of the infectious disease after convergence of the integrated Kalman filtering; and further used for prediction of other regions or time of the infectious disease and research on performance of the infectious disease.

2. The infectious disease parameter estimation method according to claim 1, characterized by, Step 5.2 The modification of the nodes, in particular, includes recording the time when each node state changes from to ; the time when each node state changes from to ; the correction of the state has the following four forms: (1) Traverse the currently infected nodes, and set their state as The neighbor nodes of the susceptible set , and select the nodes to be infected with equal probability; update the state of the newly infected nodes to the current t + 0.5, which is used to distinguish the newly infected nodes in the prediction stage; (2) Sort the current infected nodes in the order of infection time, remove the infected first, and remove the infected at the same time with equal probability; update the state of the removed node to The time is the current t + 0.5, which is used to distinguish the nodes removed in the prediction stage; (3) : Sort the current removal nodes in reverse order of removal time, and the nodes removed later are backtracked first, and the nodes removed at the same time are backtracked with equal probability; The state of the reset rollback node is emptied at the time when the state advances to r; (4) Sort the current infected nodes in reverse order of infection time, the later infected ones back to the susceptible state first, the nodes infected at the same time back to the susceptible state with equal probability; reset the state of the back nodes to the moment empty; Since the result of a single step of the system has randomness, a parameter time window L is introduced, and the state of the system is modified according to the following method: Each step updates the parameters of the system, i.e. the parameters that need to be assimilated and ; Every The other states of the system, i.e. the fraction of infected and removed people, are updated. The specific modification steps are as follows: Step 1: if the current removal proportion is greater than the value after the integrated Kalman filter correction, execute ; if less, execute ; Step 2: If the current infection number proportion is greater than the value after the integrated Kalman filter correction, execute ; if less, execute .

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