Weighted susceptible-infected-recovered-deceased epidemic model

Abstract

The present invention relates to an improved epidemic prediction model, the Weighted Susceptible-Infected-Recovered-Deceased (W-SIRD) model, designed to more accurately predict the spread of infectious diseases by incorporating individual user-specific factors such as social distance, mask usage, and age. Unlike traditional Susceptible-Infected-Recovered-Deceased (SIRD) models, which assume uniform infection rates, the W-SIRD model assigns weights to both the nodes (users) and edges (interactions between users) to account for varying susceptibilities and protective measures. The model generates more realistic predictions by simulating the effects of personal behaviors and environmental conditions on disease transmission. Experimental results demonstrate that the W-SIRD model yields lower infection rates compared to the SIRD model, as it considers the protective effects of safety measures like mask-wearing and social distancing. This invention provides a valuable tool for more effective epidemic management, offering insights that can guide public health interventions and resource allocation during pandemics.

Specification

Description:






Field of the Invention:
[001] The present invention pertains to the field of data science, particularly within the area of predictive modeling and graph analytics. It involves the development of a new, user-status-based epidemic model that enhances the predictive capability of traditional epidemic models, specifically the Susceptible-Infected-Recovered-Deceased (SIRD) model. This new model, called the Weighted Susceptible-Infected-Recovered-Deceased (W-SIRD) model, uses additional factors that influence infection rates, such as social distancing, mask usage, and age, to better simulate the dynamics of a pandemic. By incorporating these parameters, the model provides a more accurate prediction of infection, recovery, and death rates, which can be instrumental for government and healthcare decision-making. This model is particularly valuable during pandemics, where understanding and managing the rate of infection is crucial.

Background of the invention and related prior art:
[002] The background of this invention lies in the challenges faced during pandemics, where accurate predictions of infection, recovery, and death rates are essential for controlling the spread of diseases and allocating resources effectively. Traditional epidemic models, such as the Susceptible-Infected-Recovered-Deceased (SIRD) model, have been widely used to predict the progression of infections. However, these models often fail to account for crucial factors such as social distancing, mask usage, and the varying susceptibility of individuals based on age. These factors significantly influence the rate at which a disease spreads within a population. As observed during the COVID-19 pandemic, the dynamics of disease transmission are deeply impacted by personal behaviours and environmental conditions. The need for a more nuanced model that incorporates these additional parameters is evident, as it can lead to more accurate predictions, better preparedness, and more effective interventions in managing a pandemic. This invention introduces a new approach by weighting the nodes (users) and edges (relationships between individuals) in the epidemic model, thus enhancing the realism and predictive power of epidemic forecasting.

[003] A patent document US11158429B2 provides an integrated health care surveillance and monitoring system that provides real-time sampling, modeling, analysis, and recommended interventions. The system can be used to monitor infectious and chronic diseases. When faced with outbreak of an infectious disease agent, e.g., influenza virus, the system can identify active cases through pro-active sampling in high-risk locations, such as schools or crowded commercial areas. The system can notify appropriate entities, e.g., local, regional and national governments, when an event is detected, thereby allowing for proactive management of a possible outbreak. The system also predicts the best response for deployment of scarce resources.
[004] Another patent document US4003379A discloses an apparatus for dispensing drugs and other medications within the body leaves the patient ambulatory and is adapted to be entirely implanted and to dispense such substances over a long period of time, e.g., one to several years, in accordance with the actual needs of the patient. A self-powered dispensing device stores a single or plural substances in powdered, liquid, or other dispensable form and utilizes a compressible container, i.e., a bellows, for withdrawing such substances from storage and dispensing to the body. The dispensing operation may be on a fixed schedule or may be controlled by monitoring single or plural sensors implanted in the body and evaluating the sensed data in order to control both the conditions under which and the kind of dispensing which takes place. Dual dispensers and dual medication may be employed.
[005] A document US4146029A discloses a device and method for dispensing medication internally of the body utilize an implanted system which includes medication storage and dispensing control circuitry having control components which may be modified by means external of the body being treated to control the manner of dispensing the medication within such body. Coordinated pace making is also available.
[006] Another document EP4004863A1 uses aggregated health data and outbreak models to conduct differential diagnosis and provide risk assessments for a user's health status. A healthcare organization server provides diagnostic kits and health applications to participating computing devices that log user health data and registers contacts with other participating users via wireless interactions. The organization compares received data with data received from computing devices from a plurality of other users and identifies common occurrences. The server creates an outbreak model of a geographic region based on the data. The server communicates a determined health status of the user to the user computing device. The server provides the outbreak model to the user, healthcare workers, or others for use in responding to the outbreak.
[007] A patent document US20020120408A1 discloses a system and method for performing real-time infection control over a computer network. The method comprises obtaining a sample of a microorganism at a health care facility, sequencing a first region of a nucleic acid from the microorganism sample, comparing the first sequenced region with historical sequence data stored in a database, determining a measure of phylogenetic relatedness between the microorganism sample and historical samples stored in the database, and providing infection control information based on the phylogenetic relatedness determination to the health care facility, thereby allowing the health care facility to use the infection control information to control or prevent the spread of an infection.
[008] None of these above patents, however alone or in combination, disclose the present invention. The invention consists of certain novel features and a combination of parts hereinafter fully described, illustrated in the accompanying drawings, and particularly pointed out in the appended claims, it being understood that various changes in the details may be made without departing from the spirit, or sacrificing any of the advantages of the present invention.

Summary of the invention:
[009] The invention proposes a novel epidemic model called the Weighted Susceptible-Infected-Recovered-Deceased (W-SIRD) model, which enhances traditional Susceptible-Infected-Recovered-Deceased (SIRD) models by incorporating additional user-specific factors such as social distance, mask usage, and age to more accurately predict the spread of infections during a pandemic. This model accounts for the varying risk of infection based on individual behaviors and interactions, providing a more realistic representation of disease transmission. Experimental results using both synthetic and real-world datasets, including the Haslemere and Copenhagen datasets, show that the W-SIRD model yields lower infection rates compared to the SIRD model, as it considers the protective effects of safety measures like mask-wearing and social distancing. By simulating different scenarios, the invention demonstrates how these parameters significantly influence the infection, recovery, and death rates, offering valuable insights for pandemic management and policy-making.

Detailed description of the invention with accompanying drawings:
[010] For the purpose of facilitating an understanding of the invention, there is illustrated in the accompanying drawing a preferred embodiment thereof, from an inspection of which, when considered in connection with the following description, the invention, its preparation, and many of its advantages should be readily understood and appreciated.
[011] The principal object of the invention is to develop weighted susceptible-infected-recovered-deceased epidemic model.

Methodology
In this invention, we use a temporal network. It is a weighted network represented as G = [G0, G1, ..., GT−1] where G is the set of the graphs at the different time steps and T is the total number of timesteps. Each G(t) = (V (t), E(t),W (t), F(t)) where t represents the timestep of the graph, V (t) is a set of nodes at that timestep, E(t) ∈ V (t) × V (t) is the set of undirected edges in G(t), W (t) is the set of edge weights and F(t) is the set attribute vectors, which indicates the attributes associated with a node.
In this invention, we denote the edge weight to be the social distance maintained. Whereas, attribute vector consists of user's age and status of mask use. To design the proposed W-SIRD model for a temporal network we will use SIRD model. The detailed descriptions are illustrated below.
Figure 1. A simple temporal model with 4 nodes and 3 timesteps according to the embodiment of the present invention.
Temporal networks
A temporal network, also known as a time variant network, is a network whose links are active only at certain points in time. They are useful in describing how the network evolves temporally. In a static network, a node, if infected, can infect any other node that it comes in contact with, regardless of the time of contact and hence over-predict the infection rates. But in a dynamic network, temporal ordering is preserved. That is, if an individual A comes in contact with a person B before B and C interacted, then A faces no risk from C. A simple example of a temporal network is shown in FIG. 1.
Susceptible - Infected - Recovered - Deceased model
It defines four states of a node - susceptible(S), infected(I), recovered(R) and deceased(D) as shown in FIG. 2. Initially, we assume that all the nodes are susceptible to the disease and initialize some of the nodes as infected. If the infected nodes come in close contact with a susceptible node, there is a possibility that the node becomes infected, with an infection probability α. An infected node can either get recovered with a recovery rate β or become deceased with case fatality rate µ. We also assume that a recovered node does not go back to being susceptible.

Weighted - Susceptible - Infected - Recovered - Deceased model
The proposed W-SIRD method is an extension of the SIRD model. The simple SIRD model does not consider any node or edge attribute to calculate susceptibility, infection, and death rate.
However, the node and edge attributes have a different significance and function in a pandemic. The considered attributes of nodes and edges of the proposed W-SIRD model are discussed below.
Figure 2. Compartmentalized SIRD model according to the embodiment of the present invention.
Social Distance
For a susceptible node to get infected, it must have come close in contact with an infected node, i.e, the two nodes must be in close proximity. If they do not maintain social distancing, the chances of getting infected is higher. The social distance between the two nodes (u, v) is denoted by d ∈ R+ where R+ is the set of all positive real numbers.
Age
Some age groups have a higher chance of getting infected than the others. This can be calculated by using the Bayes' theorem, P(InfectionRate | Age) = P(Persons in age group being infected) / P(total persons in age group). The age for a node is age ∈ Z+ where Z+ is the set of all positive integers.
Mask
The usage of a mask reduces the chances of getting infected significantly (Leung, N.H.L. 2020). For a node n, the usage of mask is a Boolean, mask ∈ {True, False}. If the mask is being used, then mask = True otherwise mask = False
Hypothesis 1 Probability of infection, given the age, social distance and usage of mask depends on the separate probabilities of infection given each of the different attributes - age, social distance and usage of mask. Mathematically,
P(A BCD) = P(A|B)P(A|C)P(A|D)
To calculate the infection rate given the age group of the node, whether or a mask is being used and if they have come in close contact with an infected node, we use some basic probability theorems. Usage of mask, user's social distance and their age are conditionally independent events.
Considering A, B, C, D to be four events and let B, C, D be conditionally independent events. By using independence,
P(BCD) = P(B)P(C)P(D)

conditionally independence,
P(BCD|A) = P(B|A)P(C|A)P(D|A)

and Bayes' theorem,


P(A B) = P(B|A)P(A)
P(B)


we can prove the hypothesis as below.

P(A BCD) = P(BCD|A)P(A)
P(BCD)

= P(B|A)P(C|A)P(D|A)P(A)
P(B)P(C)P(D)

P(A|B)P(B) P(A|C)P(C) P(A|D)P(D)P(A)

= P(A)

P(A)

P(A)

P(B)P(C)P(D)

= P(A|B)P(B)P(A|C)P(C)P(A|D)P(D) P2(A)P(B)P(C)P(D)

= P(A|B)P(A|C)P(A|D)
P2(A)
Hence from Hypothesis 1, P (Infection Rate | Age, Mask use status, Social Distance), which is a discrete variable, for a susceptible node can be calculated as,


P(I AMD) = P(I|A)P(I|M)P(I|D)
P2(I)

(1)



Algorithm 1 Weighted SIRD

procedure W-SIRD(G(V, E,W, F), numO f Nodes, T )
for t [0, T ] do for u G do
if u = infected then r random() if r < β then
u recovered
else if r < µ then
u deceased
else
u remains infected
else if u = deceased recovered then
u remains in the same state
else ▷ u = susceptible
r random()
for v neighbors(u) do
α Calculate the infection rate given u[mask], u[age], d.
if r < α then
u infected
break
else u remains in the same state return count of nodes in each of these states
during the time period

Where I is Infection Rate, M is Usage of Mask, D is the distance and A is age.
To predict the number of infections, recovered and deaths, a Monte Carlo simulation is run 1000 times. A susceptible node becomes infected with probability of α = P(I|AMD). An infected node becomes a recovered with a probability β or deceased with probability µ.
W-SIRD Algorithm for a static network
A static network refers to the case when T = 0. The algorithm describes a Monte Carlo simulation for a time T. Consider a node u ∈ V , if the node u is in the infected state, then it can go to the recovered|deceased state depending on β and µ, where β is the rate of recovery and µ is the case fatality rate.
If the node u is in the susceptible state, then, the probability of getting infected depends on the attributes - social distance, age and usage of mask, as calculated in (1). If u has a neighbor
v ∈ infected, the social distance between u and any such v is considered as the distance. The infection rate is calculated from the equation 1, considering the distance, age and mask usage of u.
Nodes in the recovered and deceased states remain in the same states. The process will be terminated when time reaches the limit T. The final output will be the count of nodes in each of these states during the time period. The algorithm for the W − SIRD is mentioned in Algorithm 1.
W-SIRD Algorithm for a temporal network
The network is temporal when T >= 1. The algorithm is similar to the static W-SIRD algorithm, except that here we also consider the incubation period of the corona virus. The incubation period of the disease refers to the time between the infection and the display of symptoms. Hence, the infected node remains in the network without going to the recovered or the deceased state for this period. Only after the onset of symptoms would the node have a possibility of moving to the recovered or deceased state. In one time step, only the immediate neighbors of the infected node can be infected.
We do not account for secondary infections within one time step. In the next time step, the neighbors of the nodes that were infected in the previous time step will be infected with a probability α calculated using 1 and recovery rate of β and case fatality rate of µ. The node gets recovered/deceased only after the incubation period.




FIG. 3: A small schematic network with |V| = 10 and |E| = 13. We considered three types of scenarios of attributes values, (a) general case (b) best case and (c) worst case. Nodes in red colour are considered to be in the infected state and is according to the embodiment of the present invention.
Experimental Setup & Result
To verify the significance and effectiveness of the proposed method, we have conducted two experiments. Experiment 1 verifies the algorithmic correctness of the infection rate and Experiment 2 compares the performance of the proposed method with the traditional SIRD method.
FIG. 4: The results obtained from the SIRD and W-SIRD models. (a) Result for a general case W-SIRD model; (b) Best case scenario of W-SIRD; (c) Worst case scenario of W-SIRD; (d) Result for SIRD model according to the embodiment of the present invention.
In the experiment, we have considered two types of networks. For better visualization, the first network, FIG. 3A, is a short schematic network (|V| = 10, |E| = 13) and second network is a large network (|V| = 5,000, |E| = 24,975) which is generated using a Baraba´si-Albert model. Initially, we considered 500 nodes to be infected in the large network. For both the networks, we considered attributes of edges and vertices, which are randomly chosen. Such as social distance between two users, user's age, and mask use status. We also considered same infection rate, α = 13%, recovery rate, β = 55% and case fatality rate, µ = 2.8%. In addition, for the small network we take T = 10 days and for the large network, T = 60 days. The attack rate of the virus for different ages is shown in Table 1. We modelled these scenarios with 1000 iterations of Monte Carlo simulations to obtain our results.

Table 1 Attack rate of COVID-19 by age
Age Group Attack rate(per 1,000,000)
0-9 6.1
10-19 12.9
20-29 40.5
30-39 48.5
40-49 50.1
50-59 64.9
60-69 61.8
70-79 53.
≥80 40.9
Experiment 1: Verification of the algorithmic correctness of the infection rate
To verify the algorithmic correctness in terms of change of the infection rate of the proposed W-SIRD model, we conducted this experiment. Here, we considered three types of scenarios for the attribute values: general case, best case, and worst case. FIG. 3A, describes the general case, where random values for age, mask usage and distances are used. Similarly, FIG. 3B describes the best case where each user in a network maintains a safe social distance greater than 3 ft, mask use status is true for all the nodes. Since a society encompasses people of all age groups, the ages of the users have been kept consistent in each of these cases. In the worst case, FIG. 3C, all the edge weights are lesser than 3 ft since no user maintains a social distance. The mask usage status for all the users is false.
We ran the W-SIRD model for these above scenarios and the experimental results for the three cases are as shown in FIG. 4A, FIG. 4B, FIG. 4C. From FIG. 4, we can say that since the recovery rate β is quite high, the infected nodes get recovered quite fast.
Since the infection rate α depends on the social distance, mask usage, and the age, we can observe different infection rates for the general (FIG. 4A), best(FIG. 4B) and worst( FIG. 4C) cases. In the general case, after 10 days, the total number of infections reaches 7. All the infected nodes get recovered after 6 days. In the best case, no further node gets infected since they all maintain social distance and use masks. And the ones that were infected, get recovered. In the worst case, all the nodes got infected as none used masks or maintained a social distance. The total number of infections reached 10 as compared to 9 in the general case and only 2 in the best case.
Hence, we can conclude that the usage of masks and maintaining a social distance by every user in the network reduces the infection rate greatly.
FIG. 5: The results obtained from running the SIRD and W-SIRD models on the Hasslemere dataset (a) Output for SIRD model; (b) Output for a W-SIRD model according to the embodiment of the present invention;
FIG. 6: The results obtained from running the SIRD and W-SIRD models on the Copenhagen dataset (a) Output for SIRD model; (b) Output for a W-SIRD model according to the embodiment of the present invention;
Experiment 2: Comparison of the performance of the proposed method and the existing SIRD model on real datasets
We considered two temporal networks Hasslemere dataset and Copenhagen dataset and ran simulations of the W-SIRD algorithm on single day snapshots of this dynamic network. We considered β= 55% and case fatality rate, µ = 2.8%.
The incubation period of the corona virus has a mean of 5-6 days. This means that it would take 5-6 days to diagnose the person with the disease. We considered the incubation period = 5 days assuming each timestep to be a single day, we ran the experiments on the two datasets.
Hasslemere dataset
This dataset had |V | = 470 and T = 24. We randomly assumed the mask use status, age and randomly assigned 10 nodes to be infected in the first timestep. We compared the temporal W-SIRD with the corresponding temporal SIRD FIG. 5. It can be concluded that the temporal W-SIRD gave a smaller count of infections than its corresponding SIRD model.
Copenhagen dataset
This dataset had |V | = 845 and T = 17. We randomly assumed the mask use status, age and randomly assigned 10 nodes to be infected in the first timestep. As before, we compared the temporal W-SIRD with the corresponding temporal SIRD FIG. 6. It can be concluded that the temporal W-SIRD gave a smaller count of infections than its corresponding SIRD model.
[012] Without further elaboration, the foregoing will so fully illustrate my invention, that others may, by applying current of future knowledge, readily adapt the same for use under various conditions of service. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention.


Advantages over the prior art
[013] Weighted susceptible-infected-recovered-deceased epidemic model proposed by the present invention has the following advantages over the prior art:
1. Increased Accuracy: By considering factors such as social distance, mask usage, and age, the W-SIRD model provides more accurate predictions of infection rates, recoveries, and deaths compared to the traditional SIRD model, which only accounts for broad population dynamics.
2. Realistic Simulation: The model better reflects real-world conditions, where individual behaviors and interactions significantly influence disease spread. This allows for more realistic forecasting of how diseases spread in various settings and scenarios.
3. Improved Risk Assessment: The model helps identify high-risk individuals and populations based on personal factors (e.g., age, mask usage), enabling targeted interventions to reduce transmission, such as tailored public health campaigns or protective measures.
4. Scenario Testing: The ability to simulate different scenarios (e.g., best case, worst case) allows governments and healthcare providers to prepare for a range of possible outcomes, optimizing resource allocation and response strategies.
5. Enhanced Decision-Making: The model provides valuable insights into how specific actions, such as increasing social distancing or mask usage, can impact the spread of a disease, assisting policymakers in making data-driven decisions during pandemics.
6. Adaptability: The W-SIRD model can be applied to both static and temporal (time-varying) networks, making it adaptable to different kinds of data, such as real-time monitoring of disease spread in dynamic environments.
[014] In the preceding specification, the invention has been described with reference to specific exemplary embodiments thereof. It will be evident that various modifications and changes may be made thereunto without departing from the broader spirit and scope of the invention as set forth in the claims that follow. The specification and drawings are accordingly to be regarded in an illustrative rather than restrictive sense. Therefore, the aim in the appended claims is to cover all such changes and modifications as fall within the true spirit and scope of the invention. The matter set forth in the foregoing description and accompanying drawings is offered by way of illustration only and not as a limitation. The actual scope of the invention is intended to be defined in the following claims when viewed in their proper perspective based on the prior art.
, Claims:We claim:

1. Weighted susceptible-infected-recovered-deceased epidemic model comprising of:
- providing a set of user-specific parameters, wherein each user is characterized by their social distance, mask usage, and age;
- generating a weighted graph representing a population, where nodes represent users and edges represent interactions between users;
- calculating a weighted infection rate for each user based on the user's parameters and the distance to other users, incorporating the user's mask usage and age;
- determining the number of infections, recoveries, and deaths over time using a modified Susceptible-Infected-Recovered-Deceased (SIRD) model that incorporates the weighted infection rates of the users and their interactions.
2. The method of claim 1, wherein the set of user-specific parameters further includes other individual health factors or environmental parameters influencing the infection rate.
3. The method of claim 1, wherein the population is represented as a static network, where the user-specific parameters are fixed and do not change over time.
4. The method of claim 1, wherein the population is represented as a temporal network, where user-specific parameters and network interactions change over time based on real-world data, allowing for dynamic predictions.
5. The method of claim 1, wherein the weighted infection rate is calculated by considering the inverse of the distance between users, with higher weights assigned to users who are closer together and do not maintain a safe social distance.
6. The method of claim 1, wherein the model is configured to simulate different epidemic scenarios, including a best-case scenario where all users maintain a safe social distance and wear masks, and a worst-case scenario where no users maintain social distance or wear masks.
7. The method of claim 1, wherein the infection rate is influenced by the age of the users, with specific age groups assigned different infection weights based on known susceptibility to the disease.
8. The method of claim 1, wherein the predictions generated by the model are used to inform public health decisions, such as the allocation of resources and the implementation of safety measures like mask mandates or social distancing guidelines.
9. A system for predicting the spread of an epidemic, comprising:
- a processor configured to generate a weighted graph of a population based on user-specific parameters, including social distance, mask usage, and age;
- a module to calculate weighted infection rates for each user based on the user's parameters and interactions with other users;
- a simulation engine configured to use the weighted infection rates to predict the number of infections, recoveries, and deaths over time according to the modified SIRD model.
10. The system of claim 9, wherein the processor is further configured to simulate the spread of an epidemic under various scenarios, including best-case, worst-case, and general-case conditions.

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