Susceptible-infected-recovered Model Research Articles

A mathematical model of an epidemic with an arbitrary law of recovery

The paper proposes a mathematical model of the epidemic process, taking into account the dependence of the rates of cure and loss of immunity on time. Today, mathematical models of epidemics based on the basic Kermack–McKendrick model have become widespread. The most famous models are Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR). These models are based on dividing the population into separate groups that are in different epidemic conditions. The description of the models is based on differential equations similar to the equations of birth and death in the process of radioactive transformations of elements in a radioactive chain. However, this approach does not take into account the dependence of the probabilities of the transition of the population from group to group on time spent in the treatment process and in the process of loss of acquired immunity. The known models do analyze the nature of the course of the epidemic for long periods of time, when the process can enter a stationary state. The paper proposes a mathematical model based on dividing the population into separate groups. The first group consists of healthy people susceptible to infection due to contact with members of the second group, which includes the infected population. Members of the third group are being treated, the fourth group includes members of the society who have recovered with antibodies and are vaccinated. The fifth group consists of deceased members of society. In contrast to the SIR and SEIR models, the proposed approach takes into account that immunity is lost over time, and people who survived again move to the group susceptible to infection. The dependences of the probabilities of transition from group to group on time spent both during the treatment process and in the loss of acquired immunity have been taken into account. Thus, the proposed mathematical model is based on five integro-differential equations, two of which are partial differential equations. A new mathematical model has been formulated that makes it possible to take into account the dependence of the cure rate and the probability of transition from the vaccinated state to the initial state on time spent in the corresponding state. It is shown that the proposed model is autocatalytic. With increasing time, a state of bistability is observed, when, under the same boundary conditions, two stable states coexist. Switching between states is determined by the epidemic spread control parameter found in the work. One of the stable states is stationary and leads to the end of the epidemic, the second one leads to the population’s extinction. It has been shown that, for the stationary regime, the form of the distribution function in terms of treatment time and exposure time in the vaccinated state does not affect the final result in any way. The conditions for suppressing the epidemic for managing the process of its development are formulated. Numerical experiments were carried out to simulate the epidemic spreading process, taking into account the constancy of all transition probabilities. Integration of the original system of equations was carried out using the Radaus algorithm for stiff differential equations. The results of numerical simulations have confirmed that the experimental data agree with the theory of the control parameter. The results of the work can be used to organize the management of the epidemic spreading process in order to suppress it as soon as possible by changing the value of the control parameter.

ДИФФЕРЕНЦИАЛЬНОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССОВ РАСПРОСТРАНЕНИЯ И ЛИКВИДАЦИИ ИНФЕКЦИОННЫХ ЗАБОЛЕВАНИЙ ЛЕСНЫХ НАСАЖДЕНИЙ

The destabilization of the sustainable state of forests and the associated deterioration of the sanitary climate are mainly associated with their diseases. One of the main tasks of forest pathology monitoring is the detection of foci of disease of stands and the fight against their spread. In this paper, we propose a mathematical description of the processes of the spread and elimination of infectious diseases of forest plantations using the deterministic SIR (Susceptible, Infected, Recovered) model, the main provisions of which are basic in the interpretation of various kinds of infectious diseases, often taking the form of epidemics. Assuming the constancy of the total number of healthy infected subjects during the spread of the disease and the direct proportional dependence of the infection rate on the number of infected, the results obtained indicate that: a) if at the initial moment of time the number of infected individuals does not exceed a certain deterministic value, then plants do not become infected over time; b) in cases if this number is greater than a fixed value, then the part of individuals susceptible to the disease will become infected. It is shown that despite the abstract nature and the impossibility of probabilistic interpretation of the detection of foci of the disease, the proposed differential model can be effective in solving the problems of the spread, localization and prediction of tree diseases and is used, practically, for any closed biological and ecological system

A method to compute the communicability of nodes through causal paths in temporal networks

We present a method aimed to compute the communicability (broadcast and receive) of nodes through causal paths in temporal networks. The method considers all possible combinations of chronologically ordered products of adjacency matrices of the network snapshots and by means of a damping procedure favors the paths that have high communication efficiency. We apply the method to four real-world networks of face-to-face human contacts and identify the nodes with high communicability. The accuracy of the method is proved by studying the spread of an epidemic in the networks using the susceptible–infected–recovered model. We show that if a node with high broadcast is chosen as the origin of the outbreak of infection then the epidemic spreads early while it is delayed and inhibited if the origin of infection is a node with low broadcast. Receiving nodes can be treated as broadcasters if the arrow of time is reversed.

On the Supervision of a Saturated SIR Epidemic Model with Four Joint Control Actions for a Drastic Reduction in the Infection and the Susceptibility through Time.

This paper presents and studies a new epidemic SIR (Susceptible–Infectious–Recovered) model with susceptible recruitment and eventual joint vaccination efforts for both newborn and susceptible individuals. Furthermore, saturation effects in the infection incidence terms are eventually assumed for both the infectious and the susceptible subpopulations. The vaccination action on newborn individuals is assumed to be applied to a fraction of them while that on the susceptible general population is of linear feedback type reinforced with impulsive vaccination actions (in practice, very strong and massive vaccination controls) at certain time points, based on information on the current levels of the susceptible subpopulation. Apart from the above vaccination controls, it is also assumed that the average of contagion contacts can be controlled via intervention measures, such as confinements or isolation measures, social distance rules, use of masks, mobility constraints, etc. The main objectives of the paper are the achievement of a strictly decreasing infection for all time periods and that of the susceptible individuals over the initial period if they exceed the disease-free equilibrium value. The monitoring mechanism is the combined activation of intervention measures to reduce the contagion contacts together with the impulsive vaccination to reduce susceptibility. The susceptibility and recovery levels of the disease-free equilibrium point are suitably prefixed by the design of the regular feedback vaccination on the susceptible subpopulation.

Online Rumor Diffusion Model Based on Variation and Silence Phenomenon in the Context of COVID-19.

In the era of mobile internet, information dissemination has made a new leap in speed and in breadth. With the outbreak of the coronavirus disease 2019 (COVID-19), the COVID-19 rumor diffusion that is not limited by time and by space often becomes extremely complex and fickle. It is also normal that a piece of unsubstantiated news about COVID-19 could develop to many versions. We focus on the stagnant role and information variants in the process of rumor diffusion about COVID-19, and through the study of variability and silence in the dissemination, which combines the effects of stagnation phenomenon and information variation on the whole communication system in the circulation of rumors about COVID-19, based on the classic rumor SIR (Susceptible Infected Recovered) model, we introduce a new concept of “variation” and “oyster”. The stability of the new model is analyzed by the mean field equation, and the threshold of COVID-19 rumor propagation is obtained later. According to the results of the simulation experiment, whether in the small world network or in the scale-free network, the increase of the immure and the silent probability of the variation can effectively reduce the speed of rumor diffusion about COVID-19 and is conducive to the dissemination of the truth in the whole population. Studies have also shown that increasing the silence rate of variation can reduce COVID-19 rumor transmission more quickly than the immunization rate. The interesting discovery is that at the same time, a higher rumor infection rate can bring more rumors about COVID-19 but does not always maintain a high number of the variation which could reduce variant tendency of rumors. The more information diffuses in the social group, the more consistent the version and content of the information will be, which proves that the more adequate each individual information is, the slower and less likely rumors about COVID-19 spread. This consequence tells us that the government needs to guide the public to the truth. Announcing the true information publicly could instantly contain the COVID-19 rumor diffusion well rather than making them hidden or voiceless.

A methodology to quantify the risk propagation of hazardous events for ship grounding accidents based on directed CN

An innovative methodology is proposed to address the coupling and correlation involved in the hazardous events contributing to ship grounding accidents. Based on the principle of a directed complex network (CN), the advantages of event tree analysis (ETA), the improved K-shell algorithm and the susceptible infected recovered (SIR) model are integrated into this methodology. The ETA is first applied to extract event chains involved in each accident report (39 reports in this study), resulting in the development of a directed CN, which is analysed from two perspectives: criticality calculation with the static application of the improved K-shell decomposition algorithm and sensitivity analysis by simulating SIR model. The risk propagation capacity of each objective node is evaluated by a combination of criticality and sensitivity. The results show that the proposed methodology is applicable for evaluating the risk propagation capacity of hazardous events. The trigger events located at the beginning of event chains should not receive too much attention, while the control of contributing events is essential for accident prevention. In addition, the criticality and sensitivity of the objective node are not always consistent; therefore, the strategies defending the risk propagation process may vary according to the different classifications of hazardous events.

Calibration and prediction for the inexact SIR model.

A Susceptible Infective Recovered (SIR) model is usually unable to mimic the actual epidemiological system exactly. The reasons for this inaccuracy include observation errors and model discrepancies due to assumptions and simplifications made by the SIR model. Hence, this work proposes calibration and prediction methods for the SIR model with a one-time reported number of infected cases. Given that the observation errors of the reported data are assumed to be heteroscedastic, we propose two predictors to predict the actual epidemiological system by modeling the model discrepancy through a Gaussian Process model. One is the calibrated SIR model, and the other one is the discrepancy-corrected predictor, which integrates the calibrated SIR model with the Gaussian Process predictor to solve the model discrepancy. A wild bootstrap method quantifies the two predictors' uncertainty, while two numerical studies assess the performance of the proposed method. The numerical results show that, the proposed predictors outperform the existing ones and the prediction accuracy of the discrepancy-corrected predictor is improved by at least 49.95%.

Analysis of Covid 19 disease with SIR model and Taylor matrix method

<abstract><p>Covid 19 emerged in Wuhan, China in December 2019 has continued to spread by affecting the whole world. The pandemic has affected over 328 million people with more than 5 million deaths in over 200 countries which has severely disrupted the healthcare system and halted economies of the countries. The aim of this study is to discuss the numerical solution of the SIR model on the spread of Covid 19 by the Taylor matrix and collocation method for Turkey. Predicting COVID-19 through appropriate models can help us to understand the potential spread in the population so that appropriate action can be taken to prevent further transmission and prepare health systems for medical management of the disease. We deal with Susceptible–Infected–Recovered (SIR) model. One of the proposed model's improvements is to reflect the societal feedback on the disease and confinement features. We obtain the time dependent rate of transmission of the disease from susceptible $ \beta(t) $ and the rate of recovery from infectious to recovered $ \gamma $ using Turkey epidemic data. We apply the Taylor matrix and collocation method to the SIR model with $ \gamma $, $ \beta(t) $ and Covid 19 data of Turkey from the date of the first case March 11, 2020 through July 3, 2021. Using this method, we focus on the evolution of the Covid 19 in Turkey. We also show the estimates with the help of graphics and Maple.</p></abstract>

A Networked Competitive Multi-Virus SIR Model: Analysis and Observability

This paper proposes a novel discrete-time multi-virus SIR (susceptible-infected-recovered) model that captures the spread of competing SIR epidemics over a population network. First, we provide a sufficient condition for the infection level of all the viruses over the networked model to converge to zero in exponential time. Second, we propose an observation model which captures the summation of all the viruses’ infection levels in each node, which represents the individuals who are infected by different viruses but share similar symptoms. We present a sufficient condition for the model to be locally observable. We propose a Luenberger observer for the system state estimation and show via simulations that the estimation error of the Luenberger observer converges to zero before the viruses die out.

A physics-informed neural network to model COVID-19 infection and hospitalization scenarios

In this paper, we replace the standard numerical approach of estimating parameters in a mathematical model using numerical solvers for differential equations with a physics-informed neural network (PINN). This neural network requires a sequence of time instances as direct input of the network and the numbers of susceptibles, vaccinated, infected, hospitalized, and recovered individuals per time instance to learn certain parameters of the underlying model, which are used for the loss calculations.The established model is an extended susceptible-infected-recovered (SIR) model in which the transitions between disease-related population groups, called compartments, and the physical laws of epidemic transmission dynamics are expressed by a system of ordinary differential equations (ODEs). The system of ODEs and its time derivative are included in the residual loss function of the PINN in addition to the data error between the current network output and the time series data of the compartment sizes. Further, we illustrate how this PINN approach can also be used for differential equation-based models such as the proposed extended SIR model, called SVIHR model.In a validation process, we compare the performance of the PINN with results obtained with the numerical technique of non-standard finite differences (NSFD) in generating future COVID-19 scenarios based on the parameters identified by the PINN. The used training data set covers the time between the outbreak of the pandemic in Germany and the last week of the year 2021.We obtain a two-step or hybrid approach, as the PINN is then used to generate a future COVID-19 outbreak scenario describing a possibly next pandemic wave. The week at which the prediction starts is chosen in mid-April 2022.

A martingale formulation for stochastic compartmental susceptible-infected-recovered (SIR) models to analyze finite size effects in COVID-19 case studies

<p style='text-indent:20px;'>Deterministic compartmental models for infectious diseases give the mean behaviour of stochastic agent-based models. These models work well for counterfactual studies in which a fully mixed large-scale population is relevant. However, with finite size populations, chance variations may lead to significant departures from the mean. In real-life applications, <i>finite size effects</i> arise from the variance of individual realizations of an epidemic course about its fluid limit. In this article, we consider the classical stochastic Susceptible-Infected-Recovered (SIR) model, and derive a martingale formulation consisting of a deterministic and a stochastic component. The deterministic part coincides with the classical deterministic SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of <i>finite size effects</i>. Our theory is supported by quantitative and direct numerical simulations of theoretical infinitesimal variance. Case studies of coronavirus disease 2019 (COVID-19) transmission in smaller populations illustrate that the theory provides an envelope of possible outcomes that includes the field data.</p>

A death, infection, and recovery (DIR) model to forecast the COVID-19 spread

BackgroundThe SARS-Cov-2 virus (commonly known as COVID-19) has resulted in substantial casualties in many countries. The first case of COVID-19 was reported in China towards the end of 2019. Cases started to appear in several other countries (including Pakistan) by February 2020. To analyze the spreading pattern of the disease, several researchers used the Susceptible-Infectious-Recovered (SIR) model. However, the classical SIR model cannot predict the death rate. ObjectiveIn this article, we present a Death-Infection-Recovery (DIR) model to forecast the virus spread over a window of one (minimum) to fourteen (maximum) days. Our model captures the dynamic behavior of the virus and can assist authorities in making decisions on non-pharmaceutical interventions (NPI), like travel restrictions, lockdowns, etc. MethodThe size of training dataset used was 134 days. The Auto Regressive Integrated Moving Average (ARIMA) model was implemented using XLSTAT (add-in for Microsoft Excel), whereas the SIR and the proposed DIR model was implemented using python programming language. We compared the performance of DIR model with the SIR model and the ARIMA model by computing the Percentage Error and Mean Absolute Percentage Error (MAPE). ResultsExperimental results demonstrate that the maximum% error in predicting the number of deaths, infections, and recoveries for a period of fourteen days using the DIR model is only 2.33%, using ARIMA model is 10.03% and using SIR model is 53.07%. ConclusionThis percentage of error obtained in forecasting using DIR model is significantly less than the% error of the compared models. Moreover, the MAPE of the DIR model is sufficiently below the two compared models that indicates its effectiveness.

The approximation of an isolated epidemic process wave using a combination of exponents

The most commonly used methods for the medium- and long-term forecasting of epidemic processes are based on the classical SIR (susceptible – infected – recovered) model and its numerous modifications. In this approach, the dynamics of the epidemic is approximated using the solutions of differential or discrete equations. The forecasting methods based on the approximation of data by functions of a given class are usually focused on obtaining a short-term forecast. They are not used for the long-term forecasts of epidemic processes due to their insufficient efficiency for forecasting nonstationary processes. In this paper, we formulated a hypothesis that the primary waves of the COVID-19 pandemic, which took place in a number of European countries, including the Republic of Belarus, in the spring-summer of 2020 are isolated and therefore can be regarded as processes close to stationary. On the basis of this hypothesis, a method of approximating isolated epidemic process waves by means of generalized logistic functions with an increased number of exponents was proposed. The developed approach was applied to predict the number of infected people in the Republic of Belarus for the period until August 2020 based on data from the beginning of the epidemic until June 12, 2020.

An algorithm for the direct estimation of the parameters of the SIR epidemic model from the I(t) dynamics.

The discrete SIR (Susceptible–Infected–Recovered) model is used in many studies to model the evolution of epidemics. Here, we consider one of its dynamics—the exponential decrease in infected cases I(t). By considering only the I(t) dynamics, we extract three parameters: the exponent of the initial exponential increase gamma ; the maximum value I_{max }; and the exponent of the final decrease varGamma . From these three parameters, we show mathematically how to extract all relevant parameters of the SIR model. We test this procedure on numerical data and then apply the methodology to real data received from the COVID-19 situation in France. We conclude that, based on the hospitalized data and the ICU (Intensive Care Unit) cases, two exponentials are found, for the initial increase and the decrease in I(t). The parameters found are larger than reported in the literature, and they are associated with a susceptible population which is limited to a sub-sample of the total population. This may be due to the fact that the SIR model cannot be applied to the covid-19 case, due to its strong hypotheses such as mixing of all the population, or also to the fact that the parameters have changed over time, due to the political initiatives such as social distanciation and lockdown.

New and simple mathematical description of epidemics including consecutive waves

AbstractThe Bateman linear differential equations describing in time nuclear levels interconnected by transitions (in general, each with its own rate) are adapted to the epidemiology case where only two infection and recovery rates are used. Time‐dependent functions describing successive generations of infected people are thus derived. Closed mathematical expressions allow an easy calculation/prediction of the properties of these generations and therefore of the whole epidemics. The main limitation consists in reaching the point when the number of susceptible to the infection people substantially decreases. As testing ground, reasonable descriptions of the newly registered COVID‐19 cases per day is achieved for Europe and the United States till December 2020, including several so‐called Pandemic waves. The approach of the present work, by using transmission from one generation of infected people to a new generation allows a much more detailed tracking of the Pandemic spread than the approaches used so far. For example, the most widely used Susceptible Infectious Recovered (SIR) models (and their variants) use instead transitions from one big compartment of susceptible people to another new one of infected people followed by recovery leading to a third compartment. Clear correlations are found between features of the Pandemic process and the emergence of new SARS‐CoV‐2 variants, with some of them (especially the variant) immediately influencing the spread of the infection disease in the Pandemic wave of Autumn of 2020.

Epidemics on networks with preventive rewiring

AbstractA stochastic SIR (susceptible infective recovered) model is considered for the spread of an epidemic on a network, described initially by an Erdős–Rényi random graph, in which susceptible individuals connected to infectious neighbors may drop or rewire such connections. A novel construction of the model is used to derive a deterministic model for epidemics started with a positive fraction initially infected and prove convergence of the scaled stochastic model to that deterministic model as the population size . For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as , the final fraction infected is discontinuous in the infection rate at its threshold , thus not converging to 0 as . The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that jumps from 0 to 1 as passes through .

Epidemiological spread models of Parkinson’s disease support the prion‐like role of alpha‐synuclein

AbstractBackgroundParkinson’s disease (PD) is a progressive neurodegenerative condition marked by brain atrophy in multiple cortical and subcortical regions. It is now known to be caused by the prion‐like propagation of misfolded alpha‐synuclein, supporting Braak’s hypothesis. The pattern of neurodegeneration is now thought to be explained by a combination of brain connectivity and local vulnerability.MethodWe used MRI and clinical data from the Parkinson Progression Markers Initiative to map the progression of brain atrophy over one, two and four years and related it to brain functional and structural connectivity, cell type expression and gene ontology enrichment analyses. We use two methods to measure atoprhy: cortical thickness measures and deformation based morphometry. We will compare these results to pathology in wild‐type mice sacrificed at different time‐points after injection of alpha‐synuclein preformed fibrils. In both cases we use a spreading model inspired by infectious‐disease epidemiology to map the propagation of misfolded alpha‐synuclein. This Susceptible‐Infectious‐Recovered (SIR) model assumes that alpha‐synuclein molecules become misfolded, propagate along the connectome, and cause tissue damage when they accumulate.Result1) The pattern of brain atrophy in de novo PD is compatible with a propagating process with an epicenter in the substantia nigra. 2) The SIR model recapitulates the observed atrophy pattern, and supports connectivity‐based propagation. 3) In addition, the SIR model suggests that brain areas with higher alpha‐synuclein concentration appear to be more susceptible to injury but also more likely to act as disease propagators. 4) Progression over four years is also explained by connectivity. 5) Gene ontology analyses suggest that synapses are especially targeted by neurodegeneration. 6) Studies in mice, where the injection site of alpha‐synuclein preformed fibrils is determined by the investigator, also confirm the connectivity by local vulnerability model.ConclusionHuman and animal studies support the Braak hypothesis of a “toxic agent” propagating from the brainstem to the cortex, accounting for the stereotyped progression of PD manifestations. Data from gene expression studies and animal experiments confirm the important role of prion‐like alpha‐synuclein in PD pathogenesis.

Epidemic Spreading in a Social Network With Facial Masks Wearing Individuals

In this paper, we present a susceptible-infected-recovered (SIR) model with individuals wearing facial masks and individuals who do not. The disease transmission rates, the recovering rates and the fraction of individuals who wear masks are all time dependent in the model. We develop a progressive estimation of the disease transmission rates and the recovering rates based on the COVID-19 data published by John Hopkins University. We determine the fraction of individual who wear masks by a maximum likelihood estimation, which maximizes the transition probability of a stochastic susceptible-infected-recovered model. The transition probability is numerically difficult to compute if the number of infected individuals is large. We develop an approximation for the transition probability based on central limit theorem and mean field approximation. We show through numerical study that our approximation works well. We develop a bond percolation analysis to predict the eventual fraction of population who are infected, assuming that parameters of the SIR model do not change anymore. We predict the outcome of COVID-19 pandemic using our theory.

2SIR-VD Model to Compare Idealized COVID-19 Vaccine Distribution Strategies in the Philippines

COVID-19 is a novel respiratory disease first identified in Wuhan, China, that is caused by the novel coronavirus, SARS-CoV-2. It has triggered a global pandemic of historic proportions. The government of the Philippines began its national vaccine drive on 01 Mar 2021, with the goal of vaccinating 70 million of its citizens by the end of the calendar year. To determine the optimum geographical distribution strategy in the Philippines for the limited supply of vaccines that is currently available, we developed and adapted a basic SIR (susceptible-infected-recovered) model that allows us to understand the evolution of a pandemic when public health authorities are vaccinating two susceptible populations within a country with different vaccine rates. Our analysis with our 2SIR-VD (two-population susceptible-infected-recovered-vaccinated-deceased) model of an idealized pandemic scenario revealed that prioritizing vaccine deployment to the National Capital Region (NCR) of the Philippines minimized the number of COVID-19 cases in the country. We, therefore, recommend deploying 80–90% of the available vaccine supply to the NCR to mitigate viral transmission there. The remaining doses would allow the rest of the archipelago to vaccinate all of their medical frontliners, senior citizens, and adults with comorbidities – thus shielding this vulnerable population against severe disease and death from COVID-19.

Model with transmission delays for COVID-19 control: Theory and empirical assessment.

The paper focuses on modeling of public health measures to control the COVID‐19 pandemic. The authors suggest a flexible integral model with distributed lags, which realistically describes COVID‐19 infectiousness period from clinical data. It contains susceptible–infectious–recovered (SIR), susceptible–exposed–infectious–recovered (SEIR), and other epidemic models as special cases. The model is used for assessing how government decisions to lockdown and reopen the economy affect epidemic spread. The authors demonstrate essential differences in transition and asymptotic dynamics of the integral model and the SIR model after lockdown. The provided simulation on real data accurately describes several waves of the COVID‐19 epidemic in the United States and is in good correspondence with government actions to curb the epidemic.