A Restricted SIR Model with Vaccination Effect for the Epidemic Outbreaks Concerning COVID-19

This paper presents a restricted SIRmathematicalmodel to analyze the evolution of a contagious infectious disease outbreak(COVID-19)using available data.The new model focuses on two main concepts:first,it can present multiple waves of the disease,and second,it analyzes how far an infection can be eradicated with the help of vaccination.The stability analysis of the equilibrium points for the suggested model is initially investigated by identifying the matching equilibrium points and examining their stability.The basic reproduction number is calculated,and the positivity of the solutions is established.Numerical simulations are performed to determine if it is multipeak and evaluate vaccination’s effects.In addition,the proposed model is compared to the literature already published and the effectiveness of vaccination has been recorded.

Dynamics of a Nonautonomous SIR Model with Time-Varying Impulsive Release and General Nonlinear Incidence Rate in a Polluted Environment

In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.

Analysis and Prediction of the COVID-19 Pandemic in Senegal Using the SIR Model

In this study, the mathematical SIR model (Susceptible-Infected-Recovered (cured and deceased)) was applied to the case of Senegal during the first two waves of the COVID-19 pandemic. During this period, from March 1, 2020, to March 30, 2021, the transmission and recovery rates as well as the number of reproduction were calculated and analyzed for the impact of the decisions taken by the Senegalese government. In both waves, the variation of the basic reproduction number as a function of time, with values below one towards the end of each study period, confirms the success of the Senegalese government in controlling the epidemic. The results show that the solution of mandatory mask-wearing is the best decision to counter the spread of the disease. Indeed, the mean number of reproduction is 2.11 in the first wave, and the second wave has a lower mean value of 1.23, while the decisions are less restrictive during this latter wave. Also, a short-term prediction model (about 4 months) was validated on the second wave. The validation criteria of this model reveal a good match between the results of the simulated model and the COVID-19 data reported via the Ministry of Health, Solidarity, and Social Action of Senegal.

Research and Establishment of SIR Model Based on COVID-19

In the context of the COVID-19 sweeping the world,countries around the globe have adopted different approaches to control the spread of disease,and in order to find better control methods,this paper explores the influence of people’s awareness on SIR model.On the basis of the SIR model,this paper studies the SEIR model with the exposure period parameter,calculates the feasible region R-naught disease-free point,and analyzes the method of controlling the spread of the disease according to R-naught,which shows that lockdown has a significant effect on the control of COVID-19.In addition,this paper also established a model affected by disease awareness,adding a factor of news media and religious awareness.The feasible region is calculated,and the reality situation based on India is analyzed.The conclusion proved that people’s awareness has a greater influence on the spread of diseases.

A Visual Analysis and Prediction System for Infectious Diseases Based on Improved SIR Model

To effectively track the impact of population migration between regions on the spread of infectious diseases, this paper proposes a visualized analysis and prediction system of infectious diseases based on the improved SIR model. The research contents including: using the multi graph link interaction mode, visualizing the space-time distribution and development trend of infectious diseases;The LightGBM model is used to track the changes of infection rate and recovery rate, and the Mi/Mo SIR model is constructed according to the initial data of different populations;Mi/Mo SIR model is used to predict infectious diseases in combination with visual panel, providing users with tools to analyze and explain the space-time characteristics and potential laws of infectious diseases. The study found that the closure of cities and the restriction of personnel mobility were necessary and effective, and the system provided an important basis for the prediction and early warning of infectious diseases.

Epidemic Propagation: An Automaton Model as the Continuous SIR Model

The use of the SIR model to predict the time evolution of an epidemic is very frequent and has spatial information about its propagation which may be very useful to contrast its spread. In this paper we take a particular cellular automaton model that well reproduces the time evolution of the disease given by the SIR model;setting the automaton is generally an annoying problem because we need to run a lot of simulations, compare them to the solution of the SIR model and, finally, decide the parameters to use. In order to make this procedure easier, we will show a fast method that, in input, requires the parameters of the SIR continuous model that we want to reproduce, whereas, in output, it yields the parameters to use in the cellular automaton model. The problem of computing the most suitable parameters for the reticular model is reduced to the problem of finding the roots of a polynomial Equation.

SIR Model of Spread of Zika Virus Infections: ZIKV Linked to Microcephaly Simulations

An SIR model of Zika virus (ZIKV) spread is formulated that includes ZIKV infections to newborns. Analytically, the model has one disease free and one endemic equilibrium point. The free one is stable for some conditions when R0 and unstable when R0>1. In Brazil, when R0≈2>1 ZIKV infections expand and for R0 = 0.875R0) of the model. There are parameters for human-mosquito transmission and some for sexual-transmission factor. It appears that controlling spread of ZIKV infections by human-mosquito transmission may greatly reduce the value of R0.

Analysis of a Delayed SIR Model with Exponential Birth and Saturated Incidence Rate

In this paper, a delayed SIR model with exponential demographic structure and the saturated incidence rate is formulated. The stability of the equilibria is analyzed with delay: the endemic equilibrium is locally stable without delay;and the endemic equilibrium is stable if the delay is under some condition. Moreover the dynamical behaviors from stability to instability will change with an appropriate?critical value. At last, some numerical simulations of the model are given to illustrate the main theoretical results.

Delayed Dynamics of SIR Model for COVID-19

This paper presents a new modified SIR model which incorporates appropriate delay parameters leading to a more precise prediction of COVID-19 real time data. The efficacy of the newly developed SIR model is proven by comparing its predictions to real data obtained from four counties namely Germany, Italy, Kuwait, and Oman. Two included delay periods for incubation and recovery within the SIR model produce a sensible and more accurate representation of the real time data. In the absence of the two-delay period () the dynamical behavior of the model will not correspond to today’s picture and lag the detection of the epidemic peak. The reproductive number R0 is defined for the model for values of recovery time delay of the infective case. The effect of recovery time may produce second wave, and/or an oscillation which could destabilize the behavior of the system and a periodic oscillation can arise due to Hopf bifurcation phenomenon.

An Analytic Approximate Solution of the SIR Model

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

Asymptotic Behavior and Stability of Stochastic SIR Model with Variable Diffusion Rates

In this paper, we propose random fluctuation on contact and recovery rates in deterministic SIR model with disease deaths in nonparametric manner and derive a new stochastic SIR model with distributed time delay and general diffusion coefficients. By analysis of the introduced model, we obtain the sufficient conditions for the regularity, existence and uniqueness of a global solution by means of Lyapunov function. Moreover, we also investigate the stochastic asymptotic stability of disease free equilibria and endemic equilibria of this model. Finally, we illustrate our general results by applications.

Identifying Critical Parameters in SIR Model for Spread of Disease

Calculating analytical approximate solutions for non-linear infectious disease models is a difficult task. Such models often require computational tools to analyse analytical approximate methods which appear in some theoretical and practical applications in systems biology. They represent key critical elements and give some approximate solutions for such systems. The SIR epidemic disease model is given as the non-linear system of ODE’s. Then, we use a proper scaling to reduce the number of parameters. We suggest Elzaki transform method to find analytical approximate solutions for the simplified model. The technique plays an important role in calculating the analytical approximate solutions. The local and global dynamics of the model are also studied. An investigation of the behaviour at infinity was conducted via finding the lines and singular points at infinity. Model dynamic results are computed in numerical simulations using Pplane8 and SimBiology Toolbox for Mathlab. Results provide a good step forward for describing the model dynamics. More interestingly, the simplified model could be accurate, robust, and used by biologists for different purposes such as identifying critical model elements.

基于SIR模型的城市路网拥堵传播分析

研究城市道路交通拥堵传播规律对缓解交通拥堵问题有着积极作用,为此建立了基于SIR的城市道路交通拥堵传播模型,用以分析城市道路交通拥堵传播过程。首先,基于城市实际路网构建路网对偶拓扑网络,并依据SIR建立交通拥堵传播模型。其次,结合道路网络的复杂网络特征和道路自身的相关属性,引入随机森林算法计算相关权重,确定拥堵模型中的传播速率等关键参数。最后,以南京市秦淮区某区域路网为例,构建有69个节点,163条连线的城市路网对偶拓扑网络进行仿真实验。结果表明:道路节点度和道路饱和度是影响道路拥堵传播的关键因素,道路节点度的影响相对较小,传播范围增长在5%以内,恢复时间影响在10%左右;道路饱和度的影响相对较大,随着道路饱和度的增长,传播范围增长最大可至40%,恢复时间影响在20%左右。

A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19

Background:Under-reporting and,thus,uncertainty around the true incidence of health events is common in all public health reporting systems.While the problem of underreporting is acknowledged in epidemiology,the guidance and methods available for assessing and correcting the resulting bias are obscure.Objective:We aim to design a simple modification to the Susceptible e Infected e Removed(SIR)model for estimating the fraction or proportion of reported infection cases.Methods:The suggested modification involves rescaling of the classical SIR model producing its mathematically equivalent version with explicit dependence on the reporting parameter(true proportion of cases reported).We justify the rescaling using the phase plane analysis of the SIR model system and show how this rescaling parameter can be estimated from the data along with the other model parameters.Results:We demonstrate how the proposed method is cross-validated using simulated data with known disease cases and then apply it to two empirical reported data sets to estimate the fraction of reported cases in Missoula County,Montana,USA,using:(1)flu data for 2016e2017 and(2)COVID-19 data for fall of 2020.Conclusions:We establish with the simulated and COVID-19 data that when most of the disease cases are presumed reported,the value of the additional reporting parameter in the modified SIR model is close or equal to one,so that the original SIR model is appropriate for data analysis.Conversely,the flu example shows that when the reporting parameter is close to zero,the original SIR model is not accurately estimating the usual rate parameters,and the re-scaled SIR model should be used.This research demonstrates the role of under-reporting of disease data and the importance of accounting for underreporting when modeling simulated,endemic,and pandemic disease data.Correctly reporting the“true”number of disease cases will have downstream impacts on predictions of disease dynamics.A simple parameter adjustment to the SIR modeling framework can help alleviate bias and uncertainty around crucial epidemiological metrics(e.g.:basic disease reproduction number)and public health decision making.

Location of emergency rescue center based on SIR epidemiological model

In view of the pressure time of emergency rescue against the infectious diseases,a mathematical model to optimize the location of emergency rescue centers is proposed.The model takes full account of the spread function of infectious diseases,the cycle of pulse vaccination,the distance between the demand area and the emergency rescue centers,as well as the building and maintenance cost for the emergency rescue center,and so on.At the same time,the model integrates the traditional location selection models which are the biggest cover model,the p-center model and the p-median model,and it embodies the principles of fairness and efficiency for the emergency center location.Finally,a computation of an example arising from practice provides satisfactory results.

DYNAMICS FOR AN SIR EPIDEMIC MODEL WITH NONLOCAL DIFFUSION AND FREE BOUNDARIES

This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.

Revisiting classical SIR modelling in light of the COVID-19 pandemic

Background:Classical infectious disease models during epidemics have widespread usage,from predicting the probability of new infections to developing vaccination plans for informing policy decisions and public health responses.However,it is important to correctly classify reported data and understand how this impacts estimation of model parameters.The COVID-19 pandemic has provided an abundant amount of data that allow for thorough testing of disease modelling assumptions,as well as how we think about classical infectious disease modelling paradigms.Objective:We aim to assess the appropriateness of model parameter estimates and preiction results in classical infectious disease compartmental modelling frameworks given available data types(infected,active,quarantined,and recovered cases)for situations where just one data type is available to fit the model.Our main focus is on how model prediction results are dependent on data being assigned to the right model compartment.Methods:We first use simulated data to explore parameter reliability and prediction capability with three formulations of the classical Susceptible-Infected-Removed(SIR)modelling framework.We then explore two applications with reported data to assess which data and models are sufficient for reliable model parameter estimation and prediction accuracy:a classical influenza outbreak in a boarding school in England and COVID-19 data from the fall of 2020 in Missoula County,Montana,USA.Results:We demonstrated the magnitude of parameter estimation errors and subsequent prediction errors resulting from data misclassification to model compartments with simulated data.We showed that prediction accuracy in each formulation of the classical disease modelling framework was largely determined by correct data classification versus misclassification.Using a classical example of influenza epidemics in an England boarding school,we argue that the Susceptible-Infected-Quarantined-Recovered(SIQR)model is more appropriate than the commonly employed SIR model given the data collected(number of active cases).Similarly,we show in the COVID-19 disease model example that reported active cases could be used inappropriately in the SIR modelling framework if treated as infected.Conclusions:We demonstrate the role of misclassification of disease data and thus the importance of correctly classifying reported data to the proper compartment using both simulated and real data.For both a classical influenza data set and a COVID-19 case data set,we demonstrate the implications of using the“right”data in the“wrong”model.The importance of correctly classifying reported data will have downstream impacts on predictions of number of infections,as well as minimal vaccination requirements.

Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated.

A stochastic SIR model for analysis of testosterone suppression of CRH-stimulated cortisol in men

A stochastic SIR influenza vertical transmission model is examined in this paper where vaccination and an incidence rate that is not linear are considered.To determine whether testosterone regulates lower sintering HPA axis function in males,we used a stochastic SIR epidemic procedure with divergent influences on ACTH and cortisol.The suppressive effects on cortisol can be attributed to a peripheral(adrenal)locus.Following that,we came to the conclusion that experimental solutions have been discovered and the requisite statistical findings have been examined.Finally,we deduce that the given mathematical model and the results are relevant to medical research.In the future,this research can be further extended to simulate more results in the medical field.

SIR model with time-varying contact rate

The contact rate is defined as the average number of contacts adequate for disease transmission by an individual per unit time and it is usually assumed to be constant in time.However,in reality,the contact rate is not always constant throughout the year due to different factors such as population behavior,environmental factors and many others.In the case of serious diseases with a high level of infection,the population tends to reduce their contacts in the hope of reducing the risk of infection.Therofore,it is more realistic to consider it to be a function of time.In particular,the study of models with contact rates decreasing in time is well worth exploring.In this paper,an SIR model with a time-varying contact rate is considered.A new form of a contact rate that decreases in time from its initial value till it reaches a certain level and then remains constant is proposed.The proposed form includes two important parameters,which represent how far and how fast the contact rate is reduced.These two parameters are found to play important roles in disease dynamics.The existence and local stability of the equilibria of the model are analyzed.Results on the global stability of disease-free equilibrium and transcritical bifurcation are proved.Numerical simulations are presented to illustrate the theoretical results and to demonstrate the effect of the model parameters related to the behavior of the contact rate on the model dynamics.Finally,comparisons between the constant,variable contact rate and variable contact rate with delay in response cases are presented.