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 ...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.展开更多
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 relea...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.展开更多
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,...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.展开更多
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 influ...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.展开更多
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 improve...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.展开更多
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 particu...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.展开更多
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 conditi...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.展开更多
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 stab...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.展开更多
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 ...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 (<img src="Edit_8ce6d5c5-9b59-4640-9c0e-334e3948d11c.png" width="67" height="20" alt="" /><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">)</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the dynamical behavior of the model will not correspond to today’s picture and lag the detection of the epidemic peak. The reproductive number <i></i></span></span></span><i><span><span><i><span style="font-family:Verdana;">R</span></i></span></span><span><span><span style="font-family:;" "=""><i><span style="font-family:Verdana;"><sub>0</sub></span></i><span style="font-family:Verdana;"></span></span></span></span></i> <span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">is defined for the model for values of recovery time delay <i></i></span></span></span><i><span style="font-family:Verdana;"><span style="font-family:Verdana;"><i><span style="font-family:Verdana;"><img src="Edit_882b068a-f7fa-478e-9fb9-4d78388010f3.png" width="25" height="20" alt="" /></span></i></span></span></i><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><sub></sub></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> of the infective case. The effect of recovery time <img src="Edit_882b068a-f7fa-478e-9fb9-4d78388010f3.png" width="25" height="20" alt="" /></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">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.</span></span></span>展开更多
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...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 <em>exact</em> (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 <em>exact</em> 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 <em>basic reproduction ratio</em>, 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.展开更多
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 g...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.展开更多
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 t...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.展开更多
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 ...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.展开更多
In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied....In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied.The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters.We discuss in detail some codimension-one bifurcations such as transcritical,period-doubling and Neimark–Sacker bifurcations,and a codimension-two bifurcation with 1:2 resonance.In addition,the phase portraits,bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis.It is found that the numerical results are consistent with the theoretical analysis.More interestingly,we also found other bifurcations in the model during the numerical simulation,such as codimension-two bifurcations with 1:1 resonance,1:3 resonance and 1:4 resonance,generalized period-doubling and fold-flip bifurcations.The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model.Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases,but also make a great contribution to the prevention and control of infectious diseases.展开更多
In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start wi...In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start with a deterministic model, then add random perturbations on the contact rate using white noise to obtain a stochastic model. We first show that the delayed stochastic differential equation that describes the model has a unique global positive solution for any positive initial value. Under the condition R<sub>0</sub> ≤ 1, we prove the almost sure asymptotic stability of the disease-free equilibrium of the model.展开更多
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 fre...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.展开更多
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 HP...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.展开更多
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 no...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.展开更多
基金Research Partnership Program no.RP-21-09-06 from the Deanship of Scientific Research of Imam Mohammad Ibn Saud Islamic University(IMSIU).
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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 (<img src="Edit_8ce6d5c5-9b59-4640-9c0e-334e3948d11c.png" width="67" height="20" alt="" /><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">)</span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> the dynamical behavior of the model will not correspond to today’s picture and lag the detection of the epidemic peak. The reproductive number <i></i></span></span></span><i><span><span><i><span style="font-family:Verdana;">R</span></i></span></span><span><span><span style="font-family:;" "=""><i><span style="font-family:Verdana;"><sub>0</sub></span></i><span style="font-family:Verdana;"></span></span></span></span></i> <span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">is defined for the model for values of recovery time delay <i></i></span></span></span><i><span style="font-family:Verdana;"><span style="font-family:Verdana;"><i><span style="font-family:Verdana;"><img src="Edit_882b068a-f7fa-478e-9fb9-4d78388010f3.png" width="25" height="20" alt="" /></span></i></span></span></i><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><sub></sub></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> of the infective case. The effect of recovery time <img src="Edit_882b068a-f7fa-478e-9fb9-4d78388010f3.png" width="25" height="20" alt="" /></span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">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.</span></span></span>
文摘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 <em>exact</em> (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 <em>exact</em> 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 <em>basic reproduction ratio</em>, 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.
文摘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.
文摘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.
基金supported by National Institute of General Medical Sciences of the National Institutes of Health,United States(Award Numbers P20GM130418,U54GM104944).
文摘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.
基金supported by the NSF of Shandong Province(ZR2021MA016,ZR2019MA034,ZR2018BF018)the China Postdoctoral Science Foundation(2019M652349)the Youth Creative Team Sci-Tech Program of Shandong Universities(2019KJI007).
文摘In this paper,a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method.The existence and stability of fixed points in this model are well studied.The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters.We discuss in detail some codimension-one bifurcations such as transcritical,period-doubling and Neimark–Sacker bifurcations,and a codimension-two bifurcation with 1:2 resonance.In addition,the phase portraits,bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis.It is found that the numerical results are consistent with the theoretical analysis.More interestingly,we also found other bifurcations in the model during the numerical simulation,such as codimension-two bifurcations with 1:1 resonance,1:3 resonance and 1:4 resonance,generalized period-doubling and fold-flip bifurcations.The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model.Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases,but also make a great contribution to the prevention and control of infectious diseases.
文摘In this paper, we treat the spread of COVID-19 using a delayed stochastic SVIRS (Susceptible, Infected, Recovered, Susceptible) epidemic model with a general incidence rate and differential susceptibility. We start with a deterministic model, then add random perturbations on the contact rate using white noise to obtain a stochastic model. We first show that the delayed stochastic differential equation that describes the model has a unique global positive solution for any positive initial value. Under the condition R<sub>0</sub> ≤ 1, we prove the almost sure asymptotic stability of the disease-free equilibrium of the model.
基金Zhao was supported by a scholarship from the China Scholarship Council,Li was partially supported by NSF of China(11731005)Cao was partially supported by NSF of China(11901264).
文摘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.
文摘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.
基金Authors would like to acknowledge financial support from Sultan Qaboos University,Oman and United Arab Emirates University through the Joint Research Grant No.CL/SQU-UAEU/17/01.
文摘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.