Steady State Solution and Stability of an Age-Structured MSIQR Epidemic Model

The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed transmission dynamics of epidemic diseases. However, the integration of the quarantine policy and age-structure is less addressed. In this work we propose an age-structured MSIQR (temporarily immune-susceptibles-infectives-quarantined-removed) model to study the impact of quarantine policies on the spread of epidemic diseases. Specifically, we investigate the existence of steady state solutions and stability property of the proposed model. The derived explicit expression of the basic reproductive number shows that the disease-free equilibrium is globally asymptotically stable if, and that the unique endemic equilibrium exists if. In addition, the stability conditions of the endemic equilibrium are derived.

Stability of a Delayed Stochastic Epidemic COVID-19 Model with Vaccination and with Differential Susceptibility

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 R0 ≤ 1, we prove the almost sure asymptotic stability of the disease-free equilibrium of the model.

GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DELAYS

In this article, an SIRS epidemic model spread by vectors （mosquitoes） which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point （if it exists） is globally stable with a respect ＂weak delay＂. Some known results are generalized.

GLOBAL STABILITY OF EXTENDED MULTI-GROUP SIR EPIDEMIC MODELS WITH PATCHES THROUGH MIGRATION AND CROSS PATCH INFECTION

In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch infection between different groups. As a result, we partially generalize the recent result in the article [16].

GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci- dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.

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.

Study on the Dynamics of an SIR Epidemic Model with Saturated Growth Rate

In this paper, we investigate the dynamic properties of an SIR epidemic model with saturated growth rate. Under the conditions of an arbitrary initial value, we prove that the system exists unique positive solution, and give the sufficient conditions caused by random environmental factors leading to the extinction of infectious diseases. Moreover, we verify the conditions for the persistence of infectious diseases in the mean sense. Finally, we provide the biology interpretation and some strategies to control the infectious diseases.

Extinction and Stationary Distribution of a Stochastic SIR Epidemic Model with Jumps

A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a unique global positive solution;(2) the results reveal that the solution of this epidemic model will be stochastically ultimately bounded,and the non-linear SDE admits a unique stationary distribution under certain parametric conditions;(3) the coefficients play an important role in the extinction of the diseases.

Stability Analysis of an SIR Epidemic Model with Non-Linear Incidence Rate and Treatment

We consider a SIR epidemic model with saturated incidence rate and treatment. We show that if the basic reproduction number, R0 is less than unity and the disease free equilibrium is locally asymptotically stable. Moreover, we show that if R0 > 1, the endemic equilibrium is locally asymptotically stable. In the end, we give some numerical results to compare our model with existing model and to show the effect of the treatment term on the model.

The Effect of State-Dependent Control for an SIRS Epidemic Model with Varying Total Population

Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.

A cellular automata model of epidemics of a heterogeneous susceptibility

In this paper we present a model with spatial heterogeneity based on cellular automata （CA）. In the model we consider the relevant heterogeneity of host （susceptible） mixing and the natural birth rate. We divide the susceptible population into three groups according to the immunity of each individual based on the classical susceptible-infectedremoved （SIR） epidemic models, and consider the spread of an infectious disease transmitted by direct contact among humans and vectors that have not an incubation period to become infectious. We test the local stability and instability of the disease-free equilibrium by the spectrum radii of Jacobian. The simulation shows that the structure of the nearest neighbour size of the cell （or the degree of the scale-free networks） plays a very important role in the spread properties of infectious disease. The positive equilibrium of the infections versus the neighbour size follows the third power law if an endemic equilibrium point exists. Finally, we analyse the feature of the infection waves for the homogeneity and heterogeneous cases respectively.

Simulation of Spread of Infectious Diseases and Population Mobility in a Deterministic Epidemic Patch Model

Computer simulation models are widely applied in various areas of the health care sector, including the spread of infectious diseases. Patch models involve explicit movements of people between distinct locations. The aim of the present work has been designed and explored a patch model with population mobility between different patches and between each patch and an external population. The authors considered a SIR （susceptible-infected-recovered） scheme. The model was explored by computer simulations. The results show how endemic levels are reached in all patches of the system. Furthermore, the performed explorations suggest that the people mobility between patches, the immigration from outside the system and the infection rate in each patch, are factors that may influence the dynamics of epidemics and should be considered in health policy planning.

Asymptotic Behavior of a Stochastic SIRS Model with Non-linear Incidence and Levy Jumps

A stochastic susceptible-infective-recovered-susceptible( SIRS) model with non-linear incidence and Levy jumps was considered. Under certain conditions, the SIRS had a global positive solution. The stochastically ultimate boundedness of the solution of the model was obtained by using the method of Lyapunov function and the generalized Ito's formula. At last,asymptotic behaviors of the solution were discussed according to the value of R0. If R0< 1,the solution of the model oscillates around a steady state, which is the diseases free equilibrium of the corresponding deterministic model,and if R0> 1,it fluctuates around the endemic equilibrium of the deterministic model.

SIRS epidemic modeling using fractional-ordered differential equations:Role of fear effect

In this paper,an SIRS epidemic model using Grunwald-Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases.The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number.The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter.Additionally,using the Grunwald-Letnikov approximation,the simulation is carried out to confirm the validity of analytic results graphically.Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021,we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one.Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases.

EVD Outbreak-2014: Burden of the Disease and Comprehensive Steps in Preventing the Epidemic

As of December 2nd 2015, a total of 28,601 cases have been reported with 11,300 reported deaths (not including cases where the outcome is unknown) during the current outbreak of ebola virus disease (EVD). In this paper, we mainly focus on the transmission of ebola virus disease, estimate the burden of the disease and the persistent nature by finding the basic reproduction number, and analyze the comprehensive steps to control the virulent disease. We have considered three mostly affected countries, Guinea, Liberia, and Sierra Leone respectively and collected data from various sources are used to surmise the present and future nature of the disease. Being the poorest country in the world like Guinea, Liberia, and Sierra Leone, they do not have efficient policies to fight against such kind of endemic and they have been unable to control the spread of the disease. We have found some real facts that increase the chances to be infected by this virulent disease. Since the reproduction number R0 is still above unity and if we do not take precautionary steps and work accordingly then the disease will definitely exist in the community and the burden of the disease would increase continuously.

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.

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.

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.

Analytic algorithms for Some Models of Nonlinear Age–Structured Population Dynamics and Epidemiology

Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncating the resulting convergent infinite series, we obtain numerical solutions of high accuracy for these models. Three numerical examples are given to illustrate the simplicity and accuracy of the methods.

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.