STOCHASTIC SIRS MODEL DRIVEN BY LVY NOISE

The paper establishes two stochastic SIRS models with jumps to describe the spread of network virus by cyber war, terrorism and others. First, adding random perturbations proportionally to each variable, we get the dynamic properties around the positive equilibrium of the deterministic model and the conditions for persistence and extinction. Second, giving a random disturbance to endemic equilibrium, we get a stochastic system with jumps. By modifying the existing Lyapunov function, we prove the positive solution of the system is stochastically stable.

Global Asymptotic Stability of a Kind of Stochastic SIRS Model

A detailed analysis was carried out on global asymptotic behavior of a kind of stochastic SIRS(susceptible-infective-removed-susceptible)model.This model has been obtained by introducing stochasticity into the original deterministic SIRS model via the technique of parameter perturbation which is standard in stochastic population modeling.By making corresponding Lyapunov function and using It formula,the condition for the solution of the model tending to the disease free equilibrium asymptotically was obtained.Under this condition,the epidemics will die out as time goes by.Based on this,almost surely exponential stability was analyzed.

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.

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.

TRAVELING WAVES FOR A SIRS MODEL WITH NONLOCAL DIFFUSION

In this paper, we study a delayed SIRS model with nonlocal diffusion. The well posedness of the model is investigated. Furthermore, we concern with the problem of traveling wave solutions. By using the partial quasi-monotone condition, cross-iteration scheme and fixed-point theorem, sufficient conditions are derived for the existence of traveling waves connecting the two equilibria which depends on the existence of a pair of upper solution and lower solution. We in fact construct a pair of upper solution and lower solution concretely to guarantee the existence of traveling waves.

Frac tai dimension and synchronization of the controlled Julia sets of the SIRS model

It is of crucial significance to study the infectious disease phenomenon by using the SIRS model and thoughts of Julia set.In this paper,Julia set of the discrete version of the SIRS model is established to analyze the fractal dynamics of this model.Then,controller is designed to change the Julia set.Furthermore,the box-counting dimensions of the controlled Julia sets by selecting different appropriate parameters are computed to show the complexity of the model.Finally,a nonlinear coupling method is introduced to synchronize the Julia sets with different parameters of the same system.Simulation results show the efficacy of t hese met hods.

Dynamics of an SIRS model with age structure and two delays

ln this paper,we propose and investigate an SlRS model with age structure and twodelays.Both the infected and the recovered individuals have age structure,the infectionrate(from the infective to the susceptible)and the immune loss rate(from the recoveredto the susceptible)are related to two independent time delays,respectively.We provethat the proposed age structured SIRS model is well-posed by using the Co-semigrouptheory.The basic reproduction number Ro is given,and the unique endemic equilib-rium exists when R_(0)>1,while the disease-free equilibrium always exists.A rigorousmathematical analysis for the stability of two equilibria is provided.The disease-freeequilibrium is local asymptotically stable if R_(0)<1,and the endemic equilibrium is localasymptotically stable if R_(0)>1 and τl=0.Finally,we give numerical simulations toverify our results.

Hopf Bifurcation Analysis for a Delayed SIRS Epidemic Model with a Nonlinear Incidence Rate

This paper is concerned with a delayed SIRS epidemic model with a nonlinear incidence rate. The main results are given in terms of local stability and Hopf bifurcation. Sufficient conditions for the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained by regarding the time delay as the bifurcation parameter. Further,the properties of Hopf bifurcation such as the direction and stability are investigated by using the normal form theory and center manifold argument. Finally,some numerical simulations are presented to verify the theoretical analysis.

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 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.

Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay

This paper is mainly concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIRS model with nonlocal delayed transmissions. We find that the existence and nonexistence of traveling wave solutions are determined by the critical wave speed c^＊. More specifically, we establish the existence of traveling wave solutions for every wave speed c〉c^＊ and R0 〉 1 by means of upper-lower solutions and Schauder＇s fixed point theorem. Nonexistence of traveling wave solutions is obtained by Laplace transform for any wave speed c ∈ （0, c^＊） and R0 〉 1.

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.

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.

Modeling of Computer Virus Propagation with Fuzzy Parameters

Typically,a computer has infectivity as soon as it is infected.It is a reality that no antivirus programming can identify and eliminate all kinds of viruses,suggesting that infections would persevere on the Internet.To understand the dynamics of the virus propagation in a better way,a computer virus spread model with fuzzy parameters is presented in this work.It is assumed that all infected computers do not have the same contribution to the virus transmission process and each computer has a different degree of infectivity,which depends on the quantity of virus.Considering this,the parametersβandγbeing functions of the computer virus load,are considered fuzzy numbers.Using fuzzy theory helps us understand the spread of computer viruses more realistically as these parameters have fixed values in classical models.The essential features of the model,like reproduction number and equilibrium analysis,are discussed in fuzzy senses.Moreover,with fuzziness,two numerical methods,the forward Euler technique,and a nonstandard finite difference(NSFD)scheme,respectively,are developed and analyzed.In the evidence of the numerical simulations,the proposed NSFD method preserves the main features of the dynamic system.It can be considered a reliable tool to predict such types of solutions.

Modeling the Infection Disease (Covid-19) and the Effect of Vaccination

In this paper we provide different types of approach in mathematical biology about infection disease and understanding the dynamic of epidemic mathematical models specially in COVID-19 disease which first outbroke in China and fast spread around the world. We work in the connection between the mathematical models and the solution analytically and numerically. At first, we emphasize the Susceptible-Infectious-Recovered (SIR) models’ extension for policy significance. Then, we found the improved SIER model done by research. In third section, we examine the improved model when an appropriate vaccine has been found, we introduce the model of SIR with vaccine term which ends up with discussion and conclusion about the effect of vaccinate. The comprehension of COVID-19 transmission methods, structures, and characteristics is greatly aided by these mathematical models analytically and numerically.

Dynamics and data fitting of a time-delayed SIRS hepatitis B model with psychological inhibition factor and limited medical resources

Hepatitis B is an infectious disease worthy of attention.Considering the incubation period,psychological inhibition factor,vaccine,limited medical resources and horizontal transmission,an SIRS model is proposed to describe hepatitis B transmission dynamics.In order to describe the behavior changes caused by people's psychological changes,the non-monotonic incidence rate is adopted in the model.We use the saturated treatment rate to describe the limited medical resources.Mathematical analysis shows the existence conditions of the equilibria,forward or backward bifurcation,Hopf bifurcation and the Bogdanov-Takens bifurcation.During the observation of the case data of hepatitis B in China,it is found that there are mainly three features,periodic outbreaks,aperiodic outbreaks,and periodic outbreaks turns to aperiodic outbreaks.According to the above features,we select three different representative regions,Jiangxi,Zhejiang province and Beijing,and then use our model to fit the actual monthly hepatitis B case data.The basic reproduction numbers that we estimated are 1.7712,1.4805 and 1.4132,respectively.The results of data fitting are consistent with those of theoretical analysis.According to the sensitivity analysis of Ro,we conclude that reducing contact,increasing treatment rate,strengthening vaccination and revaccinating can effectively prevent and control the prevalence of hepatitis B.

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.

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.

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].

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.