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.

TWO DIFFERENTIAL INFECTIVITY EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.

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.

Global Dynamics Analysis of a Cholera Transmission Model with General Incidence and Multiple Modes of Infection

This paper develops an SIBR cholera transmission model with general incidence rate. Necessary and sufficient conditions for local and global asymptotic stability of the equilibria are established by Routh Hurwitz criterium, Lyapunov function, and the second additive composite matrix theorem. What is more, exploiting the DED is cover simulation tool, the parameter values of the model are estimated with the 1998-2021 cholera case data in China. Finally, we perform sensitivity analysis for the basic reproduction number to seek for effective interventions for cholera control. .

Deterministic and Stochastic Analysis of a New Rumor Propagation Model with Nonlinear Propagation Rate in Social Network

This paper presents a study on a new rumor propagation model with nonlinear propagation rate and secondary propagation rate. We divide the total population into three groups, the ignorant, the spreader and the aware. The nonlinear incidence rate describes the psychological impact of certain serious rumors on social groups when the number of individuals spreading rumors becomes larger. The main contributions of this work are the development of a new rumor propagation model and some results of deterministic and stochastic analysis of the rumor propagation model. The results show the influence of nonlinear propagation rate and stochastic fluctuation on the dynamic behavior of the rumor propagation model by using Lyapunov function method and stochastic related knowledge. Numerical examples and simulation results are given to illustrate the results obtained.

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.

Complex dynamics of a discrete-time SIR model with nonlinear incidence and recovery rates

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.

Dynamical analysis of a reaction-diffusion SEI epidemic model with nonlinear incidence rate

In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions.The basic reproduction numbers are given in both heterogeneous and homogeneous environments.For spatially heterogeneous environment,by the comparison principle of the diffusion system,the infection-free steady state is proved to be globally asymptotically stable if R_(0)<1,if R_(0)>1,the system will be persistent and admit at least one positive steady state.For spatially homogenous environment,by constructing a Lyapunov function,the infect ion-free steady state is proved to be globally asymptotically stable if,R_(0)<1,and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R_(0)>1.Finally,two examples are given via numerical simulations,and then some control strategies are also presented by the sensitive analysis.

一类具有非线性发生率的SEIR传染病模型的全局稳定性分析（英文）

In this paper,an SEIR model with nonlinear incidence rates are studied.The basic reproduction number R_0 characterizes the disease transmission dynamics: if R_0≤ 1,the disease-free equilibrium is globally asymptotically stable and the disease always dies out,if R_0> 1 then there is a unique endemic equilibrium which is globally asymptotically stable and the disease persists.