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.

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.

A stochastic switched SIRS epidemic model with nonlinear incidence and vaccination:Stationary distribution and extinction

In this paper,a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases.Nonlinear incidence and vaccination strategies are also considered in the proposed model.By using the method of stochastic analysis,we point out the key parameters that determine the persistence and extinction of the diseases.Specifically,if R0^s is greater than 0,the stochastic system has a unique ergodic stationary distribution;while if R ^* is less than 0,the diseases will be extinct at an exponential rate.

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.

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.

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.

Stability analysis of an SIR epidemic model with homestead-isolation on the susceptible and infectious,immunity,relapse and general incidence rate

In this paper,we propose the global dynamics of an SIR epidemic model with distributed latent period,immunity,relapse,homestead-isolation of the susceptible and infectious individuals and general incidence rate.The resulting model has a disease-free equilibrium and if Ro>1,then the SIR epidemic model admits a unique endemic equilibrium.By using suitable Lyapunov functionals and LaSalle's invariance principle,the global stability of the disease-free equilibrium and the endemic equilibrium is established,under suitable monotonicity conditions on the incidence function.

Bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals

This study focuses on the stability and local bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals analytically,and numerically.The analytical results are obtained using thenormal form technique and numerical results are obtained using the numerical continuation method.For this model,a number of bifurcations are studied,including the transcritical(pitchfork)and fip bifurcations,the Neimark-Sacker(NS)bifurcations,and the strong resonance bifurcations.We especially determine the dynamical behaviors of the model for higher iterations up to fourth-order.Numerical simulation is employed to present a closed invariant curve emerging about an NS point,and its breaking down to several closed invariant curves and eventuality giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

Global dynamics of a fractional-order SIR epidemic model with memory

In this paper,an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley-Martin type functional response and Holling type-II treatment rate are established along the memory.The existence and stability of the equilibrium points are investigated.The sufficient conditions for the persistence of the disease are provided.First,a threshold value,Ro,is obtained which determines the stability of equilibria,then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle.The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by LI scheme which involves the memory trace that can capture and integrate all past activity.Meanwhile,by using Lyapunov functional approach,the global dynamics of the endemic equilibrium point is discussed.Further,some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained.The outcome of the study reveals that the applied LI scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics.The results show that order of the fractional derivative has a significant effect on the dynamic process.Also,from the results,it is obvious that the memory effect is zero for p=1.When the fractional-order p is decreased from 1,the memory trace nonlinearly increases from 0,and its dynamics strongly depends on time.The memory effect points out the difference between the derivatives of the fractional-order and integer order.

A new epidemic modeling approach:Multi-regions discrete-time model with travel-blocking vicinity optimal control strategy

First,we devise in this paper,a multi-regions discrete-time model which describes the spatial-temporal spread of an epidemic which starts from one region and enters to regions which are connected with their neighbors by any kind of anthropological movement.We suppose homogeneous Susceptible-Infected-Removed(SIR)populations,and we consider in our simulations,a grid of colored cells,which represents the whole domain affected by the epidemic while each cell can represent a sub-domain or region.Second,in order to minimize the number of infected individuals in one region,we propose an optimal control approach based on a travel-blocking vicinity strategy which aims to control only one cell by restricting movements of infected people coming from all neighboring cells.Thus,we show the influence of the optimal control approach on the controlled cell.We should also note that the cellular modeling approach we propose here,can also describes infection dynamics of regions which are not necessarily attached one to an other,even if no empty space can be viewed between cells.The theoretical method we follow for the characterization of the travel-locking optimal controls,is based on a discrete version of Pontryagin's maximum principle while the numerical approach applied to the multi-points boundary value problems we obtain here,is based on discrete progressive-regressive iterative schemes.We illustrate our modeling and control approaches by giving an example of 100 regions.

Threshold Dynamics of an SIR Epidemic Model with Nonlinear Incidence Rate and Non-Local Delay Effect

在这份报纸，我们涉及反应 -- 有在一个连续围住的空间领域的非线性的发生率和非局部的延期效果的散开先生流行模型。我们由下一代操作员的想法介绍模型的基本繁殖号码 R 0 。借助于动态系统和一致坚持的理论，我们以 R 0 调查模型的全球动力学。最后，我们实现数字模拟显示出我们的结果的可行性并且探索一些流行病学的卓见。

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.

Dynamic characterization of a stochastic SIR infectious disease model with dual perturbation

Environmental perturbations are unavoidable in the propagation of infectious diseases.In this paper,we introduce the stochasticity into the susceptible-infected recovered(SIR)model via thc^parameter perturbation method.The stochastic disturbances associated with the disease transmission coefficient and the mortality rate are presented with two perturbations:Gaussian white noise and Levy jumps,respectively.This idea provides an overview of disease dynamics under different random perturbation scenarios.By using new techniques and methods,we study certain interesting asymptotic properties of our perturbed model,namely:persistence in the mean,ergodicity and extinction of the disease.For illustrative purposes,numerical examples are presented for checking the theoretical study.

Inverse problem for adaptive SIR model:Application to COVID-19 in Latin America

This work presents a method for solving an Adaptive Susceptible-Infected-Removed(ASIR)epidemic model with time-dependent transmission and removal rates.Available COVID-19 data as of March 2021 are used for identifying the rates from an inverse problem.The estimated rates are used to solve the adaptive SIR system for the spread of the infectious disease.This method simultaneously solves the problem for the time-dependent rates and the unknown functions of the A-SIR system.Presented results show the spread of COVID-19 in the World,Argentina,Brazil,Colombia,Dominican Republic,and Honduras.Comparisons of the reported affected by the disease individuals from the available real data and the values obtained with the A-SIR model demonstrate how well the model simulates the dynamic of the infectious disease.

Time-varying and state-dependent recovery rates in epidemiological models

Differential equation models of infectious disease have undergone many theoretical extensions that are invaluable for the evaluation of disease spread.For instance,while one traditionally uses a bilinear term to describe the incidence rate of infection,physically more realistic generalizations exist to account for effects such as the saturation of infection.However,such theoretical extensions of recovery rates in differential equation models have only started to be developed.This is despite the fact that a constant rate often does not provide a good description of the dynamics of recovery and that the recovery rate is arguably as important as the incidence rate in governing the dynamics of a system.We provide a first-principles derivation of state-dependent and time-varying recovery rates in differential equation models of infectious disease.Through this derivation,we demonstrate how to obtain time-varying and state-dependent recovery rates based on the family of Pearson distributions and a power-law distribution,respectively.For recovery rates based on the family of Pearson distributions,we show that uncertainty in skewness,in comparison to other statistical moments,is at least two times more impactful on the sensitivity of predicting an epidemic's peak.In addition,using recovery rates based on a power-law distribution,we provide a procedure to obtain state-dependent recovery rates.For such state-dependent rates,we derive a natural connection between recovery rate parameters with the mean and standard deviation of a power-law distribution,illustrating the impact that standard deviation has on the shape of an epidemic wave.