Global dynamics of SIR model with switched transmission rate (original) (raw)

Mathematical Model of an SIR Epidemic Switching with Zero Co-Infectives

International journal of mathematical research, 2020

Contribution/Originality: This study is one of the few studies in mathematical epidemiology which have investigated the role of switching in an SIR model of two epidemics with zero co-infectives. In addition, Lyapunov functions theory and Center Manifold method is applied to the model for the global stability analysis and existence of forward bifurcation respectively.

Recurrent Epidemics Resulting from Secondary Transmission Routes in Sir Models

2011

In this article, we analyze the behavior of solutions to a variant of the SIR (susceptible, infected, recovered) model from epidemiology. The model studied includes a secondary route for susceptible individuals to be exposed to the infectious agent. This secondary route provides a feedback mechanism that, within certain parameter regimes, allows for a limit cycle; i.e., sustained periodic behavior in the solutions.

Stability Criteria for SIS Epidemiological Models under Switching Policies

We study the spread of disease in an SIS model. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.

Analysis of a Stochastic SIR Model with Vaccination and Nonlinear Incidence Rate

International Journal of Differential Equations

We expand an SIR epidemic model with vertical and nonlinear incidence rates from a deterministic frame to a stochastic one. The existence of a positive global analytical solution of the proposed stochastic model is shown, and conditions for the extinction and persistence of the disease are established. The presented results are demonstrated by numerical simulations.

Dynamical Properties of Two Diffusion Sir Epidemic Model with Markovian Switching

In this paper we consider a stochastic SIR epidemic model with treatment, nonlinear incidence rate and regime switching. For our model, we first prove the existence and uniqueness of the global positive solution. Then, we provide conditions under which the disease prevails in the population, as well as the sufficient conditions for the extinction of the disease. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use real-life data for Ebola outbreak in Sierra Leone and Covid-19 pandemic in Pakistan. AMS Mathematics Subject Classification (2010): 60H10, 92D25, 93E15

Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

Mathematical Biosciences and Engineering, 2011

We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number R 0 , confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.

Dynamics of a Stochastic SIRS Epidemic Model with Regime Switching and Specific Functional Response

Discrete Dynamics in Nature and Society

The purpose of this work is to investigate the dynamic behaviors of the SIRS epidemic model with nonlinear incident rate under regime switching. We establish the existence of a unique positive solution of our system. Furthermore, we obtain the conditions for the extinction of diseases, and we show the existence of the stationary distribution for our stochastic SIRS model under regime switching. Numerical simulations are employed to illustrate our theoretical analysis.

Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate

2015

In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. The existence of Hopf bifurcation of model is investigated by using Andronov-Hopf bifurcation theorem. Further, numerical simulations are done to exemplify the analytical studies.

Dynamical Behavior of a Stochastic SIRS Epidemic Model

Mathematical Modelling of Natural Phenomena, 2015

In this paper we study the Kernack-MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold λ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of λ, the system can globally tend towards an endemic state or a disease free state. The aim of this work is also to describe completely the ω-limit set of all positive solutions to the model. Moreover, the attraction of the ω-limit set and the stationary distribution of solutions will be shown.