Global dynamics of SIR model with switched transmission rate (original) (raw)
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.
Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models
Journal of Mathematical Biology, 1986
When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.
Dynamics of an SIR epidemic model with limited medical resources, revisited and corrected
arXiv (Cornell University), 2023
This paper generalizes and corrects a famous paper (more than 200 citations) concerning Hopf and Bogdanov-Takens bifurcations due to L. Zhou and M. Fan, "Dynamics of an SIR epidemic model with limited medical resources revisited", in which we discovered a significant numerical error. Importantly, unlike the paper of Zhou and Fan and several other papers which followed them, we offer a notebook where the reader may recover all the results, and also modify them for analyzing similar models. Our calculations lead to the introduction of some interesting symbolic objects, "Groebner eliminated traces and determinants"-see (4.5), (4.6), which seem to have appeared here for the first time, and which might be of independent interest. We hope our paper might serve as yet another alarm bell regarding the importance of accompanying papers involving complicated hand computations by electronic notebooks.
2021
In this paper, we study an SIR epidemic model with ratio dependent incident rate function. We explore the impact of vaccination and treatment on the transmission dynamics of the disease. The treatment control strategies depend on the availability of maximal treatment capacity: treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. The existence and stability of the endemic equilibria are governed by the basic reproduction number and treatment control strategies. By carrying out rigorous mathematical analysis and numerical evaluations, it has been shown that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the i...
SIR model with local and global infective contacts: A deterministic approach and applications
Theoretical Population Biology
An epidemic model with births and deaths is considered on a two dimensional L × L lattice. Each individual can have global infective contacts according to the standard SIR model rules or local infective contacts with its nearest neighbors. We propose a deterministic approach to this model and verified that there is a good agreement with the stochastic simulations for different situations of the disease transmission and parameters corresponding to pertussis and rubella in the prevaccine era.