Bifurcation analysis in an SIR epidemic model with birth pulse and pulse vaccination

The dynamical behavior of an SIR epidemic model with birth pulse and pulse vaccination is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the infection-free periodic solution and the epidemic periodic solution. By using the impulsive effects, a Poincaré map is obtained. The Poincaré map, center manifold theorem, and bifurcation theorem are used to discuss flip bifurcation and bifurcation of the epidemic periodic solution. Moreover, the numerical results show that the epidemic periodic solution (period-one) bifurcates from the infection-free periodic solution through a supercritical bifurcation, the period-two solution bifurcates from the epidemic periodic solution through flip bifurcation, and the chaotic solution generated via a cascade of period-doubling bifurcations, which are in good agreement with the theoretical analysis.     

Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination

An SIR epidemic model with state dependent pulse vaccination is proposed in this paper. Using the Poincaré map, the differential inequality and the method of qualitative analysis, we prove the existence and the stability of positive order-1 or order-2 periodic solution for this model. Moreover, we show that there is no periodic solution with order larger than or equal to three. Numerical simulations are carried out to illustrate the feasibility of our main results and the suitability of state dependent pulse vaccination is also discussed.     

A delay SIR epidemic model with pulse vaccination and incubation times

In this paper, a new delay SIR epidemic model with pulse vaccination and incubation times is considered. We obtain an infection-free semi-trivial periodic solution and establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination and nonlinear incidence have significant effects on the dynamics behaviors of the model. Our results are illustrated and corroborated with some numerical experiments.     

The effect of constant and pulse vaccination on an SIR epidemic model with infectious period

In this paper, we investigate two delayed SIR models with vaccination and a generalized nonlinear incidence and obtain sufficient conditions for eradication and permanence of the disease, respectively. Our results indicate that a larger vaccination rate will lead to the eradication of a disease. Furthermore, theoretical results show that constant vaccination strategy can lead to disease eradication at relatively low values of vaccination than pulse vaccination strategy, which is different from the results in [1]. In addition, numerical simulations indicate that pulse vaccination strategy or a longer infectious period will make a larger fraction of population infected by disease.     

The differential susceptibility SIR epidemic model with stage structure and pulse vaccination

The differential susceptibility SIR epidemic model with stage structure and pulse vaccination is introduced. By the comparison theorem, some sufficient conditions for the globally attractivity of an infection-free periodic solution and the permanence of this system are presented. Two numerical simulations are also given to illustrate our main results.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.     

On pulse vaccination strategy in the SIR epidemic model with vertical transmission

The aim of this short paper is to improve a result recently given by Lu et al. on the global asymptotic stability of the eradication solution of the PVS applied to diseases with vertical transmission, by demonstrating that the condition for local stability guarantees also the global stability.     

Theoretical examination of the pulse vaccination policy in the SIR epidemic model

Based on a theory of population dynamics in perturbed environments, it was hypothesized that measles epidemics can be more efficiently controlled by pulse vaccination, i.e., by a vaccination effort that is pulsed over time [1]. Here, we analyze the rationale of the pulse vaccination strategy in the simple SIR epidemic model. We show that repeatedly vaccinating the susceptible population in a series of ‘pulses,’ it is possible to eradicate the measles infection from the entire model population. We derive the conditions for epidemic eradication under various constraints and show their dependence on the parameters of the epidemic model.     

The effect of impulsive vaccination on an SIR epidemic model

In this paper, an SIR epidemic model is constructed and analyzed. We get the result that if the parameters satisfy the condition β>α+γ+b, then the disease will be ultimately permanent. Under this condition, we consider how the impulsive vaccination affects the original system. The sufficient condition for the global asymptotical stability of the disease-eradication solution is obtained. We also get that if the impulsive vaccination rate is less than some value, the disease will be permanent, and the disease cannot be controlled. People can select appropriate vaccination rate according to our theoretical result to control diseases.     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

A delayed SIRS epidemic model with pulse vaccination

A delayed SIRS epidemic model with pulse vaccination and saturated contact rate is investigated. By using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the system. Further, by using the comparison theorem, we prove that under the condition that R₀ < 1 the infection-free periodic solution is globally attractive, and that under the condition that R′ > 1 the disease is uniformly persistent, which means that after some period of time the disease will become endemic.     

Numerical modelling of an SIR epidemic model with diffusion

A spatial SIR reaction-diffusion model for the transmission disease such as whooping cough is studied. The behaviour of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by using Lyapunov functional. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small. These results are verified numerically by constructing, and then simulating, a robust implicit finite-difference method. Furthermore, the new implicit finite-difference method will be seen to be more competitive (in terms of numerical stability) than the standard finite-difference method.     

An SIRS epidemic model with pulse vaccination and non-monotonic incidence rate

An SIRS epidemic model with pulse vaccination and non-monotonic incidence rate is introduced. Some sufficient conditions for the global attractivity of the infection-free periodic solution and permanence of this system are presented. Two numerical simulations are also given to illustrate our main results.     

Global behaviour of an SIR epidemic model with time delay

We study the stability of a delay susceptible–infective–recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods—by Lyapunov functionals, and—in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number ?₀ is less than unity, by LaSalle invariance principle, the disease‐free equilibrium E_s is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model. Copyright © 2006 John Wiley & Sons, Ltd.     

Two profitless delays for an SEIRS epidemic disease model with vertical transmission and pulse vaccination

Since the investigation of impulsive delay differential equations is beginning, the literature on delay epidemic models with pulse vaccination is not extensive. In this paper, we propose a new SEIRS epidemic disease model with two profitless delays and vertical transmission, and analyze the dynamics behaviors of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive when some parameters of the model are under appropriate conditions. Using a new modeling method, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delays, pulse vaccination and vertical transmission can bring different effects on the dynamics behaviors of the model by numerical analysis. Our results also show the delays are “profitless”. In this paper, the main feature is to introduce two discrete time delays, vertical transmission and impulse into SEIRS epidemic model and to give pulse vaccination strategies.     

Analysis of pulse vaccination strategy in SIRVS epidemic model

In this paper, we investigate the dynamics of a SIRVS epidemic model with pulse vaccination strategy and saturation incidence. We show that the disease is eradicated when the basic reproduction number is less than unity, and the disease is permanent when the basic reproduction number is greater than unity. Finally, by means of numerical simulation, we obtain the parameters reach some critical value, and the disease will go to extinction.