Infection-free Periodic Solution Research Articles

TWO DIFFERENT VACCINATION STRATEGIES IN AN SIR EPIDEMIC MODEL WITH SATURATED INFECTIOUS FORCE

Two different vaccination and treatment strategies in the SIR epidemic model with saturation infectious force are analyzed. With the continuous vaccination and treatment, it is obtained that the disease free equilibrium and endemic equilibrium are globally asymptotically stable by using Lassall theorem and Pioncare–Bendixon trichotomy. Moreover, with pulse vaccination and treatment at different time, the dynamics of the epidemic model is globally investigated by using Floquet theory and comparison theorem of impulsive differential equation and analytic method. We obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. Finally, we compare the two different vaccination and treatment strategies, and obtain that the elimination of disease is independent of treatment in the case of the pulse vaccination.

Dynamic behavior of an eco-epidemic system with impulsive birth

In the paper, we investigate an eco-epidemic system with impulsive birth. The conditions for the stability of infection-free periodic solution are given by applying Floquet theory of linear periodic impulsive equation. And we give the conditions of persistence by constructing a consequence of some abstract monotone iterative schemes. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive periodic solution. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate

In this paper, an SEIR epidemic disease model with time delay and nonlinear incidence rate is studied, and the dynamical behavior of the model under pulse vaccination is analyzed. Using the discrete dynamical system determined by the stroboscopic map, we show that there exists an infection-free periodic solution. Further, we show that the infection-free periodic solution is globally attractive when the period of impulsive effect is less than some critical value. Using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination. We show that time delay, pulse vaccination can bring different effects on the dynamic behavior of the model by numerical analysis. Our results also show the time delay is “profitless”. The main feature of this paper is to introduce time delay and impulse into the SEIR epidemic model and to give pulse vaccination strategies.

An impulsive delayed SEIRS epidemic model with saturation incidence

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using Krasnoselskii's fixed-point theorem, we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system. We define some new threshold values , and . Further, using the comparison theorem, we obtain the explicit formulae of and . Under the condition , the infection-free periodic solution is globally attractive, and that implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ* and the disease is uniformly persistent if the vaccination rate is less than θ*. Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Moreover, we prove that the disease will be permanent as .

A Delayed Epidemic Model with Pulse Vaccination

A delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this model and point out that there exists an infection‐free periodic solution which is globally attractive if R1 < 1, R2 > 1, and the disease is permanent. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient condition for the eradication of the disease. The main feature of this paper is to introduce time delay and impulse into SEIRS model and give pulse vaccination strategies.

The effects of pulse vaccination on SEIR model with two time delays

Abstract In this paper, impulsive vaccination has been applied to control the spread and transmission of an infectious disease. An SEIR epidemic model with two time delays and pulse vaccination is formulated. The exact infection-free periodic solution of the impulsive epidemic system is obtained. Moreover, we show that, if the vaccination rate is larger than θ ∗ , the infectious population disappear so the disease dies out, while if the vaccination rate is less than θ ∗ , the infectious population persist. Our results indicate that a large vaccination rate or a short period of pulsing or a long latent period of the disease is a sufficient condition for the eradication of the disease.

Analysis of a delayed SIR epidemic model with pulse vaccination

In this paper, a delayed SIR epidemic model with pulse vaccination is investigated. By the comparison theorem for impulsive differential equations, we obtain that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is permanent if the vaccination proportion is less than some critical value under appropriate condition. By Brouwer’s fixed-point theorem, we establish sufficient condition for the existence of positive periodic solution. Our results indicate that a large vaccination rate or a short period of pulsing is a sufficient condition for the eradication of the disease.

DYNAMICS OF SEIS EPIDEMIC MODELS WITH VARYING POPULATION SIZE

In this paper, SEIS epidemic models with varying population size are considered. Firstly, we consider the case when births of population are throughout the year. A threshold σ is identified, which determines the outcome of disease, that is, when σ < 1, the disease dies out; whereas when σ > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable; when σ = 1, bifurcation occurs and leads to "the change of stability". Two other thresholds σ′ and [Formula: see text] are also identified, which determine the dynamics of epidemic model with varying population size, when the disease dies out or it is endemic. Secondly, we consider the other case, birth pulse. The population density is increased by an amount B(N)N at the discrete time nτ, where n is any non-negative integer and τ is a positive constant, B(N) is density-dependent birth rate. By applying the corresponding stroboscopic map, we obtain the existence of infection-free periodic solution with period τ. Lastly, through numerical simulations, we show the dynamic complexities of SEIS epidemic models with varying population size, there is a sequence of bifurcations, leading to chaotic strange attractors. Non-unique attractors also appear, which implies that the dynamics of SEIS epidemic models with varying population size can be very complex.

Analysis of an SIR epidemic model with pulse vaccination and distributed time delay.

Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.

A delayed SIRS epidemic model with pulse vaccination☆

Abstract 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 disease is uniformly persistent, which means that after some period of time the disease will become endemic.

Analysis of a delayed epidemic model with pulse vaccination and saturation incidence

Pulse vaccination is an important strategy for the elimination of infectious diseases. An SEIRS epidemic model with time delays and pulse vaccination is formulated in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger than θ ∗ . Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than θ ∗ . The permanence of the model is investigated analytically. Our results indicate that a long latent period of the disease is sufficient condition for the extinction of the disease.

The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission

A SIR epidemic model with horizontal and vertical transmission is presented. The dynamics of this disease model under constant and pulse vaccination are analyzed. Theoretical results show that under constant vaccination, the dynamic behavior is similar to no vaccination. But under a planned pulse vaccination regime, the system converges to a stable periodic solution with the number of infectious individuals equal to zero. Further, the global stability of an infection-free periodic solution is analyzed. And the existence of a nontrivial periodic solution (endemic) is considered by using bifurcation theory. Our theoretical results are confirmed by numerical simulations. The effectiveness of constant and pulse vaccination policies are compared.