Disease-free Periodic Solution Research Articles

The Effect of Impulsive Vaccination on Delayed SEIRS Epidemic Model Incorporating Saturation Recovery

A delay SEIRS model with pulse vaccination incorporating a general media coverage function and saturation recovery is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution. Moreover, we show that the disease is uniformly persistent if vaccination rate is less thanθ*. Finally, we discuss the effect of media coverage on controlling disease.

Subcritical Bifurcation and Global Dynamics in an Imperfect Pulse Vaccination Model

In this talk we investigate an SIVS epidemic model with imperfect vaccine, thus vaccinated individuals can also contract the infection. We consider pulse vaccination... It is known that in some vaccination models, backward bifurcation occurs and multiple subthreshold endemic equilibria exist, thus the behaviour of solution depends on the initial value [1]. We know that pulse vaccination can be more effective than constant vaccination, thus it is an interesting question to study whether backward bifurcation can arise in a pulse vaccination model. We prove that backward bifurcation can occur in pulse vaccination model. First, we find a disease-free periodic solution, which is locally asymptotically stable in the whole phase space if \(R_c 1\), then the infection is strongly uniformly persistent in the population.

Disease Control of Delay SEIR Model with Nonlinear Incidence Rate and Vertical Transmission

The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate, continuous treatment, and impulsive vaccination for a class of epidemic with latent period and vertical transition. For continuous treatment, we obtain a basic reproductive number ℜ 0 and prove the global stability by using the Lyapunov functional method. We obtain two thresholds ℜ* and ℜ ∗ for impulsive vaccination and prove that if ℜ* < 1, then the disease-free periodic solution is globally attractive and if ℜ ∗ > 1, then the disease is permanent by using the comparison theorem of impulsive differential equation. Numerical simulations indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease.

A Reaction-Diffusion Lyme Disease Model with Seasonality

This paper is devoted to the study of a reaction-diffusion Lyme disease model with seasonality. In the case of a bounded habitat, we obtain a threshold result on the global stability of either disease-free or endemic periodic solution. In the case of an unbounded habitat, we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time-periodic traveling wave solutions. We also estimate parameter values based on some published data and use them to study the Lyme disease transmission in Port Dove, Ontario. Our numerical simulations are consistent with the obtained analytic results.

Threshold dynamics of a malaria transmission model in periodic environment

In this paper, we propose a malaria transmission model with periodic environment. The basic reproduction number R0 is computed for the model and it is shown that the disease-free periodic solution of the model is globally asymptotically stable when R0<1, that is, the disease goes extinct when R0<1, while the disease is uniformly persistent and there is at least one positive periodic solution when R0>1. It indicates that R0 is the threshold value determining the extinction and the uniform persistence of the disease. Finally, some examples are given to illustrate the main theoretical results. The numerical simulations show that, when the disease is uniformly persistent, different dynamic behaviors may be found in this model, such as the global attractivity and the chaotic attractor.

The Global Behavior of a Periodic Epidemic Model with Travel between Patches

We establish an SIS (susceptible‐infected‐susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratio R0 and two theorems on the global behavior: if R0 &lt; 1 the disease‐free periodic solution is globally asymptotically stable and if R0 &gt; 1, then it is unstable; if R0 &gt; 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.

THRESHOLD DYNAMICS IN A PERIODIC SVEIR EPIDEMIC MODEL

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

MIXED VACCINATION STRATEGY IN SIRS EPIDEMIC MODEL WITH SEASONAL VARIABILITY ON INFECTION

In many diseases seasonal fluctuations are observed. SIRS epidemic model with seasonal varying contact rate and mixed vaccination strategy (including first vaccination and pulse vaccination strategy) is investigated. The effects of the variation of dependent on the season of the contact rate and the vaccination strategy to eradicate infectious diseases are studied and discussed. A threshold for a disease to be extinct or endemic is established. The existence and global asymptotic stability of disease-free periodic solution and the permanence of the disease are illustrated. Finally, our theoretical results are confirmed by numerical simulations.

Dynamics of an SI epidemic model with external effects in a polluted environment

In a polluted environment, considering the biological population infected with some kinds of diseases and hunted by human beings, we formulate two SI pollution-epidemic models with continuous and impulsive external effects, respectively, and investigate the dynamics of such systems. We assume that only the susceptible population is hunted by human beings. For the continuous system, we obtain sufficient conditions of the ultimate boundedness of solutions and the global asymptotical stability of equilibria. For the impulsive system, by using the comparison theorem and the analysis method, we show that under different conditions the disease-free periodic solution is globally asymptotically attractive, or the system is permanent. Numerical simulations confirm our theoretical results.

SIR-SVS epidemic models with continuous and impulsive vaccination strategies

Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R 0 which is defined through the spectral radius of a linear integral operator. If R 0 < 1 , then the disease free periodic solution is globally asymptotically stable and if R 0 > 1 , then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537–2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.

A delayed SEIQR epidemic model with pulse vaccination and the quarantine measure

A delayed SEIQR epidemic model with pulse vaccination and the quarantine measure is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic disease-free solution. Using the comparison method, we prove that the disease-free periodic solution is globally attractive when the basic reproductive number (ℜ∗) is less than unity, and that the disease is permanent when another basic reproductive number (ℜ∗) is greater than unity. In other words, the disease will be extinct if the pulse vaccination rate is larger than a critical value θ∗ and the disease will be uniformly persistent if the vaccination rate is less than another critical value θ∗. Our results indicate that a longer latent period of the disease or a larger pulse vaccination rate will lead to the eradication of the disease, and whether the disease will be extinct or not is independent of the removal rate from the quarantined group. Furthermore, a larger fraction of susceptibles should be vaccinated against the disease unless the quarantine measure is taken. Finally, we find that the number of the infected decreases as the quarantine measure is taken. We carry out numerical simulations to verify our results.

Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates

In this paper, two delayed SEIR epidemic models with continuous and impulsive vaccination and saturating incidence are investigated. The dynamical behaviors of the disease are analyzed. For continuous vaccination, we obtain a basic reproductive number R 1 and prove that if R 1 ≤ 1 then the disease-free equilibrium is globally attractive and if R 1 > 1 then the disease is permanent by using the Lyapunov functional method. For impulsive vaccination, we obtain two thresholds R ∗ and R ∗ and prove that if R ∗ < 1 then the disease-free periodic solution is globally attractive and if R ∗ > 1 then the disease is permanent by using the comparison theorem of impulsive differential equation and the Lyapunov functional method. Lastly, we compared the effects of two vaccination strategies.

Predator-prey model with disease infection in both populations

A predator–prey model with disease infection in both populations is proposed to account for the possibility of a contagious disease crossing species barrier from prey to predator. We obtain several threshold parameters from local analysis of various equilibria of the proposed system as well as coupled conditions on these threshold parameters which determine the stability of these equilibria. One of the coupled conditions, in the form of an ecological threshold number for the predator–prey ecosystem, always determines the coexistence of predators and prey. The other condition, in the form of a disease basic reproduction number, dictates whether the disease will become endemic in the ecosystem. Under one combination of these coupled conditions, a highly infectious disease could drive the predators to extinction when predators and prey would have coexisted without the disease. For another combination of the conditions, the predation of the more vulnerable infected prey could cause the disease to be eradicated in the ecosystem, in some case even approaching a disease-free periodic solution, when the disease would have otherwise remained endemic in the prey population in the absence of predation. This indicates that the presence of disease in both predators and prey could either promote or impair coexistence, and its precise impact needs to be explored specifically in each particular situation. By considering disease infection in both populations, our model also yields more complex dynamics, allowing for the possibility of bistability and periodic oscillation, in either disease-free or endemic states, in the ecosystem for which the conditions are obtained analytically and with the help of numerical simulations.

Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission

Three different vaccination and treatment strategies in the SIR epidemic model with saturated infectious force and vertical transmission are analyzed. The dynamics of epidemic models are globally investigated by using Floquet theory and comparison theorem of impulsive differential equation. Thresholds are identified and global stability results are proved. For every treatment and vaccination strategy, the disease-free periodic solution of impulsive system has been obtained and is found to be globally asymptotically stable when the basic reproduction number is less than one, equivalently the cure rate is larger than the threshold value, whereas the disease is persistent when the basic reproduction number is larger than one. These results indicate that a large cure rate will lead to the eradication of a disease.

DIFFERENTIAL SUSCEPTIBILITY TIME-DEPENDENT SIR EPIDEMIC MODEL

A non-autonomous epidemic dynamical system, in which we include variable susceptibility, is proposed. Some threshold conditions are derived which determine whether or not the disease will go to extinction. Some new threshold values, [Formula: see text], [Formula: see text] and [Formula: see text], are deduced for this general time-dependent system such that when [Formula: see text] is greater than 0, the disease is endemic in the sense of permanence and when one of the threshold values [Formula: see text] and [Formula: see text] is less than 0, the disease will die out. As an application of these results, the basic reproductive number ℛ0will be given if all the coefficients are periodic with common period. In addition, ℛ0&lt; 1 implies the global stability of the disease-free periodic solution. Some corollaries are given for periodic and almost-periodic cases. The theoretical results are confirmed by a special example and numerical simulations.

An SIRVS epidemic model with pulse vaccination strategy

The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.

Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model

A delayed SEIRS epidemic model with pulse vaccination and bilinear incidence rate is investigated. Using Krasnoselskii’s fixed-point theorem, we obtain the existence of disease-free periodic solution (DFPS for short) of the delayed impulsive epidemic system. Further, using the comparison method, we prove that under the condition R ∗ < 1, the DFPS is globally attractive, and that R ∗ > 1 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.

Pulse vaccination delayed SEIRS epidemic model with saturation incidence

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution and the permanence of disease. Our results indicate that a long latent period of the disease or a proper pulse vaccination rate will lead to eradication of the disease.

Pulse vaccination of an SEIR epidemic model with time delay

Pulse vaccination is an effective strategy for the elimination of infectious diseases. A delayed epidemic model with pulse vaccination is formulated in this paper. It is proved that the disease-free periodic solution is globally attractive if the vaccination rate is larger than θ * , and 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 large vaccination rate or short pulse of vaccination or long latent period is sufficient condition for the extinction of the disease.