Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay

Abstract

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.

Switching Vaccination Schemes for Vector-Borne Diseases With Seasonal Fluctuations

“…21 The focus of this paper is on a generalized epidemic model in which the disease is transmitted by vectors exhibiting a finite (extrinsic) incubation period before becoming infectious, along the lines of Refs. [22][23][24][25][26][27][28][29]. Two important complications to the aforementioned modeling efforts are considered: the first is the addition of seasonal effects to the model, motivated by the discussion above, which is incorporated by considering model parameters that experience abrupt changes in time.…”

“…27, the authors Ma et al analyzed the permanence of (2.5) with birth rate not equal to death rate. Gao et al 28 investigated a pulse vaccination scheme for an SIR vector-borne disease model with distributed delays. The work on global stability of the endemic equilibrium of (2.5), with birth rate not equal to death rate, was completed by McCluskey in Ref.…”

Positivity and Boundedness of Solutionsfor a Stochastic Seasonal EpidemiologicalModel for Respiratory Syncytial Virus(RSV)

“…Some examples of such diseases are measles, diphtheria, chickenpox, cholera, rotavirus, respiratory syncytial virus (RSV ), vector-borne diseases including malaria and even sexually transmitted gonorrhoea [1]. In the modeling of the transmission of seasonal diseases, several nonlinear models of ordinary differential equations have been used [2], [3]. In these models, the variables commonly represent subpopulations of susceptibles (S), infected (I), recovered (R), latent (E), transmitted disease vectors, and so forth.…”

Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia.

“…The second approach use Lyapunov's functions global stability can sometimes be shown (Aranda et al, 2008), (Jódar et al, 2009), (Zhang and Teng, 2008). Moreover, infections disease models with pulse vaccination strategies and distributed time delay have been studied in Meng and Chen (2008), Gao et al (2007) and Jianga and Wei (2008) where sufficient conditions for global asymptotic stability, bifurcations and permanence in the solutions are presented. Now, if the system of nonlinear ordinary differential equations of the first order is non-autonomous, the above theory mentioned can not be applied because the equilibrium points depend on the time.…”

“…In this way, recently the SIR infections disease model with pulse vaccination strategy and distributed time delay has been studied in Meng and Chen (2008), Gao et al (2007), Jianga and Wei (2008) where it presents sufficient conditions of global asymptotic stability, bifurcations and permanence in the solutions. Also the SIRS epidemic model with time delay and infection-age dependence has been analyzed in Zhang and Teng (2008), Zhang and Teng (2007), where the contact rate is nonlinear.…”