Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination
Introduction
Since smallpox was successfully eradicated in 1977, interruption of poliomyelitis transmission has occurred subsequently in most countries due to high vaccine coverage [1]. A large number of vaccine-preventable diseases have been effectively controlled under routine vaccination schemes, such as measles, pertussis, influenza, Hepatitis B virus (HBV), and the human tuberculosis (TB). It was observed that vaccines rarely develop perfect immunity in populations [2], [3], [4], [5], which is dependent of the immune status of the vaccine recipients. Many compartmental models (e.g. [6], [7], [8], [9], [10], [11]) have been constructed to better understand how incomplete vaccination impacts on disease control and transmission dynamics. In order to verify the global stability of a susceptible–vaccinated–exposed–infectious–recovered (SVEIR) model, Li et al. [7] developed an approach for determining the Volterra-type Lyapunov function and verifying its time derivative comprises the arithmetic and the associated geometric means of the variables, i.e., algebraic approach [6], [12]. But this approach becomes restricted in nonlinear epidemic models. Wang et al. [11] proposed another direct approach by which can be transferred into the sum of Volterra-type functions, and successfully resolved the global stability for an SVEIR model with distributed delay and nonlinear incidence.
Traditionally, it was postulated that all the vaccine recipients can acquire the equal protection over the whole immunity period in a homogeneous population [6], [7], [8], [9], [11]. However, the protective efficacy of vaccines may decline with time since vaccination (e.g. HBV vaccine [4], BCG vaccine [13]). One most recent study [14] found that duration and effectiveness of BCG protection against TB vary with time since vaccination among Norwegian-born individuals. For slowly progressive diseases [15, (pp. 232)] like HBV, TB, it is also well known that the sojourn time for individuals in the latent stage may vary with the physical status of individuals [16], [17].
Accordingly, chronological age or class age [15], [18], as one of the most important features distinguishing individuals in a population, needs incorporating into epidemic models to describe generic sojourn time distributions in a certain state. The models thus are involved in age-structured partial differential equations (PDEs) and has acquired wide attention, including vaccination age specifying vaccine waning rate [10], [19], [20], [21], [22], age of latency [8], [16], [17], [20], age of infection [8], [9], [17], [23], [24], [25], [26], etc. Most recently, Duan et al. [20] formulated an SVEIR model with ages of vaccination and latency, and complete global stability analysis was conducted, where it was assumed that certain vaccine confers a full protection against disease. Liu et al. [16] introduced age-dependent latency and relapse into an SEIR epidemic model. It should be noted that the Volterra-type Lyapunov functions have played the key role of a tool for proving global stability of these age-dependent epidemic models with mass action law [8], [9], [10], [16], [17], [19], [20], [21], [25], [26] or nonlinear incidence [22], [23], [24], and one refers the reader to recent works [27], [28] concerning the more application of such Lyapunov functions in nonlinear age-structured models. But so far there have been few literatures exploring the effects of age-dependent latency and vaccine efficacy on global stability of epidemic models with nonlinear incidence.
As another major disease control measure, the extensive media coverage and education could effectively contain the emerging epidemics, resulting in a reduction in the frequency and probability of potentially contagious contacts among the well-informed people [29], [30], [31], [32], [33], [34], [35]. Capturing this feature, the transmission rate was introduced by Cui et al. [30] into an SEI model with logistic growth, which may exhibit periodic oscillations under weak media impact (for small ) and may admit three positive equilibria under strong media impact (for large ). It was pointed out in [31], [32] that the use of media can effectively reduce the contact rates among the population to a limited level, and the transmission rate was developed [31] to describe this inherent characteristics from media impact, where indicates the velocity at which media and individuals respond to the epidemics, and is its reduced maximum value and less than if is sufficiently large. In a simple SEIR model, the nonmonotone incidence reflecting psychological/media effects [36] and these two functions above in [30], [31], were identified by Collinson and Heffernan [37]. It was found that key measurements of an epidemic outbreak, including peak number of infectious cases, peak and end times of the epidemic and total case number, depend on the media function chosen, so authors called for more study on the impact of mass media on disease transmission dynamics.
Inspired by the ideas in [7], [16], [20], [31], our work is concerned with the effects of incomplete vaccination and media on the global dynamical behaviors though formulating an SVEIR epidemic model with media-induced nonlinear incidence, and age-dependent vaccination and latency. Moreover, to establish the global stability of the endemic equilibrium, what we have to stress is that its time derivative of the Volterra-type Lyapunov function still comprises several Volterra-type functions [11], different from the approaches in [6], [7], [8], [9], [10].
The basic structure of this paper is as follows. The next section proposes an SVEIR epidemic model with media impact and ages of vaccination and latency. In Section 3, some preliminaries for our main results are presented. Section 4 analyzes the existence of equilibria for our target model. Section 5 contains our main results on uniform persistence and global stability of the model. We close with a brief discussion section.
Section snippets
Model formulation
The total population under consideration is compartmentalized into the five disjoint subclasses: susceptible , vaccinated at time with vaccination age , latent at time with latent age , infectious , and recovered . Without vaccination, the newborns enter the susceptible population at a constant rate , where is natural death rate. Due to media impact on the epidemics, susceptible individuals are assumed to become infected at rate (e.g. [32],
Preliminaries
In this section, some primary results are presented for establishing our main conclusions.
The existence of equilibria
The results on the existence of equilibria of system (2.4) are first stated as follows. Theorem 4.1 System (2.4) has a unique endemic equilibrium (EE) iff , and there always exists a disease-free equilibrium (DFE) .
Proof For any equilibrium of (2.4), , it must satisfy Together with , integrating
Uniform persistence
This part establishes uniform persistence of system (2.4), which indicates that there always exist infectious individuals given that infection initially occurs and . The following notations are introduced to define the invariant sets of uniform persistence. Set Obviously, due to . By similar arguments of Theorem 3.1 in [20], we have Theorem 5.1 The semi-flow generated by
Discussion
Vaccination is the most effective prevention measure against diseases, but the efficacy of vaccines [4], [14] and the vaccine-induced immunity [13], [14] may decline or wane with time since vaccination. Moreover, media coverage plays a vital role in containing the emerging epidemics [29], [30], [31], [32], [33]. In this paper, we propose an SVEIR epidemic model with media impact, age-dependent vaccination and latency and study uniform persistence and the global stability of system (2.4). Our
Acknowledgments
The authors thank the editor and the anonymous reviewers for their constructive comments that help to improve the early version of this paper. The work is supported by National Natural Science Foundation of China (No. 11261017 and No. 11371048).
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