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An NSFD scheme for SIR epidemic models of childhood diseases with constant vaccination strategy

Qianqian Cui12, Jiabo Xu3, Qiang Zhang2 and Kai Wang4*

Author Affiliations

1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

2 College of Sciences, Shihezi University, Shihezi, 832000, People’s Republic of China

3 Xinjiang Institute of Engineering, Urumqi, 830091, People’s Republic of China

4 Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, 830011, People’s Republic of China

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Advances in Difference Equations 2014, 2014:172  doi:10.1186/1687-1847-2014-172

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2014/1/172


Received:10 December 2013
Accepted:4 June 2014
Published:24 June 2014

© 2014 Cui et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we construct a nonstandard finite difference (NSFD) scheme for an SIR epidemic model of childhood disease with constant strategy. The dynamics of the obtained discrete model is investigated. First we show that the discrete model has equilibria which are exactly the same as those of the continuous model. Furthermore, we prove that the conditions for those equilibria to be globally asymptotically stable are consistent with the continuous model for any size of numerical time-step. The analytical results are confirmed by some numerical simulations.

Keywords:
mathematical model; transmission dynamics; basic reproduction number; sensitivity analysis; control strategies

1 Introduction

Childhood diseases are the most common form of infectious diseases. These are the diseases such as measles, mumps, chicken pox, rubella, poliomyelitis, etc. to which children are born susceptible and usually contract within five years. Because young children are in frequent contact with each other at school or other place, such a disease can be spread very quickly. Meanwhile, the development of vaccines against infectious children diseases has been booming and protecting children from the diseases. Hence vaccination is considered to be the most effective strategy against childhood diseases, it is essential for us to predict the optimal vaccine coverage level to prevent the spread of theses diseases. A universal effort to extend vaccination coverage to all children began in 1974, when the World Health Organization (WHO) founded the Expanded Program on Immunization (EPI). Mathematical models (see [1-7]) of deterministic type have often been used to provide deeper insights into the transmission dynamics of a childhood disease and to evaluate control strategies.

In this paper, the total population that is involved in the spread of infection is split into three epidemiological classes: a susceptible class (S), an infected class (I) and a removed class (R) denoting vaccinated as well as recovered people with permanent immunity. We assume that the efficacy of vaccine is 100%, and the natural death rates μ in the classes remain unequal to births, so that the total population N is realistically not constant. Citizens are born into the population at a constant birth rate A with extremely low childhood disease mortality rate. We denote the fraction of citizens vaccinated at birth each year as p (with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M1">View MathML</a>) and assume the rest are susceptible. A susceptible individual will move into the infected group through contact with an infected individual, approximated by an average contact rate β. An infected individual recovers at rate γ, and enters the removed class. The removed class also contains people who are vaccinated. The differential equations for the SIR (see [1,4,7]) epidemic model of childhood diseases with constant vaccination strategy are as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M2">View MathML</a>

(1.1)

The biological background requires that all parameters be nonnegative. Makinde [4] employed the Adomian decomposition method to compute an approximate non-perturbative solutions of model (1.1). Yildirim and Cherruault [7] by qualitative analysis revealed the vaccination reproductive number for disease control and eradication.

However, for practical purposes, it is often necessary to discretize the continuous model. The discrete dynamical system obtained from the discretization should contain as many qualitative properties of the continuous problem as possible. It is shown that many standard methods such as Euler method, Runge-Kutta method and some other standard finite schemes implemented in a dynamical system can lead to negative solutions for spurious dynamical behaviors such as converging to wrong equilibrium point or wrong periodic cycle or numerical instabilities [8-10]. In this paper, we propose a numerical scheme to solve model (1.1) by implementing a nonstandard finite difference (NSFD) scheme. This method was originally developed by Mickens [11-16]. The nonstandard scheme relied on the following important rules: the standard denominator h in standard discrete derivative is replaced by a denominator function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M3">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M4">View MathML</a>; the nonlinear terms are approximated in a nonlocal way using more than one mesh point. Here, h is the time-step size of numerical integration. Moreover, the fundamental principle for constructing NSFD scheme for differential equations is dynamic consistency, that is, the discretized model maintain essential dynamical properties such as positivity of solutions, boundedness of solutions, monotonicity of solutions, correct number and stability of fixed-points and other special solutions of the continuous model. This method has been applied to various problems in which the resulting discrete systems preserve dynamical properties of the related continuous models [5,17-19]. In [19], the NSFD scheme has been implemented in a special class of SIR epidemic models. Mickens [5] considered a SIR epidemic model with square-root dynamics.

This paper is organized as follows. In the next section, we present several important properties of solutions to the continuous model. A particular discretization is constructed in Section 3. We illustrate the global asymptotic stability of disease-free equilibrium and endemic equilibrium in Sections 4 and 5, and provided numerical examples to verify our results in Section 6. Finally, we provide a summary of the obtained results and present a possible extension of this work.

2 Dynamical properties of the continuous model

In this section, we review some dynamical properties of the SIR epidemic model with constant vaccination strategy (1.1). It should be noted that the system of continuous equations, if added together, satisfies the conservation law [16]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M5">View MathML</a>

(2.1)

The exact solution of equation (2.1) is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M6">View MathML</a>

(2.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M7">View MathML</a>. Any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M9">View MathML</a>, of model (1.1) satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M10">View MathML</a>

(2.3)

All valid epidemic models must have this feature since negative population numbers cannot exist as a physical reality. Moreover, it is easy to verify that the domain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M11">View MathML</a>

is a compact, positively invariant set for model (1.1).

Define the basic reproduction number as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M12">View MathML</a>

(2.4)

Then the following results can be summarized in Li et al.[3].

Theorem 1Model (1.1) always has a disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M13">View MathML</a>and has a unique endemic equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M14">View MathML</a>when<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M16">View MathML</a>

Theorem 2For model (1.1), the following results hold.

(i) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17">View MathML</a>, the disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M18">View MathML</a>is globally asymptotically stable inD. On the other hand, if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M18">View MathML</a>is unstable;

(ii) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then the unique endemic equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M22">View MathML</a>is globally asymptotically stable in D.

3 The NSFD scheme

To derive the NSFD scheme, we define the following notation [16]:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M24">View MathML</a> is the constant time-step size. By applying the NSFD scheme to model (1.1), we can obtain the following discrete-time SIR epidemic model with constant vaccination strategy

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M25">View MathML</a>

(3.1)

where the denominator function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M26">View MathML</a> (see [11,12]) is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M27">View MathML</a>

It is noted that discretized model (3.1) can be considered as using the standard Euler method for the first derivative and a nonlocal expression for nonlinear terms. It can be easily rearranged to get its explicit version

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M28">View MathML</a>

(3.2)

Since all the parameters in model (3.2) are positive, it is clear that if the initial values <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M30">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M31">View MathML</a> are positive, then the numerical solutions will also be positive for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M32">View MathML</a>, namely

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M33">View MathML</a>

Defining <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M34">View MathML</a> and adding the three equations of model (3.1), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M35">View MathML</a>

(3.3)

which is the exact finite scheme for the conservation law as expressed by equation (2.1) (see [16]). A straightforward calculation gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M36">View MathML</a>

Similar to the continuous-time case, discrete model (3.1) or equivalent (3.2) also has a compact, positivity invariant set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M37">View MathML</a>

It is easy to verify that discrete model (3.1) or equivalent (3.2) has the same equilibrium as model (1.1) which is independent of h. It can be described as the following theorem.

Theorem 3For model (3.1) or equivalent (3.2), there always exists a disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M38">View MathML</a>and has a unique endemic equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39">View MathML</a>when<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M41">View MathML</a>

4 Global asymptotic stability of disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a>

In this section, we mainly discuss the global asymptotic stability of disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a>. First, we consider the local stability of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a> for model (3.1). In order to obtain the local stability of equilibria for discrete model (3.1) or equivalent (3.2), we notice that the first two equations in model (3.1) or (3.2) do not depend on the third equation, and therefore the third equation can be omitted without loss of generality properties. For the sake of simplicity, we define the following functions [19]:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M45">View MathML</a>

Obviously, the Jacobian matrix [20,21] at the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M46">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M47">View MathML</a>

(4.1)

Now, we give the following theorem about the local stability of the disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a> for model (3.1).

Theorem 4If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17">View MathML</a>, the disease-free equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a>of discrete model (3.1) or equivalent (3.2) is locally asymptotically stable in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M51">View MathML</a>. On the other hand, if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a>is unstable.

Proof Substituting the disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a> to Jacobian matrix (4.1) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M55">View MathML</a>

The eigenvalues of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M56">View MathML</a> are as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M57">View MathML</a>

Obviously <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M58">View MathML</a> for all h. From the definition of basic reproductive number (2.4), we can easily conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17">View MathML</a> is equivalent to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M60">View MathML</a>. Thus, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M17">View MathML</a>, then the magnitude of eigenvalue <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M62">View MathML</a> is also strictly less than unity irrespective of h. This completes the proof. □

Next, let us consider the global asymptotic stability of disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a> for model (3.1). Since the variable R does not appear in the first and the second equations, it is sufficient to consider the following 2-dimensional system:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M64">View MathML</a>

(4.2)

Set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M65">View MathML</a>

Similar to the Izzo et al. [[22], proof of Lemma 3.3], we obtain the following basic lemma.

Lemma 1For any solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M66">View MathML</a>of model (4.2), with the initial conditions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M68">View MathML</a>, we have that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M69">View MathML</a>

(4.3)

Furthermore, we define the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M70">View MathML</a> as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M71">View MathML</a>

then we easily get the following lemma.

Lemma 2<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M70">View MathML</a>is strictly monotone increasing function on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M73">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M74">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M75">View MathML</a>

(4.4)

Moreover, if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then there exists a unique solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M77">View MathML</a>of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M78">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M79">View MathML</a>

By applying techniques in Izzo et al.[22], we now prove the global stability of the disease-free equilibrium for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>.

Theorem 5If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>, then the disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a>of model (3.1) is globally asymptotically stable.

Proof From (4.3) in Lemma 1, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M83">View MathML</a>, there exists an integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M85">View MathML</a>

Construct the following sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M86">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M87">View MathML</a>

(4.5)

Then, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M89">View MathML</a>

Since ϵ is arbitrary, we conclude that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M90">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M91">View MathML</a>

which yields that the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M92">View MathML</a> is monotone decreasing. Therefore, there exists a nonnegative constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M94">View MathML</a>. We will prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>. In fact, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M97">View MathML</a>, then we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M98">View MathML</a>. Consequently, from (4.5), we have that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95">View MathML</a>. Meanwhile, by Lemma 1, we can conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M100">View MathML</a>. Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M101">View MathML</a>, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M102">View MathML</a>. We first transform model (4.2) into the following form:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M103">View MathML</a>

(4.6)

We claim that there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M105">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M106">View MathML</a>, the claim is evident. Now we consider the case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M107">View MathML</a>. By applying the first inequality in (4.4), we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M108">View MathML</a>. Therefore, there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M110">View MathML</a>. From the first equation of (4.6), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M111">View MathML</a>

that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M112">View MathML</a>

As <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M113">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M114">View MathML</a>

from which it is not difficult to obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M115">View MathML</a>

(4.7)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M116">View MathML</a>

(4.8)

We easily find that (4.7) implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M117">View MathML</a>. Meanwhile, combining (4.8) with (4.5), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M118">View MathML</a>. Therefore, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M95">View MathML</a>, which yields <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M120">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M121">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M122">View MathML</a>.

Finally, we will prove that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M124">View MathML</a> is uniformly stable. First, we consider the case that there exists a nonnegative integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M126">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M127">View MathML</a>. From the first equation of (3.1), we have that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M129">View MathML</a>

then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M131">View MathML</a>

which implies that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M128">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M133">View MathML</a>

(4.9)

Meanwhile, from the second equation of model (3.1), we have that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M135">View MathML</a>

From the third equation of model (3.1), it is not difficult to obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M136">View MathML</a>

which implies that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M137">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M138">View MathML</a>

(4.10)

Next, we consider the case that there exists a nonnegative integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M140">View MathML</a>. Then, by the first equation of (3.1), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M141">View MathML</a>

Thus, we have, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M143">View MathML</a>. Then, by the second equation of (3.1) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>, we have that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M146">View MathML</a>

(4.11)

Moreover, by the first equation of model (3.1), we have, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M148">View MathML</a>

which implies that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M150">View MathML</a>

(4.12)

By applying (4.11), we conclude that (4.10) holds for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M142">View MathML</a>. Thus, from (4.9)-(4.12), we conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M152">View MathML</a> is uniformly stable. Hence, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M152">View MathML</a> is globally asymptotically stable. □

5 Global asymptotic stability of the endemic equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a>

In this section, we mainly discuss the global dynamics of the endemic equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a> of model (3.1). Before we prove the stability of the endemic equilibrium, we first give the following lemma.

Lemma 3[23,24]

The quadratic equation<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M157">View MathML</a>has two roots that satisfy<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M159">View MathML</a>, if and only if the following conditions are satisfied:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M160">View MathML</a>,

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M161">View MathML</a>,

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M162">View MathML</a>.

Theorem 6If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then the endemic equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a>of discrete model (3.1) or equivalent (3.2) is locally asymptotically stable in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M51">View MathML</a>.

Proof Assuming that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a> and substituting the endemic equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a> to Jacobian matrix (4.1) lead to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M168">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M169">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M170">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M171">View MathML</a>.

The characteristic equation of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M172">View MathML</a> is given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M157">View MathML</a>, straightforward calculation gives

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M174">View MathML</a>

Obviously,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M175">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M176">View MathML</a>, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M177">View MathML</a>. Finally, using the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, it is easy to obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M179">View MathML</a>

Hence all the conditions in Lemma 3 are satisfied when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19">View MathML</a>. This proves that when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then the endemic equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39">View MathML</a> is locally asymptotically stable for any h. □

Next, we will prove the permanence of model (3.1) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>. Similar to the result of McCluskey in [25,26], we first give the following lemma.

Lemma 4If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M184">View MathML</a>holds, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M185">View MathML</a>; inversely, if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M186">View MathML</a>holds, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M187">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M188">View MathML</a>

Proof By the second equation of model (3.1), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M189">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M186">View MathML</a> holds, then we have that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M191">View MathML</a>

This completes the proof of Lemma 4. □

Theorem 7If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then for any solution of model (3.1) with the initial conditions that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M68">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M195">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M196">View MathML</a>

(5.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M197">View MathML</a>

(5.2)

where the constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M198">View MathML</a>is sufficiently large such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M199">View MathML</a>.

Proof For any positive constant ϵ, there exists a sufficiently large positive integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M201">View MathML</a>

(5.3)

By the first equation of models (3.1) and (5.3), we have that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M203">View MathML</a>

Since ϵ is arbitrary, we conclude that (5.1) holds. Now we prove that (5.2) holds. In fact, for any positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M204">View MathML</a>, it is seen that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M205">View MathML</a>

We first claim that any solution of model (3.1) does not have the following property: there exists a nonnegative integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M207">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208">View MathML</a>. Suppose on the contrary that there exists a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M209">View MathML</a> of model (3.1) and a nonnegative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M125">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M207">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208">View MathML</a>, it can be seen that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M214">View MathML</a>

Consequently, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M208">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M216">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M217">View MathML</a>, therefore, there exists a positive <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M198">View MathML</a> such that for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M219">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M220">View MathML</a>

We hence set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M221">View MathML</a>

Thus, by applying Lemma 1, we have that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M187">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M223">View MathML</a>.

Furthermore, for the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M224">View MathML</a> defined by (4.5), we have that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M225">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M226">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M227">View MathML</a>, this leads to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M228">View MathML</a>, which yields a contradiction. Hence the claim is proved.

By the claim above, we are left to consider two possibilities. First, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M229">View MathML</a> for all n sufficiently large. If this case holds, we get the conclusion of the proof. Second, we investigate the case that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M230">View MathML</a> oscillates about <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M231">View MathML</a> for all n sufficiently large. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M232">View MathML</a> be sufficiently large such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M233">View MathML</a>

By the second equation of model (3.1), we obtain, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M234">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M235">View MathML</a>

Thus, we have that for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M236">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M237">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M238">View MathML</a>, then by applying a similar discussion above, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M239">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M240">View MathML</a>. We hence prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M241">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M242">View MathML</a>. Since the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M243">View MathML</a> is arbitrarily chosen, we conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M241">View MathML</a> for all n sufficiently large. Meanwhile, since q is also arbitrary, we conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M245">View MathML</a>. This completes the proof. □

By Theorem 6, we easily obtain the permanence of model (3.1) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>. Next, by constructing the Lyapunov function, we will prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a> is globally asymptotically stable for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19">View MathML</a>. Consider the Lyapunov function as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M249">View MathML</a>

where the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M250">View MathML</a>. First, by calculating, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M251">View MathML</a>

(5.4)

In the same way, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M252">View MathML</a>

(5.5)

From (5.4) and (5.5), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M253">View MathML</a>

(5.6)

which implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M254">View MathML</a>is a monotone decreasing sequence. We easily get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M255">View MathML</a>

Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M256">View MathML</a>, which combined with (5.6) implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M257">View MathML</a>

(5.7)

From the first equation of models (3.1) and (5.7), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M258">View MathML</a>

Notice that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M259">View MathML</a>, which implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a> is uniformly stable. Finally, we hence obtain the theorem as follows.

Theorem 8If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, then the endemic equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M155">View MathML</a>for model (3.1) or equivalent (3.2) is globally asymptotically stable.

6 Numerical simulations

In this section, numerical simulations will be given to verify theoretical results obtained in the previous section. The simulation is performed using MATLAB software.

(i) We choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M263">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M264">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M265">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M266">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M267">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M268">View MathML</a>. By calculation, we have that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M269">View MathML</a> and the endemic equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M270">View MathML</a>. According to Theorem 5, the disease-free equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M42">View MathML</a> of discrete model (3.1) or equivalent (3.2) is globally stable, which is shown in Figure 1.

thumbnailFigure 1. The solutions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M272">View MathML</a>of model (3.1) or equivalent (3.2) are globally asymptotically stable and converge to the disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M273">View MathML</a>, when<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M274">View MathML</a>.

(ii) Assuming the following parameter values: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M263">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M264">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M265">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M266">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M279">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M268">View MathML</a>, by calculation, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M281">View MathML</a> and the endemic equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M282">View MathML</a>. According to Theorem 8, the endemic equilibrium <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M39">View MathML</a> of discrete model (3.1) or equivalent (3.2) is globally stable, which is depicted in Figure 2.

thumbnailFigure 2. The solutions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M272">View MathML</a>of model (3.1) or equivalent (3.2) are globally asymptotically stable and converge to the disease-free equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M285">View MathML</a>, when<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M286">View MathML</a>.

7 Conclusion

In this paper, we have proposed a discrete-time analogue of the continuous SIR epidemic model of childhood diseases with constant vaccination strategy which is derived by the NSFD scheme of Michens. In order to obtain the permanence of model (3.1) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M19">View MathML</a>, we offer Lemma 4. Applying the discrete Lyapunov functional technique (see [25,26]) for both cases <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M288">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M15">View MathML</a>, it shown that the global dynamics of this discrete-time analogue of the continuous SIR epidemic model is fully determined only by the basic reproduction number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/172/mathml/M290">View MathML</a>. This shows dynamical consistency between the discrete SIR epidemic model and its corresponding continuous model. The NSFD scheme constructed in this paper is for the SIR epidemic model with constant vaccination strategy. For our future work, we will consider an epidemic model with varying vaccination strategy.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The work was supported by the National Natural Science Foundation of P.R. China (11201399, 11301451) and the Natural Science Foundation of Shihezi University (2013ZRKXYQ-YD05).

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