Qualitative behavior of a host-pathogen model

  • Qamar Din1Email author,

    Affiliated with

    • Abdul Qadeer Khan1 and

      Affiliated with

      • Muhammad Naeem Qureshi1

        Affiliated with

        Advances in Difference Equations20132013:263

        DOI: 10.1186/1687-1847-2013-263

        Received: 23 April 2013

        Accepted: 29 July 2013

        Published: 29 August 2013

        Abstract

        In this paper, we study the qualitative behavior of a discrete-time host-pathogen model for spread of an infectious disease with permanent immunity. The time-step is equal to the duration of the infectious phase. Moreover, the local asymptotic stability, the global behavior of unique positive equilibrium point, and the rate of convergence of positive solutions is discussed. Some numerical examples are given to verify our theoretical results.

        MSC:39A10, 40A05.

        Keywords

        difference equations local stability global character

        1 Introduction

        It is a well-known fact that in the population growth, the disease is an important agent controlling the population dynamics. Many experiments show that parasites can reasonably reduce the host population and even take the host population to complete annihilation. This natural phenomenon is successfully modeled by many simple SI type host-parasite models. The most interesting properties of such models are their ability of generating host annihilation dynamics with the ideal parametric values and initial conditions. This is possible, because such models naturally contain the proportion transmission term, which is often referred to as ratio-dependent functional response in the case of predator-prey models. In the SI model, the population is subdivided into two classes, susceptibles S and infectives I. The notation SI means that there is a transfer from the susceptible to infective class, susceptibles become infective and do not recover from the infection. Thus, the transfer continues until all individuals become infected. This type of model is very simple, but may represent some complicated dynamical properties. Most of the SI type models consist of the mass action principle, i.e., the assumption that the new cases arise in a simple proportion to the product of the number of individuals which are susceptible and the number of which are infectious. However, this principle has a limited validity and in the discrete models, this principle leads to biologically irrelevant results, unless some restrictions are suggested for the parameters. It is more appropriate for discrete epidemic models to include an exponential factor in the rate of transmission. Exponential difference equations can be used to study the models in population dynamics [13]. We consider here a simple exponential discrete-time host-pathogen model for spread of an infectious disease with permanent immunity. The time-step is equal to the duration of the infectious phase. The state variables are S n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq1_HTML.gif, the number of susceptible individuals at time n, and I n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq2_HTML.gif representing the number of individuals, getting the disease (new cases) between times n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq3_HTML.gif and n,
        I n + 1 = S n ( 1 e α I n ) , S n + 1 = S n e α I n + β , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ1_HTML.gif
        (1)

        where β is the number of births between n and n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq4_HTML.gif, all added to the susceptible class and assumed to be constant over time. So, the difference equation S n + 1 = S n e α I n + β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq5_HTML.gif is just a ‘conservation of mass’ for the susceptible class. The first part I n + 1 = S n ( 1 e α I n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq6_HTML.gif of the model is just like Nicholson-Bailey; it comes from assuming that each susceptible escapes infection with probability e α I n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq7_HTML.gif; the more infectives there are, the lower the chance of escape. The model ignores mortality in the susceptible class, on the assumption that everyone gets the disease while young, and mortality occurs later in life.

        Many ecological competition models are governed by differential and difference equations. We refer to [4, 5] and the references therein for some interesting results, related to the global character and local asymptotic stability. As it is pointed out in [6, 7], the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. The study of the discrete-time models described by difference equations has now been given a great attention, since these models are more reasonable than the continuous time models when populations have non-overlapping generations. Discrete-time models give rise to more efficient computational models for numerical simulations and also show rich dynamics compared to the continuous ones. In recent years, many papers have been published on the mathematical models of biology that discussed the system of difference equations generated from the associated system of differential equations as well as the associated numerical methods. Mathematical models of epidemics have created a major area of research interest during the last few decades. Recently, theory on the effects of parasites on host population dynamics has received much attention and epidemiological models are often used to explain empirical results for host-parasites interaction system. For more details of such biological models, one can see [810].

        More precisely, our aim is to investigate local asymptotic stability of unique positive equilibrium point, the global asymptotic character of equilibrium point, and the rate of convergence of positive solutions of system (1). For more results for the systems of difference equations, we refer the reader to [1114].

        2 Boundedness and persistence

        The following theorem shows that every positive solution { ( I n , S n ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq8_HTML.gif of system (1) is bounded and persists.

        Theorem 1 Every positive solution { ( I n , S n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq9_HTML.gif of system (1) is bounded and persists.

        Proof Let { ( I n , S n ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq10_HTML.gif be any positive solution of system (1). It is easy to see that S n β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq11_HTML.gif and I n β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq12_HTML.gif for all n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq13_HTML.gif . Then, it follows that I n + 1 β ( 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq14_HTML.gif and S n + 1 S n e α β ( 1 e α β ) + β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq15_HTML.gif. Consider the following difference equation
        z n + 1 = z n e α β ( 1 e α β ) + β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equa_HTML.gif
        with initial condition z 0 S 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq16_HTML.gif, then its solution is given by
        z n = z 0 A n 1 + β ( 1 A n ) 1 A , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equb_HTML.gif
        where A = e α β ( 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq17_HTML.gif. Since e α β ( 1 e α β ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq18_HTML.gif, therefore,
        z n β 1 e α β ( 1 e α β ) as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equc_HTML.gif
        Then, by comparison we have S n z n β 1 e α β ( 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq19_HTML.gif. Hence,
        β ( 1 e α β ) I n β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equd_HTML.gif
        and
        β S n β 1 e α β ( 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Eque_HTML.gif

        for all n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq13_HTML.gif . □

        Theorem 2 Let { ( I n , S n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq9_HTML.gif be a positive solution of the system (1). Then, [ β ( 1 e α β ) , β ] × [ β , β 1 e α β ( 1 e α β ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq20_HTML.gif is an invariant set for system (1).

        Proof Let { ( I n , S n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq9_HTML.gif be a positive solution of system (1) with initial conditions I 0 I = [ β ( 1 e α β ) , β ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq21_HTML.gif and S 0 J = [ β , β 1 e α β ( 1 e α β ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq22_HTML.gif. Then, from system (1)
        I 1 = S 0 ( 1 e α I 0 ) β ( 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equf_HTML.gif
        and
        I 1 = S 0 ( 1 e α I 0 ) β 1 e α β ( 1 e α β ) ( 1 e α β ( 1 e α β ) ) = β . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equg_HTML.gif
        Similarly, we have
        S 1 = S 0 e α I 0 + β β e α β + β = β ( 1 + e α β ) β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equh_HTML.gif
        and
        S 1 = S 0 e α I 0 + β β 1 e α β ( 1 e α β ) e α β ( 1 e α β ) + β β 1 e α β ( 1 e α β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equi_HTML.gif
        Hence, I 1 I http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq23_HTML.gif and S 1 J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq24_HTML.gif. Suppose that the result is true for n = k > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq25_HTML.gif, i.e., I k I http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq26_HTML.gif and S k J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq27_HTML.gif. Then from system (1), one can easily obtain
        β ( 1 e α β ) I k + 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equj_HTML.gif
        and
        β S k + 1 β 1 e α β ( 1 e α β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equk_HTML.gif

        Hence, the proof is completed. □

        3 Linearized stability

        Let us consider two-dimensional discrete dynamical system of the form
        x n + 1 = f ( x n , y n ) , y n + 1 = g ( x n , y n ) , n = 0 , 1 , , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ2_HTML.gif
        (2)
        where f : I × J I http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq28_HTML.gif and g : I × J J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq29_HTML.gif are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution { ( x n , y n ) } n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq30_HTML.gif of system (2) is uniquely determined by initial conditions ( x 0 , y 0 ) I × J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq31_HTML.gif. An equilibrium point of (2) is a point ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq32_HTML.gif that satisfies
        x ¯ = f ( x ¯ , y ¯ ) , y ¯ = g ( x ¯ , y ¯ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equl_HTML.gif
        Let ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq32_HTML.gif be an equilibrium point of a map F ( x , y ) = ( f ( x , y ) , g ( x , y ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq33_HTML.gif, where f and g are continuously differentiable functions at ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq34_HTML.gif. The linearized system of (2) about the equilibrium point ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq35_HTML.gif is
        X n + 1 = F ( X n ) = F J X n , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equm_HTML.gif

        where X n = ( x n y n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq36_HTML.gif and F J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq37_HTML.gif is Jacobian matrix of system (2) about the equilibrium point ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq35_HTML.gif.

        Let ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq38_HTML.gif be the equilibrium point of system (1), then one has
        I ¯ = S ¯ ( 1 e α I ¯ ) , S ¯ = S ¯ e α I ¯ + β . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equn_HTML.gif
        Then, it follows that ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq39_HTML.gif is the unique positive equilibrium point of system (1). Moreover, the Jacobian matrix F J ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq40_HTML.gif of system (1) about the equilibrium point ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq38_HTML.gif is given by
        F J ( I ¯ , S ¯ ) = ( α S ¯ e α I ¯ 1 e α I ¯ α S ¯ e α I ¯ e α I ¯ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equo_HTML.gif
        The characteristic polynomial of F J ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq40_HTML.gif is given by
        P ( λ ) = λ 2 e α I ¯ ( 1 + α S ¯ ) λ + α S ¯ e α I ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ3_HTML.gif
        (3)

        Lemma 1 [15]

        Consider the second-degree polynomial equation
        λ 2 + p λ + q = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ4_HTML.gif
        (4)
        where p and q are real numbers. Then, the necessary and sufficient condition for both roots of Equation (4) to lie inside the open disk | λ | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq41_HTML.gif is
        | p | < 1 + q < 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equp_HTML.gif

        Lemma 2 [16]

        Assume that X n + 1 = F ( X n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq42_HTML.gif, n = 0 , 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq43_HTML.gif , is a system of difference equations and X ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq44_HTML.gif is the fixed point of F. If all eigenvalues of the Jacobian matrix J F http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq45_HTML.gif about X ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq44_HTML.gif lie inside the open unit disk | λ | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq46_HTML.gif, then X ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq44_HTML.gif is locally asymptotically stable. If one of them has a modulus greater than one, then X ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq44_HTML.gif is unstable.

        Theorem 3 Assume that e α β ( 1 + α β ) < 1 + e 2 α β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq47_HTML.gif. Then, the unique positive equilibrium point ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq48_HTML.gif is locally asymptotically stable.

        Proof The characteristic polynomial of F J ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq40_HTML.gif about positive equilibrium point ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq49_HTML.gif is given by
        P ( λ ) = λ 2 ( e α β + e α β α β 1 e α β ) λ + e α β α β 1 e α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ5_HTML.gif
        (5)
        Let
        f ( λ ) = λ 2 , g ( λ ) = ( e α β + e α β α β 1 e α β ) λ e α β α β 1 e α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equq_HTML.gif
        Assume that e α β ( 1 + α β ) < 1 + e 2 α β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq47_HTML.gif, and | λ | = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq50_HTML.gif. Then, one has
        | g ( λ ) | ( e α β + e α β α β 1 e α β ) + e α β α β 1 e α β = e α β + 2 e α β α β 1 e α β = e α β + 2 α β e α β 1 e 2 α β e α β < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equr_HTML.gif
        Then, by Rouche’s theorem f ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq51_HTML.gif and f ( λ ) g ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq52_HTML.gif have the same number of zeroes in an open unit disk | λ | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq41_HTML.gif. Hence, both roots
        λ 1 = 1 e α β e α β α β + 4 e α β ( e α β + e 2 α β ) α β + ( 1 e α β e α β α β ) 2 2 ( e α β e 2 α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equs_HTML.gif
        and
        λ 2 = 1 + e α β + e α β α β + 4 e α β ( e α β + e 2 α β ) α β + ( 1 e α β e α β α β ) 2 2 ( e α β + e 2 α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equt_HTML.gif

        of (5) lie in an open disk | λ | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq41_HTML.gif, and it follows from Lemma 2 that the equilibrium point ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq53_HTML.gif is locally asymptotically stable. □

        The following theorem shows the necessary and sufficient condition for the local asymptotic stability of a unique positive equilibrium point of system (1).

        Theorem 4 The unique positive equilibrium point ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq48_HTML.gif of system (1) is locally asymptotically stable if and only if 1 + α β e α β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq54_HTML.gif.

        Proof Let p = ( e α β + e α β α β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq55_HTML.gif and q = e α β α β 1 e α β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq56_HTML.gif, then (5) can be written as
        P ( λ ) = λ 2 + p λ + q . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equu_HTML.gif

        Then, | p | = e α β + e α β α β 1 e α β < 1 + e α β α β 1 e α β = 1 + q http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq57_HTML.gif and 1 + q = 1 + e α β α β 1 e α β < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq58_HTML.gif if and only if 1 + α β e α β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq59_HTML.gif. Hence, from Lemma 1, the unique positive equilibrium point ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq48_HTML.gif of system (1) is locally asymptotically stable if and only if 1 + α β e α β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq54_HTML.gif. □

        4 Global character

        The following lemma is similar to Theorem 1.16 of [15].

        Lemma 3 Let I = [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq60_HTML.gif and J = [ c , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq61_HTML.gif be real intervals, and let f : I × J I http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq28_HTML.gif and g : I × J J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq29_HTML.gif be continuous functions. Consider system (2) with initial conditions ( x 0 , y 0 ) I × J http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq31_HTML.gif. Suppose that the following statements are true:
        1. (i)

          f ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq62_HTML.gif is non-decreasing in both arguments.

           
        2. (ii)

          g ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq63_HTML.gif is non-increasing in x, and non-decreasing in y.

           
        3. (iii)
          If ( m 1 , M 1 , m 2 , M 2 ) I 2 × J 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq64_HTML.gif is a solution of the system
          m 1 = f ( m 1 , m 2 ) , M 1 = f ( M 1 , M 2 ) , m 2 = g ( M 1 , m 2 ) , M 2 = g ( m 1 , M 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equv_HTML.gif
           

        such that m 1 = M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq65_HTML.gif, and m 2 = M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq66_HTML.gif.

        Then, there exists exactly one equilibrium point ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq32_HTML.gif of the system (2) such that lim n ( x n , y n ) = ( x ¯ , y ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq67_HTML.gif.

        Theorem 5 The unique positive equilibrium point ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq48_HTML.gif of system (1) is a global attractor.

        Proof Let f ( x , y ) = y ( 1 e α x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq68_HTML.gif, and g ( x , y ) = y e α x + β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq69_HTML.gif. Then, it is easy to see that f ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq62_HTML.gif is non-decreasing in both x and y. Moreover, g ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq63_HTML.gif is non-increasing in x, and non-decreasing in y. Let ( m 1 , M 1 , m 2 , M 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq70_HTML.gif be a solution of the system
        m 1 = f ( m 1 , m 2 ) , M 1 = f ( M 1 , M 2 ) , m 2 = g ( M 1 , m 2 ) , M 2 = g ( m 1 , M 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equw_HTML.gif
        Then, one has
        m 1 = m 2 ( 1 e α m 1 ) , M 1 = M 2 ( 1 e α M 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ6_HTML.gif
        (6)
        and
        m 2 = m 2 e α M 1 + β , M 2 = M 2 e α m 1 + β . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ7_HTML.gif
        (7)
        From system (6), one has
        e α m 1 = m 2 m 1 m 2 , e α M 1 = M 2 M 1 M 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ8_HTML.gif
        (8)
        From (7), one has
        e α m 1 = M 2 β M 2 , e α M 1 = m 2 β m 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ9_HTML.gif
        (9)
        Furthermore, assuming as in the proof of Theorem 1.16 of [15], it suffices to suppose that
        m 1 M 1 , m 2 M 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equx_HTML.gif
        Using the fact that e α m 1 e α M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq71_HTML.gif, one has from (8) and (9)
        m 2 m 1 m 2 m 2 β m 2 , M 2 β M 2 M 2 M 1 M 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ10_HTML.gif
        (10)
        It follows from (10) that m 1 β M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq72_HTML.gif. Then, (7) implies that
        M 2 m 2 = β ( e α m 1 e α M 1 ) ( 1 e α m 1 ) ( 1 e α M 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ11_HTML.gif
        (11)
        Using (6) in (11), we obtain
        | M 2 m 2 | = β | M 2 M 1 m 2 m 1 | . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equy_HTML.gif

        Following the same technique as in the proof of Proposition 4.1 of [17], one has | M 2 m 2 | β M 1 | M 2 m 2 | http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq73_HTML.gif, i.e., ( 1 β M 1 ) | M 2 m 2 | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq74_HTML.gif. Thus, m 2 = M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq66_HTML.gif and, similarly, one can show that m 1 = M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq65_HTML.gif. Hence, from Lemma 3, the equilibrium point ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq75_HTML.gif of system (1) is a global attractor. □

        Lemma 4 The unique positive equilibrium point ( I ¯ , S ¯ ) = ( β , β 1 e α β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq76_HTML.gif of system (1) is globally asymptotically stable if and only if 1 + α β e α β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq77_HTML.gif.

        Proof The proof follows from Theorem 4 and Theorem 5. □

        5 Rate of convergence

        In this section, we determine the rate of convergence of a solution that converges to the unique positive equilibrium point of system (1). Similar methods can be found in [18] and [19].

        The following result gives the rate of convergence of solutions of a system of difference equations
        X n + 1 = ( A + B ( n ) ) X n , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ12_HTML.gif
        (12)
        where X n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq78_HTML.gif is an m-dimensional vector, A C m × m http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq79_HTML.gif is a constant matrix, and B : Z + C m × m http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq80_HTML.gif is a matrix function satisfying
        B ( n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ13_HTML.gif
        (13)
        as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq81_HTML.gif, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq82_HTML.gif denotes any matrix norm, which is associated with the vector norm
        ( x , y ) = x 2 + y 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equz_HTML.gif

        Proposition 1 (Perron’s theorem [20])

        Suppose that condition (13) holds. If X n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq78_HTML.gif is a solution of (12), then either X n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq83_HTML.gif for all large n or
        ρ = lim n ( X n ) 1 / n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ14_HTML.gif
        (14)

        exists and is equal to the modulus of one the eigenvalues of matrix A.

        Proposition 2 [20]

        Suppose that condition (13) holds. If X n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq78_HTML.gif is a solution of (12), then either X n = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq83_HTML.gif for all large n or
        ρ = lim n X n + 1 X n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ15_HTML.gif
        (15)

        exists and is equal to the modulus of one the eigenvalues of matrix A.

        Let { ( I n , S n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq9_HTML.gif be any solution of system (1) such that lim n I n = I ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq84_HTML.gif, and lim n S n = S ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq85_HTML.gif. To find the error terms, one has from system (1)
        I n + 1 I ¯ = S n ( 1 e α I n ) S ¯ ( 1 e α I ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equaa_HTML.gif
        and
        S n + 1 S ¯ = S n e α I n S ¯ e α I ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equab_HTML.gif
        Let e n 1 = I n I ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq86_HTML.gif, and let e n 2 = S n S ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq87_HTML.gif, then one has
        e n + 1 1 = a n e n 1 + b n e n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equac_HTML.gif
        and
        e n + 1 2 = c n e n 1 + d n e n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equad_HTML.gif
        where
        a n = S ¯ ( e α I ¯ e α I n ) I n I ¯ , b n = 1 e α I n , c n = S ¯ ( e α I n e α I ¯ ) I n I ¯ , d n = e α I n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equae_HTML.gif
        Moreover,
        lim n a n = α S ¯ e α I ¯ , lim n b n = 1 e α I ¯ , lim n c n = α S ¯ e α I ¯ , lim n d n = e α I ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equaf_HTML.gif
        Now, the limiting system of error terms can be written as
        [ e n + 1 1 e n + 1 2 ] = [ α S ¯ e α I ¯ 1 e α I ¯ α S ¯ e α I ¯ e α I ¯ ] [ e n 1 e n 2 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equag_HTML.gif

        which is similar to linearized system of (1) about the equilibrium point ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq88_HTML.gif.

        Using Proposition 1, one has the following result.

        Theorem 6 Assume that { ( I n , S n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq9_HTML.gif is a positive solution of system (1) such that lim n I n = I ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq84_HTML.gif and lim n S n = S ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq85_HTML.gif, where ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq38_HTML.gif is a unique positive equilibrium point of (1). Then, the error vector e n = ( e n 1 e n 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq89_HTML.gif of every solution of (1) satisfies both of the following asymptotic relations
        lim n ( e n ) 1 n = | λ 1 , 2 F J ( I ¯ , S ¯ ) | , lim n e n + 1 e n = | λ 1 , 2 F J ( I ¯ , S ¯ ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equah_HTML.gif

        where λ 1 , 2 F J ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq90_HTML.gif are the characteristic roots of the Jacobian matrix F J ( I ¯ , S ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq40_HTML.gif.

        6 Examples

        In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference equations (1). All plots in this section are drawn with Mathematica.

        Example 1 Let α = 0.05 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq91_HTML.gif, and let β = 0.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq92_HTML.gif. Then, system (1) can be written as
        x n + 1 = y n ( 1 e 0.05 x n ) , y n + 1 = y n e 0.05 x n + 0.3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ16_HTML.gif
        (16)

        with initial conditions x 0 = 0.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq93_HTML.gif, y 0 = 20 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq94_HTML.gif.

        In this case, the unique equilibrium point ( β , β 1 e α β ) = ( 0.3 , 20.1504 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq95_HTML.gif. Moreover, in Figure 1, the plot of x n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq96_HTML.gif is shown in Figure 1(a), the plot of y n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq97_HTML.gif is shown in Figure 1(b), and an attractor of system (16) is shown in Figure 1(c). The basic reproductive number of system (16) is R 0 = 1 + α β e α β = 0.999889 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq98_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Fig1_HTML.jpg
        Figure 1

        Plots for system ( 16 ).

        Example 2 Let α = 0.2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq99_HTML.gif, and let β = 1.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq100_HTML.gif. Then, system (1) can be written as
        x n + 1 = y n ( 1 e 0.2 x n ) , y n + 1 = y n e 0.2 x n + 1.5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ17_HTML.gif
        (17)

        with initial conditions x 0 = 1.4 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq101_HTML.gif, y 0 = 5.7 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq102_HTML.gif.

        In this case, the unique equilibrium point ( β , β 1 e α β ) = ( 1.5 , 5.78744 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq103_HTML.gif. Moreover, in Figure 2, the plot of x n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq96_HTML.gif is shown in Figure 2(a), the plot of y n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq97_HTML.gif is shown in Figure 2(b), and an attractor of system (16) is shown in Figure 2(c). The basic reproductive number of system (17) is R 0 = 1 + α β e α β = 0.963064 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq104_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Fig2_HTML.jpg
        Figure 2

        Plots for system ( 17 ).

        Example 3 Let α = 0.9 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq105_HTML.gif, and let β = 0.01 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq106_HTML.gif. Then, system (1) can be written as
        x n + 1 = y n ( 1 e 0.9 x n ) , y n + 1 = y n e 0.9 x n + 0.01 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ18_HTML.gif
        (18)

        with initial conditions x 0 = 0.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq107_HTML.gif, y 0 = 1.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq108_HTML.gif.

        In this case, the unique equilibrium point ( β , β 1 e α β ) = ( 0.01 , 1.11612 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq109_HTML.gif. Moreover, in Figure 3, the plot of x n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq96_HTML.gif is shown in Figure 3(a), the plot of y n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq97_HTML.gif is shown in Figure 3(b), and an attractor of system (18) is shown in Figure 3(c). The basic reproductive number of system (18) is R 0 = 1 + α β e α β = 0.99996 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq110_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Fig3_HTML.jpg
        Figure 3

        Plots for system ( 18 ).

        Example 4 Let α = 0.01 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq111_HTML.gif, and let β = 0.05 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq112_HTML.gif. Then, system (1) can be written as
        x n + 1 = y n ( 1 e 0.01 x n ) , y n + 1 = y n e 0.01 x n + 0.05 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ19_HTML.gif
        (19)

        with the initial conditions x 0 = 0.02 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq113_HTML.gif, y 0 = 100 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq114_HTML.gif.

        In this case, the unique equilibrium point ( β , β 1 e α β ) = ( 0.05 , 100.025 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq115_HTML.gif. Moreover, in Figure 4, the plot of x n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq96_HTML.gif is shown in Figure 4(a), the plot of y n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq97_HTML.gif is shown in Figure 4(b), and an attractor of the system (19) is shown in Figure 4(c). The basic reproductive number of system (19) is R 0 = 1 + α β e α β = 0.9999998750416589 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq116_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Fig4_HTML.jpg
        Figure 4

        Plots for system ( 19 ).

        Example 5 Let α = 0.0001 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq117_HTML.gif, and let β = 50 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq118_HTML.gif. Then, system (1) can be written as
        x n + 1 = y n ( 1 e 0.0001 x n ) , y n + 1 = y n e 0.0001 x n + 50 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Equ20_HTML.gif
        (20)

        with the initial conditions x 0 = 40 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq119_HTML.gif, y 0 = 9 , 000 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq120_HTML.gif.

        In this case, the unique equilibrium point ( β , β 1 e α β ) = ( 50 , 10 , 025 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq121_HTML.gif. Moreover, in Figure 5, the plot of x n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq96_HTML.gif is shown in Figure 5(a), the plot of y n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq97_HTML.gif is shown in Figure 5(b), and an attractor of system (20) is shown in Figure 5(c). The basic reproductive number of system (20) is R 0 = 1 + α β e α β = 0.9999875415886458 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq122_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_Fig5_HTML.jpg
        Figure 5

        Plots for system ( 20 ).

        Conclusion and future work

        This work is related to the qualitative behavior of an exponential discrete-time host-pathogen model for spread of an infectious disease with permanent immunity. We proved that system (1) has a unique positive equilibrium point, which is locally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In the paper, a general result for global character for such type of systems is proved. Due to the simplicity of our SI-type model, we have carried out a systematic local and global stability analysis of it. The most important finding here is that the unique positive equilibrium point can be a global asymptotic attractor for model (1). Moreover, the rate convergence of positive solutions has also been investigated. In such models, there is a threshold parameter that might tell whether a population will increase or die out, or whether an infectious disease will persist or die out within a population. This parameter is commonly known as the basic reproductive number and is denoted by R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq123_HTML.gif. In epidemiology, this number R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq123_HTML.gif is defined as the number of newly infected individual, produced by a single infected individual in its period of infectivity. In case of system (1), the basic reproductive number is given by R 0 = 1 + α β e α β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq124_HTML.gif. From our investigations, it is obvious that the unique positive equilibrium point of system (1) is globally asymptotically stable if R 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq125_HTML.gif, and unstable if R 0 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-263/MediaObjects/13662_2013_Article_599_IEq126_HTML.gif. Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions. The qualitative behavior of the general model, where there is host mortality at some constant rate, will be our next aim to study.

        Declarations

        Acknowledgements

        The authors would like to thank the anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.

        Authors’ Affiliations

        (1)
        Department of Mathematics, University of Azad Jammu and Kashmir

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