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Multidimensional discrete Halanay inequalities and the global stability of the disease free equilibrium of a discrete delayed malaria model
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 113 (2016)
Abstract
In this paper, we establish two multidimensional discrete Halanaytype inequalities. As an application, we consider a malaria transmission model with two delays and employ the two discrete Halanaytype inequalities to establish the global (exponential) asymptotical stability of the disease free equilibrium of the model. It is further shown that the disease free equilibrium of the delayed model is globally asymptotically stable when the basic reproduction number of the model is less than 1.
1 Introduction
The Halanay inequality was first introduced in [1] and many generalizations of this inequality have been obtained due to their significance in the analysis of delayed dynamical systems [2â€“9], and especially in proving the global exponential stability of the equilibria of mathematical models proposed in neural networks and biology with time delays [8â€“11]. The original form of the Halanay inequality and some of its generalizations are for continuous dynamical systems with only one unknown scalar function involved in the system [1, 6, 9], i.e., the onedimensional case. In [9], Halanay inequality was generalized to a multidimensional form to deal with the stability of the equilibria of continuous dynamical systems in neural networks with impulses.
Along with the development of Halanaytype inequalities for continuoustime dynamical systems, discretetime Halanay inequalities have also been established to handle the stability of discrete dynamical systems with delays [12â€“20]. For example, the following result was obtained in [15].
Theorem 1.1
(TheoremÂ 1 in [15])
Let \(r>0\) be a natural number, and let \(\{x(n)\}_{n\ger}\) be a sequence of real numbers satisfying the inequality
where Î” is the forward difference operator and \(\Delta x(n)=x(n+1)x(n)\). If \(0< b< a\le1\), then there exists a constant \(\lambda_{0}\in(0,1)\) such that
where \(\lambda_{0}\) is the smallest root in the interval \((0,1)\) of the equation
Based on TheoremÂ 1.1, different generalizations of discrete Halanaytype inequalities have been developed [12, 13, 16, 19]. However, the generalizations are mostly dealing with onedimensional functions, such as the sequence \(\{x(n)\}\) in TheoremÂ 1.1; while generalizations of discrete Halanaytype inequalities for multidimensional sequences \(\{X(n)\}=\{(x_{1}(n), x_{2}(n), \ldots, x_{m}(n))^{T}\}\) are hardly developed. Motivated by this, the first aim of our present work is to develop certain multidimensional discrete Halanaytype inequalities.
On the other hand, the global stability of the equilibria of epidemic models is a key research topic in the quantitative analysis of the transmission of infectious diseases. Much research on this topic has been done on compartmental models of infectious diseases such as influenza, malaria, dengue, cholera, etc. [21â€“25]. Usually, a compartmental infectious disease model has a disease free equilibrium, which is globally asymptotically stable when the basic reproduction number \(R_{0}\) of the model is less than 1, and it has a positive equilibrium which is globally asymptotically stable when \(R_{0}>1\) [26]. Time delays are frequently involved in these infectious disease models [21â€“23, 27] and their presence poses great difficulty in stability analysis, especially in analyzing the global asymptotical stability of the equilibria of these models. For infectious disease models with two or more delays involved, the local asymptotical stability of the equilibrium can be obtained through the analysis of the eigenvalues of the Jacobian matrix of the linearized model at the corresponding equilibrium, but the global asymptotical stability of the equilibria of some of these infectious disease models with several delays remains unsolved at present [22, 23]. Hence, another aim of this work is to establish the global asymptotical stability of the disease free equilibrium of a malaria transmission model with two delays. We shall tackle this by applying the multidimensional discrete Halanay inequalities developed in this paper. Moreover, we shall show that the disease free equilibrium of the model is globally asymptotically stable when the basic reproduction number \(R_{0}\) is less than 1, which is well consistent with the threshold property of the basic reproduction number.
The paper is organized as follows. In SectionÂ 2, we present two generalizations of TheoremÂ 1.1 to multidimensional case. Applications of these two generalizations are given in SectionÂ 3 to obtain the global asymptotical stability of the disease free equilibrium of a malaria transmission model with two delays. Finally, some concluding remarks are given in SectionÂ 4.
2 Generalized discrete Halanaytype inequalities
In this section, we shall establish two generalizations of TheoremÂ 1.1 to multidimensional case. First, we introduce some notations as follows. Throughout, we denote
Let \(X(n)=(x_{1}(n), x_{2}(n), \dots, x_{m}(n))^{T}\) where \(\{x_{i}(n)\}\), \(i\in\mathbb{I}\) is a real sequence. Define
and
Moreover, as usual the forward difference operator Î” is defined by \(\Delta X(n)=X(n+1)X(n)\). Let \(Y(n)=(y_{1}(n), y_{2}(n), \dots, y_{m}(n))^{T}\). The notion \(X \le (\ge) \, Y\) means \(x_{i}(n) \le (\ge)\, y_{i}(n)\) for \(i\in\mathbb{I}\) and \(n\in\mathbb{S}\). Let \(E=\operatorname{diag}(1, 1, \ldots, 1)\) be the unitary matrix of suitable order and \(\delta_{ij}\) be the Kronecker symbol, i.e., \(\delta_{ij}=1\) if \(i=j\) and \(\delta_{ij}=0\) if otherwise.
Our first generalization of TheoremÂ 1.1 is the following result.
Theorem 2.1
Let \(\{X(n)\}_{n\ger}\) be a nonnegative sequence satisfying
and the initial conditions (also known as initial strings in [15])
where \(A=(a_{ij})_{m\times m}\) and \(B=(b_{ij})_{m\times m}\) are two matrices satisfying
and
Then there exists \(\lambda_{i}\in(0,1)\), \(i\in\mathbb{I}\), such that
where \(\lambda_{i}\in(0,1)\) is the root of the equation
for each \(i\in\mathbb{I}\).
Proof
Consider the following system:
with initial values
such that
For each \(i\in\mathbb{I}\), the ith component of (2.1) gives
while the ith component of (2.6) is
Noting that \(a_{ij}+\delta_{ij}\ge0\), \(b_{ij}\ge 0\), \(i,j\in\mathbb{I}\) (from (2.3)) and the initial values satisfy (2.8), by comparison arguments [28] we get
Now, define for each \(i\in\mathbb{I}\),
We observe that \(F_{i}(\lambda)\) is continuous in the interval \([0, 1]\) (with respect to Î»). Further, in view of (2.3) and (2.4),
and
Consequently, there exists \(\lambda_{i}\in(0, 1)\) such that \(F_{i}(\lambda_{i})=0\), i.e., \(\lambda_{i}\) is a root of equation (2.5).
It is also noted that \(F_{i}(\lambda)=0\) is the characteristic equation of (2.10) when \(0<\lambda<1\). Hence, \(y_{i}(n)=K\lambda_{i}^{n}\), \(n\ge0\) is a solution of (2.10), where K is a constant. In particular, we set \(K=\max_{s\in\mathbb{S}_{1}}\{0, x_{i}^{(s)}\}\) such that (2.8) is fulfilled for \(s\in\mathbb{S}_{1}\). Thus, it follows from (2.11) that
The proof is complete.â€ƒâ–¡
Corollary 2.2
Let \(\{X(n)\}_{n\ger}\) be a nonnegative sequence satisfying (2.1) and (2.2). Suppose that A and B also satisfy (2.3) and (2.4). Then there exists \(\mathcal{K}=(k_{1}, k_{2}, \ldots, k_{m})^{T}\), where \(k_{i}, i\in\mathbb{I}\) is a positive constant, and \(\lambda\in(0, 1)\) such that
Proof
Set \(k_{i}=\max_{s\in\mathbb{S}_{1}} \{0, x_{i}^{(s)}\}\), \(i\in\mathbb{I}\) and \(\lambda=\max\{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{m}\}\), where \(\lambda_{i}\) has the same meaning as in TheoremÂ 2.1. It is immediate from TheoremÂ 2.1 that \(X(n)\le\mathcal{K}\lambda^{n}\), \(n\ge0\).â€ƒâ–¡
Corollary 2.3
Let \(\{X(n)\}_{n\ge n_{0}r}\) be a nonnegative sequence satisfying
and the initial conditions
where \(A=(a_{ij})_{m\times m}\) and \(B=(b_{ij})_{m\times m}\) are two matrices satisfying (2.3) and (2.4). Then there exists \(\mathcal{K}=(k_{1}, k_{2}, \ldots, k_{m})^{T}\), where \(k_{i}=\max_{s\in\mathbb{S}_{2}}\{0, x_{i}^{(s)}\}\), \(i\in\mathbb{I}\), and \(\lambda\in(0,1)\) such that
Proof
Set \(nn_{0}=\bar{n}\) and it follows from TheoremÂ 2.1 and CorollaryÂ 2.2 that
i.e., \(X(n)\le\mathcal{K}\lambda^{nn_{0}}\), \(n\ge n_{0}\).â€ƒâ–¡
In the sequel, we shall generalize TheoremÂ 1.1 to another multidimensional case by applying the theory of nonsingular Mmatrix. The following lemma on nonsingular Mmatrix [29] will be needed later.
Lemma 2.4
[29]
Suppose that the matrix \(C=(c_{ij})_{m\times m}\) satisfies \(c_{ij}\le0\), \(i\neq j\), \(i, j\in\mathbb{I}\). The following statements are equivalent to C is a nonsingular Mmatrix.

(1)
All the successive principal minors of C are positive.

(2)
\(c_{ii}>0\) and there exists a positive vector \(z>0\) such that \(Cz>0\).

(3)
\(C=DM\) and \(\rho(D^{1}M)<1\), where \(M\ge0\), \(D=\operatorname{diag}(d_{1}, d_{2}, \dots, d_{m})\) and \(\rho(D^{1}M)\) is the spectral radius of the matrix \(D^{1}M\).
Remark 2.5
For a nonsingular Mmatrix C, we denote
The set \(\Gamma(C)\) is not empty in view of (2) of LemmaÂ 2.4. Also, if \(z\in\Gamma(C)\), then \(kz\in\Gamma(C)\) for any constant \(k>0\).
The second generalization of TheoremÂ 1.1 is stated as follows.
Theorem 2.6
Let \(\{X(n)\}_{n\ger}\) be a nonnegative sequence satisfying (2.1), where \(A=(a_{ij})_{m\times m}\) and \(B=(b_{ij})_{m\times m}\) are two matrices satisfying
and \(C=(A+B)\) is a nonsingular Mmatrix. Moreover, suppose the nonnegative sequence \(\{X(n)\}_{n\ge r}\) satisfies
where \(z=(z_{1}, z_{2}, \ldots, z_{m})^{T}>0\) and \(0<\lambda<1\) is the solution of
Then
Proof
Since \(C=(A+B)\) is a nonsingular Mmatrix, from (2) of LemmaÂ 2.4 there exists \(z>0\) such that \((A+B)z>0\), or equivalently,
Noting that \(A\lambda^{r}+\lambda^{r}E+B\lambda^{r+1}E=A+B\) when \(\lambda=1\), it follows from (2.18) that (2.16) has a solution \(z>0\) when \(\lambda=1\). Consequently, (2.16) has a solution \(z>0\) when \(0<\lambda<1\) due to the continuity.
We now claim that
for any \(\epsilon>0\) arbitrarily given, where \(z>0\) and \(\lambda\in(0, 1)\) satisfy (2.16).
Suppose that (2.19) is not true, then there exist a positive integer h, \(1\le h\le m\), and a positive integer N such that
and
Using (2.9) (when \(i=h\) and \(n=N\)) and (2.21), we find
On the other hand, the hth component of (2.16) gives
or equivalently
Using the above inequality in the earlier inequality, we get
which contradicts the second inequality of (2.20). Therefore, (2.19) holds for \(n\ge0\). Now letting \(\epsilon\to0\) in (2.19), we obtain
The proof is complete.â€ƒâ–¡
Corollary 2.7
Let \(\{X(n)\}_{n\ge n_{0}r}\) be a nonnegative sequence satisfying (2.12), where \(A=(a_{ij})_{m\times m}\) and \(B=(b_{ij})_{m\times m}\) are two matrices satisfying (2.14) and \(C=(A+B)\) is a nonsingular Mmatrix. Moreover, suppose the nonnegative sequence \(\{X(n)\}_{n\ge n_{0}r}\) satisfies
where \(z=(z_{1}, z_{2}, \dots, z_{m})^{T}>0\) and \(0<\lambda<1\) satisfy (2.16). Then
Proof
By setting \(\bar{n}=nn_{0}\), the result is an immediate consequence of TheoremÂ 2.6.â€ƒâ–¡
Remark 2.8
A similar version of CorollaryÂ 2.7 can be found in [20]. Under the assumptions
it is established in TheoremÂ 1 of [20] that if the nonnegative sequence \(\{U(n)\}\) satisfies
and
where \(\tau(n)=(\tau_{ij})_{m\times m}\), \(0\le\tau_{ij}\le\tau\), \(i,j\in\mathbb{I}\), Ï„ is a nonnegative integer and the constant \(\mu>0\), the vector \(z>0\) are determined by
then \(U(n)\le z\exp(\mu(nn_{0}))\), \(n\ge n_{0}\).
Comparing (2.23) with (2.1), we see that \(P=A+E\) and \(Q=B\). If we set \(C=(A+B)\), \(D=E\), and \(M=E((A+B))\), then \(D^{1}M=M=E+A+B=P+Q\). Hence, the condition \(\rho(P+Q)<1\) is equivalent to the condition that \((A+B)\) is a nonsingular Mmatrix (according to (3) of LemmaÂ 2.4).
The next theorem shows that the estimation of \(\{X(n)\}\) similar to (2.17) still holds without the assumption (2.15), which imposes a condition on the initial values of (2.1).
Theorem 2.9
Let \(\{X(n)\}_{n\ger}\) be a nonnegative sequence satisfying (2.1) and (2.2). Suppose that A and B also satisfy (2.14) and \((A+B)\) is a nonsingular Mmatrix. Further, \(z>0\) and \(0<\lambda<1\) satisfy (2.16). Then there exists a constant \(\omega>0\) such that
Proof
As in the proof of TheoremÂ 2.6, we can choose \(z=(z_{1}, z_{2}, \ldots, z_{m})^{T}>0\) and \(\lambda\in(0, 1)\) satisfying (2.16). Noting the conditions of TheoremÂ 2.6, we only need to show
Setting
it is easy to verify that (2.26) holds for \(\omega=\omega_{1}\omega_{2}\). Noting RemarkÂ 2.5, \(\omega z>0\) and \(\lambda\in(0, 1)\) also fulfill (2.16). Thus, (2.25) is obtained as in the proof of TheoremÂ 2.6 by setting \(u_{i}(n)\) in (2.21) as \(u_{i}(n)=(1+\epsilon)\omega z_{i}\lambda^{n}\).â€ƒâ–¡
Remark 2.10
The results of Theorems 2.1, 2.6, and 2.9 can be applied to obtain the global exponential stability of the zero solution of (2.1). As such, we can make use of these theorems to analyze the global exponential stability [27, 30] of the equilibrium of a discrete dynamical system with time delays. It is well known that the equilibrium is globally asymptotically stable if it is globally exponentially stable [30]. Hence, the global asymptotical stability of the equilibria of some discrete dynamical systems with time delays may be established via the results obtained in this section. We shall give such applications in the next section.
3 Global asymptotical stability of equilibrium
In this section, we shall obtain the global stability of the disease free equilibrium of an infectious disease model with two delays that describes malaria transmission. Our proofs employ the generalizations of Halanaytype inequalities obtained in SectionÂ 2.
In [22], the following delayed RossMacdonald model was established to describe the malaria transmission between human and mosquito:
In model (3.1), \(x(y)\) is the ratio of the number of infected human (mosquito) to the total number of human (mosquito) population at time t (days). The total number of human (mosquito) population is H \((M)\) and H \((M)\) is a constant. Define \(m=M/H\). Moreover, Î³ is the average recovery rate of human from malaria infection, Î¼ is the natural death rate of mosquito, a is the average number of bites of a mosquito per human per day, b is the rate of a susceptible human becoming infectious after being bitten by an infected mosquito, c is the rate of a susceptible mosquito becoming infectious after it bites an infected human, and \(\tau_{1}\) and \(\tau_{2}\) are the average incubation times of the parasites in the body of human and mosquito, respectively. For more details of (3.1), one can refer to [22].
We now consider a discrete version of (3.1) as follows:
In model (3.2), \(n\ge0\) and \(x(n)\) (\(y(n)\)) is the ratio of the number of infected humans (mosquitos) to the total number of humans (mosquitos) of the population on the nth day. Moreover, \(r_{1}\) and \(r_{2}\) are now positive integers to reflect the incubation time of the parasites in the body of human and mosquito, respectively. The implications of the other parameters are the same as in (3.1) and a, b, c, m, Î³, and Î¼ are all positive constants.
Denote \(r=\max\{r_{1}, r_{2}\}\). The initial values of (3.2) are
In view of the practical background of model (3.2), it is natural to assume that
Remark 3.1
The basic reproduction number \(R_{0}\) of model (3.2) (refer to [22, 23]) is defined as
where \(\alpha=\exp(\gamma r_{1})\) and \(\beta=\exp(\mu r_{2})\). If we denote \(R_{01}=abm\alpha/\gamma\) and \(R_{02}=ac\beta/\mu\), then \(R_{0}=R_{01}R_{02}\).
Remark 3.2
Since Î³ is the recovery rate of human from malaria infection, it follows that \(1/\gamma\) is the average recovery time of human from the infection. As it takes 730 days for a human to recover from the infection [22, 23], it is reasonable to assume that
Moreover, since the natural lifespan of mosquito is 1460 days [22, 23], it also makes sense to assume that
Lemma 3.3
Suppose that (3.5), (3.6) are satisfied and
Then the solutions \(\{x(n), y(n)\}\) of (3.2) with (3.3) satisfy
Proof
We shall first prove that \(0< x(n)<1\) for \(0< n\le r\). Suppose that there exists \(0\le N_{1}< r\) such that \(x(N_{1})> 0\) and \(x(N_{1}+1)\le 0\). From (3.2), noting (3.3) and (3.5) we find
which is a contradiction. Hence, \(x(n)>0\) for \(0< n\le r\).
Next, suppose that there exists \(0\le N_{2}< r\) such that \(x(N_{2})< 1\) and \(x(N_{2}+1)\ge1\). In view of (3.3), (3.5), and (3.7), it follows from (3.2) that
This is a contradiction and hence \(x(n)<1\) for \(0< n\le r\).
Similarly, we can prove that \(0< y(n)<1\) for \(0< n\le r\). By assuming that \(0< x(n)<1\) and \(0< y(n)<1\) for \((k1)r< n\le kr\), using a similar technique we can show that \(0< x(n)<1\) and \(0< y(n)<1\) for \(kr< n\le (k+1)r\). Hence, it is shown by induction that \(0< x(n)<1\) and \(0< y(n)<1\) for \(n\ge0\).â€ƒâ–¡
Remark 3.4
LemmaÂ 3.3 gives sufficient conditions to guarantee that the solutions of (3.2) with (3.3) are positive and bounded. As pointed out in [31], sometimes these sufficient conditions may be quite rigorous. Indeed, noting RemarkÂ 3.1 we see that condition (3.7) implies \(R_{0}<1\), but in fact numerical simulations show that solutions of (3.2) with (3.3) can be positive and bounded even when \(R_{0}>1\). In view of this, we shall introduce a set Î© of parameters that ensures the positivity and boundedness of the solutions of (3.2) with (3.3), let
The following lemma is about the existence of the equilibrium of model (3.2), which is obtained by direct computation.
Lemma 3.5

(1)
There exists only the disease free equilibrium \(E_{0}=(0, 0)\) of (3.2) if \(R_{0}\le 1\).

(2)
There exist two equilibria of (3.2) if \(R_{0}>1\), namely the disease free equilibrium \(E_{0}\) and the positive equilibrium \(E^{*}=(x^{*}, y^{*})\), where
$$ x^{*}=\frac{R_{0}1}{R_{0}+R_{01}},\qquad y^{*}=\frac{R_{0}1}{R_{0}+R_{02}}. $$(3.9)
We shall now employ TheoremÂ 2.1 to obtain the global asymptotical stability of the disease free equilibrium \(E_{0}\) of model (3.2).
Theorem 3.6
Suppose that (3.5), (3.6), and (3.7) hold. Then the disease free equilibrium \(E_{0}\) of (3.2) with (3.3) is globally asymptotically stable.
Proof
We rewrite (3.2) as
Since the conditions of LemmaÂ 3.3 are satisfied, we have \(x(n)>0\) and \(y(n)>0\) for \(n\ger\). Hence, we get
which clearly leads to
for \(n\ge0\).
Comparing (3.10) with (2.1), we see that
It is obvious that condition (2.3) is fulfilled. Next, noting from (3.7) that \(R_{01}=abm\alpha/\gamma<1\) and \(R_{02}=ac\beta/\mu<1\), we have
Hence, condition (2.4) is also satisfied. Now, by TheoremÂ 2.1, there exist \(\lambda_{1},\lambda_{2}\in(0,1)\) such that
for \(n\ge0\), where \(k_{1}=\max_{s\in\mathbb{S}_{1}}\{x^{(s)}\}\) and \(k_{2}=\max_{s\in\mathbb{S}_{1}}\{y^{(s)}\}\).
In view of (3.12), the zero solution of (3.2), which is the disease free equilibrium \(E_{0}\), is globally exponentially stable. Noting RemarkÂ 2.10, it follows that \(E_{0}\) is also globally asymptotically stable.â€ƒâ–¡
Remark 3.7
When the conditions of TheoremÂ 3.6 are satisfied, noting (3.12) the exponential convergence rate of the solutions of (3.2) with (3.3) to the disease free equilibrium is \(\min\{\ln\lambda_{1}, \ln\lambda _{2}\}\).
Remark 3.8
Condition (3.7) implies that \(R_{0}=R_{01}R_{02}<1\), but \(R_{0}<1\) may not imply (3.7). In the literature of compartmental infectious disease models, usually \(E_{0}\) is globally asymptotically stable when \(R_{0}<1\). This expected result is not obtained when TheoremÂ 2.1 is applied to model (3.2). As such, in the sequel we shall apply TheoremÂ 2.9 to model (3.2) to see whether \(E_{0}\) is globally asymptotically stable when \(R_{0}<1\).
Theorem 3.9
Suppose that (3.5), (3.6) are satisfied and
Then the disease free equilibrium \(E_{0}\) of (3.2) with (3.3) is globally asymptotically stable inÂ Î©.
Proof
Noting the definition of Î© (refer to (3.8)), (3.10) also holds true. With A and B defined as in (3.11), we see that condition (2.14) is satisfied. Moreover, since
we have \(\gamma>0\) and \(\det((A+B))=\gamma\mua^{2}bcm\alpha\beta=\gamma\mu(1R_{0})>0\) due to (3.13), i.e., the successive principal minors of \((A+B)\) are both positive. Therefore, according to (1) of LemmaÂ 2.4, \((A+B)\) is a nonsingular Mmatrix. The conditions of TheoremÂ 2.9 are all satisfied, hence the disease free equilibrium \(E_{0}\) of (3.2) with (3.3) is globally exponentially stable and consequently it is globally asymptotically stable in Î©.â€ƒâ–¡
We shall now give an example to illustrate TheoremÂ 3.9 and RemarkÂ 3.4.
Example 3.10
Consider a special case of (3.2) (with day as the time unit)
Here, the parameters are
By direct computation, we have \(R_{01}=0.2624\), \(R_{02}=1.0570\), and \(R_{0}=0.2774\). According to TheoremÂ 3.9, the disease free equilibrium \(E_{0}\) of (3.14) is globally asymptotically stable since (3.5), (3.6), and (3.13) are satisfied. Indeed, the dynamics of ExampleÂ 3.10 is depicted in FigureÂ 1 and we observe that both \(x(n)\) and \(y(n)\) tend to 0 as \(n\to+\infty\), i.e., the disease free equilibrium \(E_{0}\) is globally asymptotically stable.
In this example, we note that (3.7) is not satisfied since \(R_{02}>1\). However, it is observed from FigureÂ 1 that the solution of (3.14) is positive and bounded. This illustrates RemarkÂ 3.4 and shows that even if (3.7) does not hold, the set Î© (see (3.8)) may not be empty.
4 Concluding remarks
Halanaytype inequalities, whether continuous or discrete, have been widely applied to obtain the global exponential (asymptotical) stability of the equilibria of dynamical systems with several delays, especially dynamical systems of neural networks. In this paper, we have derived two generalizations of multidimensional discrete Halanaytype inequalities. Further, the generalizations are applied to a discrete malaria transmission model with two delays. We have shown that the disease free equilibrium is globally asymptotically stable when the basic reproduction number \(R_{0}\) is less than 1, which is well consistent with the threshold property of the basic reproduction number.
The global asymptotical stability of the equilibria of infectious disease models with time delays is usually obtained via the construction of suitable Lyapunov functionals together with Razumikhintype theorem and/or LaSalle invariant sets. However, a suitable Lyapunov functional is somewhat difficult to construct for a delayed dynamical system. Hence, it is reasonable to try other methods to obtain the global asymptotical stability of equilibria of dynamical systems with time delays. From our present work, we have observed that it is direct and simple to obtain the global asymptotical stability of equilibria of dynamical systems with time delays via Halanaytype inequalities.
When using Halanaytype inequalities established in this paper to obtain the global asymptotical stability of the equilibrium of a discrete dynamical system with time delays, the positivity of the solutions should be initially guaranteed. This can be proved for many kinds of compartmental infectious disease models. Hence, it is direct to obtain the global asymptotical stability of the disease free equilibria of dynamical systems with time delays by applying Halanaytype inequalities. It is well known that, in order to obtain the global asymptotical stability of the positive equilibrium \((x^{*}, y^{*})\) of a dynamical system with time delays, the change of variables \(\bar{x}=xx^{*}\) and \(\bar{y}=yy^{*}\) is usually applied to transfer the global asymptotical stability of the positive equilibrium to the global asymptotical stability of the zero solution of the system with respect to xÌ„ and È³. However, after the change of variables, the positivity of xÌ„ and È³ cannot be guaranteed. Hence, if one intends to employ Halanaytype inequalities to obtain the global asymptotical stability of the positive equilibrium, new techniques are needed to deal with xÌ„ and È³. This remains as future work.
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The research of Chunqing Wu is sponsored by the Jiangsu Overseas Research & Training Program. The authors would like to thank the referees for their comments which help to improve the paper.
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Wu, C., Wong, P.J. Multidimensional discrete Halanay inequalities and the global stability of the disease free equilibrium of a discrete delayed malaria model. Adv Differ Equ 2016, 113 (2016). https://doi.org/10.1186/s1366201608404
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DOI: https://doi.org/10.1186/s1366201608404