On the global stability of a delayed epidemic model with transport-related infection.

We study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain J. Math. 38 (5) (2008) 1525-1540] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport.

Introduction

Population dispersal by human transportation currently plays an important role in the spread of infectious disease around the world. SARS (severe acute respiratory syndrome) spread along the routes of international air travel and infection was carried to many places [1]. Khan et al. [2] pointed out a correlation between inter-regional spread of a novel influenza A (H1N1) virus and travelers. From these observations a number of authors have proposed epidemic models describing disease transmission dynamics among multiple locations due to the population dispersal (see [3], [4], [5], [6] and the references therein).

Cui et al. [7] have proposed an epidemic model which models a phenomenon where individuals in a population suffer from diseases and possibly become infected during the movement between two regions. The model is given as a system of ordinary differential equations based on an SIS epidemic model. Takeuchi et al. [8] analyzed local and global stability of equilibria as well as uniform persistence of the system. They found that there is a possibility of endemic situation due to the transport-related infection. However, as pointed out by Liu et al. [9] the epidemic models proposed in [7], [10], [8] implicitly assumed that the transportation between two regions occurs instantaneously. This motivated Liu et al. [9] to rigorously describe the disease transmission dynamics during the transportation by introducing the time needed to complete the use of the transportation. They assumed that it takes units of time to complete one-way transport between two regions. They obtained the following delay differential equations: with Here and denote the numbers of susceptible and infected individuals at time in region , respectively, where . is the total number of newborns per unit time, is the natural death rate and is the recovery rate. Disease is transmitted by , where is the disease transmission coefficient in each region. Susceptible and infected individuals leave a region towards another region at a per capita rate . Thus the numbers of susceptible and infected individuals leaving region per unit time are given by and , respectively. and for and denote the numbers of susceptible and infected individuals arriving in region from region per unit time at time . They leave region at time and spend units of time in transportation, where disease transmission occurs. We denote by the transmission coefficient in the transportation. It is assumed that every parameter is positive.

Following [9], we explain how to derive and for and . We denote by and the numbers of susceptible and infected individuals leaving region per unit time at time and spend units of time in transportation to region , where . Then considering the number of susceptible and infected individuals leaving region to per unit time at time we obtain that We assume that the individuals in the population do not die in the transportation. Then the disease dynamics in the transportation from region to can be described as where is the transmission coefficient in the transportation. Since it holds that we obtain that By solving (1.4) using (1.2) as the initial condition, we obtain the expression for . Then it is easy to see that (1.3) gives the expression for .

Liu et al. [9] identified the basic reproduction number and analyzed the stability property of the equilibria and uniform persistence of the system; the basic reproduction number is given as (1.1) always has a disease-free equilibrium. They proved that the disease-free equilibrium is globally asymptotically stable if . If then (1.1) admits a unique endemic equilibrium. By analyzing an associated characteristic equation they proved that the endemic equilibrium is locally asymptotically stable if  [9, Theorem 4.1]. Moreover, by the uniform persistence theorem [11], they obtained that the disease eventually persists if  [9, Theorem 4.2].

However, the problem of global stability of the endemic equilibrium remains unsolved. Stability analysis for epidemic models is helpful for obtaining insight into the disease transmission dynamics. Global stability of equilibria, in particular, makes the model dynamics clear and enhances our understanding of the mathematical models. In this paper we prove that the endemic equilibrium is globally asymptotically stable if . Our proof is based on constructing a Lyapunov functional and using LaSalle’s invariance principle. Our mathematical results suggest that, when the infectious disease is transferred between two regions by human transportation, the endemic steady state is globally asymptotically stable regardless of the length of the travel time if the basic reproduction number exceeds 1.

The paper is organized as follows. In the next section, first, we discuss the global dynamics of the total populations in both regions and show that a unique positive equilibrium is globally asymptotically stable. Using this result we obtain a limit system for (1.1). Constructing a Lyapunov functional for this reduced system, we prove that an unique positive equilibrium of the system is globally asymptotically stable if in Theorem 2.6. This implies that the endemic equilibrium of (1.1) is also globally asymptotically stable if . In Section 3, we offer a discussion.

Global stability of the endemic equilibrium

To investigate the dynamics of (1.1), we set a suitable phase space. We denote by the Banach space of continuous functions mapping the interval into equipped with the sup-norm. The nonnegative cone of is defined as . From the biological meanings, the initial conditions for (1.1) are where .

First, we consider the global dynamics of total populations in both regions. Let us define Then from (1.1), (2.1) we have with the initial conditions and for , where and . We prove that (2.2) has a unique positive equilibrium which is globally asymptotically stable.

See [9, Lemmas 2.1 and 2.2]

The solution of (1.1) with initial conditions (2.1) is nonnegative for all . Moreover, there exist and such that and for and .

(2.2) has a unique positive equilibrium which is globally asymptotically stable.

Since the equilibrium satisfies we easily obtain the existence of the unique positive equilibrium.

Let us consider the asymptotic stability of the equilibrium. We define , which is the closure of , is positively invariant for (2.2).

We denote by the positive equilibrium of (2.2). Consider the following functional defined on : where and , is any solution of (2.2). It is clear that is continuous on and that for any (the boundary of ), the limit , exists or is . We consider the time derivative of along the solution of (2.2).

First of all, we see that Then, from , it follows that It follows that Therefore, we obtain that Hence we obtain that From (2.3), (2.4), the positive equilibrium is stable.

Let us define that and that is the largest subset in that is invariant with respect to (2.2). Then, we see that Consider any initial function . Then it holds that , where and through . Hence, . By an extension of LaSalle’s invariance principle [12, Lemma 3.1], any solution tends to (see also [13], [14]). Hence, the positive equilibrium is globally asymptotically stable.  □

Next using the result in Lemma 2.2 we derive a limit system of (1.1). We denote by the unique positive equilibrium of (2.2). Since , (1.1) has the following limit system: where From now on we consider the reduced system (2.5) in order to understand the asymptotic behavior of the solution of (1.1) (see [16], [17]).

Suzuki and Matsunaga [15] established the necessary and sufficient conditions for asymptotic stability of a class of linear delay differential equations. Their result is applicable to (2.2) and it also yields Lemma 2.2 [15, Example 2].

Liu et al. [9] proved that (1.1) has a unique endemic equilibrium if and only if . We denote the endemic equilibrium by where every component is strictly positive. Then it is easy to prove that is a unique positive equilibrium of (2.5) if and only if . We study the global stability of the positive equilibrium of (2.5) in order to establish the global stability of the endemic equilibrium of (1.1).

We give an elementary result to prove the global asymptotic stability of the endemic equilibrium. Let us define

Let us assume that . Then it holds that and for .

Direct computation gives for . Since and are monotone increasing functions we obtain (2.6). (2.6) implies that Then (2.7) holds. The proof is complete. □

We prove the global asymptotic stability of the positive equilibrium of (2.5).

The property of (2.7) in Lemma 2.4 is also found in [18], [19] and it is used to analyze global dynamics of epidemic models having a nonlinear incidence rate.

Let us assume that there exists such that . Let . Then the positive equilibrium of (2.5) is globally asymptotically stable.

Since there exists a unique positive equilibrium of (2.5) it holds that

Then (2.5) becomes the following:

We define , the closure of , is positively invariant for (2.8). Moreover, there exists an such that every solution of (1.1) with for some satisfies (see [9, Theorem 5.1]). This implies that is also positively invariant for (2.8).

Consider the following functional defined on : where , are any solutions of (2.8). Then it is clear that is continuous on and that for any (the boundary of ), the limit is .

We consider the time derivative of along the solution of (2.8). First, we see that

Next it follows that

From (2.10), (2.11) we obtain that where We compute as follows: Then, from (2.7) in Lemma 2.4, we see that . Therefore, we obtain from (2.12).

Let us define that and that is the largest subset in that is invariant with respect to (2.8). Then, we see that Consider any initial function . Then it holds that , where through . Hence, . By an extension of LaSalle’s invariance principle [12, Lemma 3.1], any solution tends to (see also [13], [14]) and hence, the positive equilibrium of (2.5) is globally asymptotically stable.  □

Discussion

In this paper, we have studied the global dynamics of a delayed epidemic model (1.1) proposed by Liu et al. [9]. The model describes disease transmission dynamics due to transport-related infection [7], [10], [8] and captures the time needed to complete the use of the transportation.

Liu et al. [9] established that the disease-free equilibrium of (1.1) is globally asymptotically stable if and that (1.1) admits a unique endemic equilibrium, which is locally asymptotically stable, if . However, the problem of the global stability of the endemic equilibrium remained unsolved and was an open problem.

For this problem, we considered the global dynamics of the limit system (2.5). (2.5) is derived from (1.1) using that the positive equilibrium of (2.2) is asymptotically stable. Constructing a Lyapunov functional and using LaSalle’s invariance principle for the reduced system, we prove that the positive equilibrium of (2.5) is globally asymptotically stable if (Theorem 2.6). This implies that the endemic equilibrium of (1.1) is also globally asymptotically stable if . The mathematical result suggests that, when the disease is endemic, in a situation where two regions are connected to each other by transportation, the endemic steady state is globally asymptotically stable regardless of the length of the travel time. However, one can see that in (1.1) it is assumed that the two regions share an identical parameter set. It may be necessary to consider two different population sizes and different dispersal rates in order to discuss precisely the impact of the transport-related infection on the disease dynamics. We leave this to future work.