Abstract
A two-city SIR epidemic model with transport-related infections is proposed. Some good analytical results are given for this model. If the basic reproduction number , there exists a disease-free equilibrium which is globally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the basic reproduction number . We also show the permanence of this SIR model. In addition, sufficient conditions are established for global asymptotic stability of the endemic equilibrium.
1. Introduction
Epidemiology is the study of the spread of disease in time and space, aiming at tracing factors that give rise to their occurrence. Since Kermark and Mckendrick in [1] built up a system to study epidemiology in 1927, the concept of “compartment modeling” is widely used until now. From then on, many great epidemic models are proposed and researched [2–5], which assume that population lives in the fixed region, without travel. However, in fact, people usually travel among different regions; thus models involving dispersal are indispensable. To control the spread of an infectious disease, we have to know how the growth and spread of the disease affect its outbreak. And there are many factors that lead to the dynamics of an infectious disease of humans, such as human behavior as population dislocations, living styles, sexual practices, and rising international travel. On the other hand, climate change enables diseases and vectors to expand their range. Since the first AIDS case was reported in the United States in June 1981, the number of cases and deaths among persons with AIDS increased rapidly during the 1980s followed by substantial declines in new cases and deaths in the late 1990s. In 2003, SARS began in Guangdong province of China; however, it broke out at last in almost all parts of China and some other cities in the world due to dispersal [6]. Recently, some epidemic models have been proposed to understand the spread dynamics of infectious disease.
Ahmed et al. in [7] introduced a model with travel between populations. In addition, Sattenspiel and Herring considered the same type of model but applied it to travel between populations in the Canadian subarctic, which can be thought of as a closed population where travel is easily quantified [8]. Ding et al. [9] and Sattenspiel et al. [10, 11] have also discussed other models for the spread of a disease among two patches and patches. In [12], Wang and Mulone studied an SIS model with standard incidence rate on population dispersal among patches. Wang and Zhao [13] proposed an SEIR epidemic model, assuming that the susceptible and exposed individuals have constant immigration rates. What is more, Wang and Zhao [14] formulated a general SEIRS for multispecies on multipatches, and the role of quarantine in the form of travel restriction was discussed.
All these investigations ignore the possibility for the individuals to become infective during travel. In paper [15], Allen et al. have proposed the following SIS epidemic model to understand the effect of transport-related infection on disease spread for the first time:
For many diseases (e.g., influenza, measles, chickenpox, etc.), after recovery, the individuals have immunity to the disease. Thus, an SIR or SIRS model is more suitable for this kind of disease. In this paper, we will study the effect of transport-related infection. Our results show that transport-related infection can make the disease endemic even if both the isolated regions are disease free.
We consider a model with state variables , , that represent the number of susceptible, infected, and removed individuals in city , respectively. The basic assumptions underlying the dynamics of the system are as follows.(i)We assume that both cities are identical.(ii)All newborns, denoted by , join into the susceptible class per unit time.(iii)Natural death rate for susceptible, infected, and removed individuals is a constant per capita rate .(iv)Disease is transmitted with the standard form incidence rate within city . The transmission rate within a city is a constant .(v)We may assume that a susceptible individual goes into the infected part after infection.(vi)Susceptible, infected, and removed individuals of every city leave for city at a per capita rate . We assume that two cities are connected by the direct transport such as airplanes or trains.(vii)When the individuals in city travel to city , disease is transmitted with the incidence rate with a transmission rate .(viii)The rate constant for recovery is denoted by , and the per capita mortality rate for infected individual is . Since this includes both natural and disease induced mortality, we have .(ix)We suppose that individuals who are traveling do not give birth and do not take death. Further we assume that removed individuals do not lose immunity during travel.
These assumptions lead to a model of the following form:
From the biological point of view, the term represents the susceptible leaving city and denote individuals in becoming infected during travel from city to . Hence, should be nonnegative. Therefore, we always suppose in the following discussion.
The paper is organized as follows. In next section, we will research the existence of equilibria and their local stability. In Section 3, we will discuss permanence of the SIR model and some sufficient conditions for global stability of equilibrium in Section 4. In the final section, we will discuss our results and give some numerical simulations.
2. Local Stability
The assumption that both cities are identical, that is, demographic parameters are the same for each city, has enabled us to obtain an analytic expression for the equilibria. It is easy to check that system (2) has a disease-free equilibrium for all parameter values, where . According to the concept of next generation matrix in Li et al. [16] and reproduction number presented in Liu et al. [17], we can define
Hence the reproduction number for system (2) is Here represents the spectral radius of the matrix . When , system (2) has unique positive equilibrium , where
The Jacobian matrix for the right hand of system (2) is given by where
Firstly, we study stability of the disease-free equilibrium .
Theorem 1. If , then is locally asymptotically stable, and if , then is unstable.
Proof. Evaluating (6)–(10) at , we have the following Jacobian matrix:
where
By Allen et al. [15], the eigenvalues of are identical to those of and since
The eigenvalues of are , , and the eigenvalues of are , , . By , we have ; we can conclude that all six eigenvalues of are negative. When , we have which implies , so we obtain that at least one eigenvalue of is positive. Hence is locally asymptotically stable if and is unstable if . This completes the proof.
Next, we research the stability of . Evaluating (6)–(10) at and using (5), we have the following Jacobian matrix for : where
Similar to the proof of Theorem 1, to calculate the eigenvalues of is equivalent to calculating the eigenvalues of matrix and , where
However, different from the case for , the eigenvalues of matrices and cannot be calculated explicitly. We will use the Routh-Hurwitz Theorem to study the stability of . Note that and have the same form as follows: The characteristic polynomial of matrix is , where , , and and , , and . For convenience, we state the Routh-Hurwitz Theorem for above matrix.
Lemma 2. is stable (i.e., each eigenvalue of has negative real part) if and only if the following conditions hold:(i),(ii),(iii).
Using Lemma 2, we have the following stability result for .
Theorem 3. If , then is locally asymptotically stable.
Proof. Consider the matrices and in , the Jacobian matrix of system (2) at . It suffices to check that both and satisfy the conditions in Lemma 2. Firstly, we check them for as the following three steps. For simplification, we will refer the entries of as .
(i) . Obviously, and . By (4),
Since , we have . Thus .
(ii) . Obviously, , , , , , . So , , .
(iii) , . Since , , we obtain
We also have
Thus, by the fact that , , , and , , . Therefore, by Lemma 2, is stable. Next we check as follows. For convenience, we also refer the entries of as .
(i) . Obviously, . Since when and also noting that , we have
the same to (18), one has
Thus, .
(ii) , . Since , . Obviously . For , we have
can be shown as the following two cases.
Case 1 (). By (4), we obtain that
Thus, can be rewritten as
Since , it is clear that .
Case 2 (). According to , we have
Since , clearly we have . The same to above analysis, for , we also obtain that
can be shown as the following two cases.
Case 1 (). Consider
Since , we have .
Case 2 (). Consider
(iii) ,
Case 1 (). Due to , so .
Case 2 (). According to , we have
Since , clearly one has . On the other hand,
because . So , since , , .
Therefore, . By Lemma 2, is also stable. Hence is locally asymptotically stable. This completes the proof.
3. Permanence
Firstly, we consider permanence of the disease. Set initial conditions as , and for . It is easy to check that all solutions of system (2) are nonnegative (i.e., , , and for and ) under the assumption . The following result shows that system (2) is ultimately bounded above.
Theorem 4. There exists an such that for any solution of system (2) with initial values , , and , , there must be a such that , , and for and .
Proof. Let . We have Hence, by comparison theory of differential equations, it is easy to verify that there exists such that , for . Then , , and for . This completes the proof.
Theorem 5. Let . Then there exists an such that every solution of system (2) with initial values , and for satisfies
Proof. By system (2) and the fact that , we have
Hence, is always ultimately lower bounded by some positive constant; see, for example, , which is independent of initial values. And so is if both and are ultimately lower bounded by some positive constant independent of initial values. Therefore, it suffices to prove that , .
The result follows from an application of Theorem 4.6 in Arino et al. [18]. Define
It then suffices to show that system (2) is uniformly persistent with respect to .
Next, is positively invariant with respect to system (2). It is easy to verify that , for if , , and for . This is also positively invariant. Furthermore, by Theorem 4, there exists a compact set in which all solutions of system (2) initiated in will enter and remain forever after. The compactness condition in Arino et al. [18] is easily verified for this set . Denote
We now show that
Suppose that . It suffices to show for any and . If not, there exists such that or . Hence, and , which contradicts . This proves (38).
Denote the omega limit set of the solutions of system (2) starting in by (which exists by Theorem 4). Let
Restricting system (2) on gives
It is easy to verify that system (40) has a unique equilibrium , where . And thus is the unique equilibrium of system (2) in . It is easy to check that is locally asymptotically stable. Hence, it is also globally asymptotically stable since system (40) is a linear system. Thus . And is a covering of , which is isolated (since is the unique equilibrium) and is acyclic (since there exists no solution in which links to itself). Finally, the proof will be done if we show that is a weak repeller for ; that is,
where is the solution of system (2) with initial value . By the proof of Lemma 3.5 in Cui et al. [19], (41) is valid if
where denotes the stable manifold of . Suppose (42) is not valid; then there exists a solution , of system (2) with initial value , such that
Since , we can choose which is small enough such that and
Define . For , by (43) there exists such that
Hence by system (2),
for . Hence, . By (44), we have as , which contradicts (43). Thus (42) holds, which completes the proof.
4. Global Stability
First of all, we will discuss the global stability of under the condition .
Theorem 6. If , then is globally asymptotically stable.
Proof. Consider the following function: Its derivative along the solutions of system (2) is If , then If , then When , we set Restricting system (2) on the set , we have , then for . , then for . Therefore, is the largest positively invariant subset of . By Lyapunov-LaSalle theorem, is global asymptotically stable provided . This completes the proof.
Theorems 4 and 5 imply that system (2) is permanent if . Next, we consider the global asymptotic stability of . The set defined in proof of Theorem 5 will be used.
Theorem 7. Suppose that Then the endemic equilibrium point is globally asymptotically stable on for .
Proof. By Theorem let us consider the function: The time derivative of along the solution of system (2) becomes Note that which gives the following: The above quadratic form is negative definite if and only if It is easy to check that the above conditions are satisfied if and only if (52) is satisfied. Hence we can find some positive constant satisfying which shows that for any solution of system (2), we have By Lyapunov's theorem, we know that is contained in the set . Here is the -limit set of the solution of system (2) with an initial value and is the state space . On the set , we now consider the following system for , , and ; that is, It is trivial that the equilibrium of system (60) is globally asymptotically stable and the equilibrium is unstable as . This shows that the endemic equilibrium of system (2) is globally asymptotically stable. This completes the proof.
5. Numerical Simulations and Discussion
In this paper, a two-city SIR epidemic model with transport-related infections is proposed. According to Theorems 1 and 3, we obtain that there exist a disease-free equilibrium and an endemic equilibrium which are locally asymptotically stable if the basic reproduction number and , respectively. Theorems 4 and 5 provide the permanence of this SIR model. In addition, sufficient conditions are established in Theorems 6 and 7 for global asymptotic stability of the disease-free and the endemic equilibrium, severally. The following numerical simulations, we will present, are to explain the feasibility of our main results.
If we set , , , , , , and , by a simple computation, we derive Obviously, the assumptions of Theorems 1 and 6 are satisfied, so disease-free equilibrium of system (2) is globally asymptotically stable. From Figure 1, it is easy to observe that if is relatively small, then both isolated cities are disease free and the transport-related infection may not lead to the disease becoming endemic.
(a)
(b)
On the other hand, if we have , , , , , , and , we have The assumption of Theorem 3 is satisfied, and since it is not difficult to prove Theorem 7. Hence, the endemic equilibrium of system (2) is globally asymptotically stable. Figure 2 shows that if is relatively large, then the disease is endemic in the two isolated cities and the transport-related infection will surely lead to the disease becoming endemic.
(a)
(b)
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province (PKLHB1302) and the soft science research project of Hubei Province (2012GDA01309) and the key discipline of Hubei province-Forestry.