Advanced Nonlinear Dynamics of Population Biology and EpidemiologyView this Special Issue
The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation
We investigate a stochastic SI epidemic model in the complex networks. We show that this model has a unique global positive solution. Then we consider the asymptotic behavior of the model around the disease-free equilibrium and show that the solution will oscillate around the disease-free equilibrium of deterministic system when . Furthermore, we derive that the disease will be persistent when . Finally, a series of numerical simulations are presented to illustrate our mathematical findings. A new result is given such that, when , with the increase of noise intensity the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system.
Epidemiology is the science to study the distribution of disease and influencing factors, so as to explore the etiology, clarify the popular rule of the disease, and formulate the countermeasures and measures for preventing, controlling, and eliminating the disease. Many mathematical models of diseases spreading help us to understand the propagation of diseases [1, 2]. The transmission of diseases can be influenced by many factors, such as the age and social structure of the population, the contact network among individuals, and the metapopulation characteristics. So it is difficult to establish an accurate epidemic model which is completely consistent with the real world. In recent years, a lot of compartmental epidemic models have been studied by many researchers [3–5], and complex networks also have been used to study the spread of diseases [6–17].
In this paper we consider an SI model with the birth and death in complex networks. As mentioned in the paper [6, 13], the birth and death do not affect the degree of nodes. Suppose are the number of the healthy and infected nodes with the degree at time ; the mean-field equations can be written as where . For system (1), it can be written as the following form: We denote , so we obtain It always has the disease-free equilibrium , where , . If is irreducible and , then is globally stable in , while if , is unstable and there is an endemic equilibrium belonging to which is globally asymptotically stable in ; here and denotes the spectral radius of .
The deterministic models have some limitations in describing the spread of disease. The accident in the process of disease transmission can not be reflected by the deterministic models. This is because of the fact that the deterministic models ignore the effect of the environmental noise. In an ecosystem, the environmental noise is inevitably in the real world; thus stochastic models are more realistic. In the research of stochastic epidemic models, many researchers make a lot of contributions [17–26].
In this paper, we consider the following stochastic system: where , , , are independent standard Brownian motions with , and , , , represent the intensities of .
The remaining parts of this paper are as follows. In the next section we show the existence and uniqueness of a global positive solution of model (5). In Section 3, we analyze the asymptotic behavior around the disease-free equilibrium. In Section 4, we study the dynamic of system (5) around the endemic of the deterministic model. In Section 5, numerical simulations and conclusions are carried out.
2. Global Positive Solution
When we study a dynamical behavior, a global solution is important for the system. In this section we show that the solution of system (5) is global and nonnegative. As we know, for a stochastic differential equation, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition. It is a sufficient condition for a stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any given initial value [27, 28]. Although the coefficients of system (5) satisfy locally Lipschitz continuous, they are not satisfied with the linear growth condition, so the solution of system (5) may explode at a finite time. In this section, Lyapunov analysis method (mentioned in ) is used to show that the solution of system (5) is positive and global.
Theorem 1. For any given initial value , there is a unique positive solution of model (5) on and the solution will remain in with probability , namely, for all a.s.
Proof. Due to the fact that the coefficients of the system (5) are locally Lipschitz continuous, for any given initial value , it has a unique local solution on , where is the explosion time . If we show that a.s., it suggests that this solution is global. Let be sufficiently large so that , all lie within the interval . For each integer , defining the stopping time we set (as usual denotes the empty set). Obviously, is increasing as . Set ; therefore a.s. If a.s. is true, then a.s. and a.s. for . In other words, to complete the proof it is required to show that a.s. If this statement is false, then there is a pair of constants and such that . Thus there is an integer , such that Define a -function as follows: Applying Itô’s formula, we obtain We can now integrate both sides of (9) from to and then take the expectations Let for and, by (7), . Note that, for every , there is at least one of and , , that equals either or , and therefore is not less than either Hence, It then follows from (7) and (10) that where is the indicator function of . Letting , we have that is a contradiction. So we must have . Therefore, it implies , , , will not explode in a finite time with probability one.
3. Asymptotic Behavior around the Disease-Free Equilibrium
As mentioned in the Introduction, is the disease-free equilibrium of system (3), and when , is globally stable, which means that the disease will be extinct in the limited time. In this section, we will study the asymptotic behavior around of system (5).
Lemma 2. If is nonnegative and irreducible, then the spectral radius of is a simple eigenvalue, and has a positive eigenvector corresponding to . Besides, if , then . (This lemma can be found in .)
Theorem 3. Assume is irreducible. If and the following condition is satisfied: then for any given initial value , the solution of system (5) has the property where
Proof. First change the variables , ; then , and system (5) can be written as
Let , where , . Define
then it is nonnegative and irreducible. By Lemma 2, there is a positive eigenvector of corresponding to , such that
Define a -function by
where , , are positive constants. Then the function is positive definite, and
Choose , ; then . And we note that
where the last equality is derived from (20). Since , then , and so according to Lemma 2. Besides, , and then . Therefore
Integrating both sides of (26) from to , and taking expectation, yields
that is, If we let then as the theorem is proved.
Remark 4. From Theorem 3, we can get the conclusion that the solution of the stochastic system will oscillate around the disease-free equilibrium of the deterministic model; the values of and have bearing on the intensity of turbulence. If the stochastic perturbations become small, the solution of system (5) will be close to the disease-free equilibrium of system (3).
Besides, if , then is also the disease-free equilibrium of system (5). From the proof of Theorem 3, we can obtain Therefore, is globally asymptotically stable.
In the deterministic model, if , there exists the endemic equilibrium . But is not the endemic equilibrium of stochastic system (5), because there is no endemic equilibrium for the stochastic system (5). In fact, we still want to find the relation between the solution of stochastic system and .
Given a weighted digraph with vertices, where is the weight matrix, whose entry equals the weight of arc if it exists, and 0 otherwise, the Laplacian matrix of is defined as Let denote the cofactor of the th diagonal element of , and we have the following results.
Theorem 5. Assume is irreducible and . For any given initial value , the solution of system (5) has the property where is the endemic equilibrium of system (3) and , , denote the cofactor of the th diagonal element of , and , , , , are positive constants defined as in the proof.
Proof. Since is the endemic equilibrium of system (3), we have Define where , , , , are positive constants to be determined later. From the property (1) of Lemma A.2 (see ), we know , . Hence is positive definite. Let be the generating operator of system (5). Then we get By property (2) of Lemma A.2 (see ), we know Besides, note that for ; then According to (41) and (42), we get Substituting (41) and (43) into (38), we get Substituting (44) into (39), we get Therefore, Choose , , , ; then Therefore, Integrating both sides of (49) from to yields Let , , which are local continuous martingale, and . Moreover By Lemma A.4 (see ), we obtain which together with (50) implies Consequently, Thus Theorem 5 is proved.
Remark 6. Theorem 5 shows that the solution of system (5) fluctuates around the certain level which is relevant to of system (3) and , , . The distance between the solution and of system (3) has the following form: where is a positive constant and . Although the solution of system (5) does not have stability as the deterministic system, we can draw a conclusion that system (5) is persistent on the basis of the result of Theorem 5, which also accounts for the fact that the disease is prevalent.
5. Simulations and Conclusions
5.1. Numerical Simulations
In order to confirm the results above, we numerically simulate the solution of system (5) with , , , , and initial value , , . Using Milstein’s Higer Order Method , we get the discretization equation: where and , , , are the independent Gaussian random variables .
From Theorem 3 and Remark 4, it is shown that the expectations of , , , are converging under some conditions, and the solution of system (5) will oscillate around the disease-free equilibrium of system (3). In Figure 1, we choose parameters ,