Abstract
We investigate an SIR epidemic model with stochastic perturbations. We assume that stochastic perturbations are of a white noise type which is directly proportional to the distances of three variables from the steady-state values, respectively. By constructing suitable Lyapunov functions and applying Itô’s formula, some qualitative properties are obtained, such as the existence of global positive solutions, stochastic boundedness, and permanence. A series of numerical simulations to illustrate these mathematical findings are presented.
1. Introduction
Almost all mathematical models for the transmission of infectious diseases descend from the classical susceptible-infective-removed (SIR) model of Kermack and McKendrick [1]. The dynamic behavior of different epidemic models and a lot of their extensions is well investigated by a number of scholars; see [2–11]. The basic and important research subjects for recent studies are the existence of the threshold values which distinguish whether the disease dies out, the stability of the disease-free and the endemic equilibria, permanence, and extinction [12]. During the last few decades, a number of realistic transmission functions have become the focus of considerable attention, and many authors are interested in the formulation of nonlinear incidence rate (see [13–17]). A nonlinear incidence rate can arise from saturation effects that if the proportion of the infection in a population is very high, so that exposure to the disease agent is virtually certain, then the transmission rate may respond more slowly than linear to the increase in the number of infection [18]. For example, Capasso and Serio [19] introduced a saturated transmission rate , where measures the infection force of the disease and measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. To be biologically feasible, the function of the incidence rate is a positive continuous and differentiable and satisfies the conditions for all . It is easy to know that the function is concave with respect to the variable ; that is, which implies that when the number of infections is very high that the exposure to the disease agent is virtually certain, the incidence rate will respond more slowly than linearly to the disease in .
In the real world, population dynamics is inevitably subjected to environmental noise, which is an important component in an ecosystem. Most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time [20, 21]. Recent advances in stochastic differential equations enable a lot of authors to introduce randomness into deterministic model of physical phenomena to reveal the effect of environmental variability, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters; see [12, 13, 22–30]. Of them, Tuckwell and Williams [28] investigated the properties of a simple discrete time stochastic epidemic model. A classical model of an SIRS epidemic in an open population was considered by El Maroufy et al. [12]. They established the global stability of disease-free and endemic equilibrium points for both the deterministic and stochastic models. Based on the theory of stochastic differential equation, Cai et al. [13] studied the dynamics of an SIRS epidemic model with a ratio-dependent incidence rate. In [29], the authors extended the classical SIRS epidemic model incorporating media coverage from a deterministic framework to a stochastic differential equation and focused on how environmental fluctuations of the contact coefficient affect the extinction of the disease.
To the best of our knowledge, a small amount of work has been done with stochastic perturbation on an SIR epidemic model with a saturated transmission rate . The purpose of this paper is to study that the stochastic factor has a significant effect on the dynamics of SIR epidemic model with a saturated incidence rate. The organization of this paper is as follows. In the next section, we present the formulation of mathematical model with environmental noise. We give some properties about deterministic model (4) and carry out the analysis of the dynamical properties of stochastic model (3), respectively. Finally, we give a concluding section.
2. Model and Dynamics Analysis
Let be the number of susceptible individuals, the number of infective individuals, and the number of removed individuals at time , respectively. Motivated by [31], we assume that stochastic perturbations are of white noise type, which are directly proportional to distances from the steady-state values of and influence on , , respectively. In this way, an SIR epidemic model with a saturated transmission rate and stochastic fluctuations will be reduced to the following form: All parameters are positive constants, is the recruitment rate of the population, is the natural death rate of the population, is the proportionality constant, is the parameter that measures the psychological or inhibitory effect, is the rate at which recovered individuals lose immunity and return to the susceptible class, and is the natural recovery rate of the infective individuals. Note that , , and are real constants and known as the intensity of the stochastic environment and is standard Brownian motion.
2.1. Dynamics of the Deterministic Model
In this subsection, when , we consider the deterministic SIR epidemic model:
Because of the biological meaning of the components , we focus on the model in the first quadrant . Model (4) always has a disease-free equilibrium , which corresponds to the extinction of the disease.
Define the basic reproduction number as which denotes the number of individuals infected by a single infected individual placed in a totally susceptible population.
Theorem 1. From model (4), it follows that(i)if , there is no positive equilibrium;(ii)if , there is a unique endemic equilibrium , which corresponds to the coexistence of , , and and is given by In other words, when , the disease can invade a totally susceptible population and the number of cases will increase, whereas when , the disease will always fail to spread.
Lemma 2. The plane is a manifold of model (4), which is attracting in the first octant.
Proof. Summing up the three equations in (4) and denoting , we obtain
It is clear that is a solution of (7) and for any , the general solution of (7) is
Hence,
which implies the conclusion.
Theorem 3. The endemic equilibrium point is globally asymptotically stable in .
Proof. The Jacobian matrix at equilibrium point is given by
where
The characteristic equation at the interior equilibrium point is
where
It is clear that
Here , and .
Now . Therefore, model (4) is globally stable at the equilibrium .
2.2. Dynamics of the Stochastic Model
Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and increasing while contains all -null sets. Denote and the norm . And denote as the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in .
We define the differential operator associated with three-dimensional stochastic differential equation as where
If acts on a function , then we denote where means transposition.
In this subsection, we first show the existence of a unique positive global solution of the stochastic model (3).
Theorem 4. For model (3) and any given initial value , there is a unique solution on and will remain in with probability one.
Proof. Since the coefficients of model (3) satisfy the local Lipschitz condition, there is a unique local solution on , where is the explosion time. Therefore, by Itô’s formula, the unique local solution of model (3) is positive. Next, let us show that this solution is global; that is, a.s.
Let be sufficiently large for , , and lying with the interval . For each integer , define a sequence of stopping times by
where we set ( represents the empty set) in this paper. Since is nondecreasing as , there exists the limit
Then a.s. Now, we need to show a.s. If this statement is violated, then there exist and such that
Thus, there is an integer such that
Define a -function by
which is a nonnegative function. If , by using Itô’s formula, we compute
where is a positive constant. Integrating both sides of the above inequality from to , we get
where . Then taking the expectations leads to
Set for and from (23), we have . For every , there are some such that equals either or for ; hence is no less than . Then we obtain
where is the indicator function of . Letting leads to the contradiction . This completes the proof.
Theorem 4 shows that the solution to model (3) will remain in . The property makes us continue to discuss how the solution varies in in more detail. Here, we present that the definition of stochastic ultimate boundedness [32] is one of the important topics in population dynamics and is defined as follows.
Definition 5. The solutions of model (3) are said to be stochastically ultimately bounded, if for any , there is a positive constant , such that for any initial value , the solution to model (3) has the property that
Theorem 6. The solutions of model (3) are stochastically ultimately bounded for any initial value .
Proof. From Theorem 4, the solution will remain in for all almost surely. Define a function
for and . By Itô’s formula we obtain
where is a suitable constant.
Based on Theorem 4 and from (31), we have
Letting yields
which implies
Note that
Then we get
which means
Therefore, there exists a positive constant such that
For any , set , then by Chebyshev’s inequality,
Thus, we obtain
which yields the required assertion.
Generally speaking, the nonexplosion property, the existence, and the uniqueness of the solution are not enough but the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence [33] will be given below.
Definition 7. The solutions of model (3) are said to be stochastically permanent, if for any , there exists a pair of positive constants and such that for any initial value , the solution to model (3) has the properties
Theorem 8. Assume and for any initial value , the solution satisfies where is an arbitrary positive constant satisfying in which is an arbitrary positive constant satisfying
Proof. Define a function
for ; using Itô’s formula, we get
Choosing a positive constant that satisfies (43) and applying Itô’s formula, we obtain
where
Using the facts that
then,
Let be sufficiently small such that it satisfies (45), by Itô’s formula; then
where and have been defined in the statement of the theorem. Thus,
Therefore we obtain
For , we know that ; consequently,
which completes the proof.
Considering Chebyshev inequality, Theorems 6 and 8, we immediately obtain the following result.
Theorem 9. Assume ; then the solutions of model (3) are stochastically permanent.
Proof. From Theorem 6, we have which implies
This follows that
By Theorem 8, we get
For any , let ; then
Hence,
which follows that
The proof is complete.
3. Conclusions
In this paper, we propose an SIR epidemic model with a nonlinear incidence rate of the form . We extend to consider and analyze the epidemic model with stochastic perturbations. The value of this study lies in two aspects. First, it presents existence and global stability analysis of the endemic equilibrium for the deterministic model (4). Second, it verifies some relevant properties of the corresponding stochastic model (3) and reveals the effect of environmental noise on the epidemic model.
To study the effect of environmental noise on the deterministic model (4), we stochastically perturb model (4) with respect to white noise around its endemic equilibrium. By constructing suitable Lyapunov functions and applying Itô’s formula, we obtain that there is a unique positive solution to model (3) for any positive initial value and derive that the solution is stochastically bounded and permanent under some conditions. These conditions depend on the intensities of noise , , and . When the intensities of noise satisfy some conditions and are not sufficiently large, the population of the stochastic model may be stochastically permanent.
As an example, we perform some numerical simulations to illustrate the analytical results of stochastic model (3) by referring to the method mentioned in Higham [34]. Then model (3) can be rewritten as the following discretization equations: where is the Gaussian random variables .
Figure 1 shows time-series plots for model (3) with and without stochastic perturbations. The parameters are taken as , , , , , and and initial value . In this case, model (4) has the endemic point . The only difference between conditions of Figures 1(a) and 1(b) is that the values of environmental noise intensities , , and are different. In Figure 1(a), with , , and and in Figure 1(b), with , , and , the condition of Theorem 9 is satisfied. That is, the solutions of model (3) are stochastically permanent. From Figures 1(a) and 1(b), one can see that with increasing the noise intensities, the solutions of model (3) will be oscillating strongly around the endemic point of model (4).
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(b)
To study the effect of noise in model (3) further, in Figure 2(a), we choose , which satisfies the condition of Theorem 9, while in Figure 2(b), , that does not satisfy the condition of Theorem 9. From Figure 2(a), one can see that the infective population will be oscillating slightly around , and both the susceptible and the removed population will be affected by the noise but the effect is very small. From Figure 2(b), when the condition of Theorem 9 is not satisfied, the noise can force the population to become largely fluctuating. In this case, the solution of model (3) is not stochastically permanent.
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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks the editor and the anonymous referees for very helpful suggestions and comments which led to the improvement of the original paper.