Modeling and analysis of SIR epidemic dynamics in immunization and cross-infection environments: Insights from a stochastic model*

Mathematical modeling is an important tool that can help us understand the transmission of an infectious disease. Many scholars [1–3, 6–11, 13, 15, 16, 18, 21, 24, 30] have put forward mathematical models and have made contributions to disease control.

As far as we know, a classic and important SIR model was early investigated by Kermazk and McKendrick [11] in 1927. In classical SIR models, the infected patients can recover health with treatment. Many scholars have also investigated the SIR model in different situations. Capasso et al. [3] summarized Kermazk–McKendrick model and took into account nonlinear incidence phenomena for large numbers of infectives. Meanwhile, Capasso et al. expanded the threshold theory and laid a foundation for solving the stochas- tic nonlinear threshold. Hethcote [10] gave a qualitative analysis of nonlinear incidence SIR model, which is appropriate for viral agent diseases and considered social impact. Liu [18] investigated a deterministic and modified nonlinear SIR model with periodic solutions. In [18], the author also explored the corresponding stochastic epidemic models and the asymptotic behavior of the solution. Ghosh et al. [7] gave an SIR model with nonmonotonic incidence and logistic growth. In [7], authors studied the condition for backward bifurcation and Hopf bifurcation and solved the optimal control problem. Dieu et al. [6] classified a stochastic SIR model and developed ergodicity of the underlying system.

In classical epidemiological models, bilinear and standard incidence rates [28] are suitable for a small number of people in a short time, so many scholars use nonlinear incidence [2, 6, 7, 10, 16, 18, 21]. Logistic model [7] is more in line with the law of social population growth. In references [16, 21], the authors introduced the stochastic epidemic model with cross-infection of diseases. It has become a common phenomenon that people are infected with different diseases at the same time. On the basis of reference [2, 6, 7, 10, 16,18,21], a deterministic SIR model with cross-infection and permanent immunization is proposed in which the nonlinear incidence is used. The corresponding model is as follows:

where and with the natural mortality rate and , respectively, are the susceptible class and the removed class, respectively and are the individuals with cross-infection at time and are the contact rates. . is the proportion of patients infected by two diseases. and are the mortality of cross-infection diseases, which include natural mortality and mortality due to diseases. and are recovery rates of cross-infection diseases, respectively. is constant vaccination rate of susceptible class.

The susceptible class will have permanent immunity after vaccination. All parameters are positive. Functions represent nonlinear incidence rates for cross-infection diseases.

However, the spread of diseases is often affected by environmental noise [1, 4, 8, 9, 13, 15,16,21,24,28,30], a lot of literatures add randomness to reflect real life more accurately. The properties for stationary distribution of random variables were proved in [8]. [15] gave an SIR epidemic model that a susceptible person is infected with a disease and tem- porarily immunized. All references [1, 4, 9, 13, 24, 28, 30] are stochastic epidemic models with nonlinear incidence that has been used in chemostat model [25]. We assume that the mortality rates of are disturbed by white noise in system (1), then we have

Thus, we can consider the following system:

In the next, we only consider the dynamic properties of system (2) through differential equation theories.

Some notations of stochastic differential equations can be seen in [19]. Let be a complete probability space, which has a filtration and satisfies the usual conditions. The functions are Brownian motion defined on this complete probability space. We define

Let be an integrable function on and define . This paper mainly studies mathematical modeling and theoretical proof as well as the influence of parameter changes on the model.

We arrange the article as follows: Section 2 proves that system (2) has a global positive solution, which is unique. In Section 3, stochastic boundedness of the solution is explored. Section 4 investigates the extinction and persistence conditions of the stochastic system (2). We show the maximum value at point and the existence of a unique ergodic stationary distribution in Section 5. At last, we give the numerical simulations and a brief conclusion.

Theorem 1.For any given initial value , there is a unique positive solutionof system (2), which belongs to with probability one.

Proof. By standard arguments, there is a unique positive local solution on, provided that . Here denotes the explosion time. Next, we need to prove the global property of the solution.

Define a function as

Since for any , we have it followa that is a positive difine fuction. Through formula, we have

Where

Here is a positive constant. Through [20], we can get the global property of the unique positive solution

Theorem 2. The solution of system (2) with any initial value , is stochastically ultimate bounded.

Proof. Let , where constant will be given later. Then

where

Choose such that

then we obtain

and

Then we get

where

Thus, one has

where

Here

Following (3), we obtain

Consequently, we have

For any small constant and letting , the Chebyshev’s inequality [28] implies that

Then

Consequently,

so is ultimately bounded. Therefore, are ultimately bounded.

From system (2) we get

We consider the stochastic equation

From stochastic comparison theory we know that

From Pasquali [22] we get Lemma 1.

Lemma 1. Define , then we have , system (4) has a unique ergodic stationary distribution withprobability density

where is an integrable function with measure .

Proof. From system (4) we get the stationary Fokker–Plank equation

with probability density , then we can simplify theequation in the following form:

Then we can get

where is a constant ,

Then we can get

where , is a constant. We can calculate

where and are constants. We have

From the conditions we integrate the above formula,and let.

We can get

For the following proof, we define

Lemma 2. (See [29].) If is the solution of system (2), then we have

and

Lemma 3. (See [14, 29].) Assume is the solution of (2), then

Theorem 3. The diseases are said to be extinctive if , respectively.

Proof. We first prove the extinction of disease . Applying Itô’s formula to system (2), one has

From system (4) we learn . Then we have

Then we get the following inequality by Lemmas 1 and 3

which implies .

In the same way as in the proof of , we can get the following inequality from system (2):

where . Then we get

So we have

thus,

Theorem 4. For any given initial value , we have the following results of system (2):

(i) If will be persistent in mean, and will be extinct. Besides, we have

(ii) If , then will be extinct, and will be persistent in mean. Besides, we have

(iii) If will be persistent in mean and satisfy

Proof. From the first equations of system (2) and system (4) one gets

and

From (5) and (6) one has

that is,

(i) By Theorem 3, since . Since small enough such that

Applying Itô’s formula to the the Lyapunov function ln , it follows that

Calculating (7) directly, we can obtain

where . According to Lemma 4, one sees that and

Then one can calculate that

(ii) By Theorem 3, since . Since small enough such that

Using formula to the Lyapunov function gets

Then we get

Then one can calculate that

(iii) Define

then we have

From (8) one gets

Sorting out the above inequalities results in

In this section, we use the Has’minskii theory [12] to prove the stationary distribution of system (2).

Theorem 5. System (2) has a unique ergodic stationary distribution , if

and

Proof. System (2) has a diffusion matrix

Let such that

, then we get that all the eigenvalues of diffusionmatrix are greater than zero.

Define

Here . Defining a sufficiently large such that

where

It is clear that for , there exists a unique minimum point . Then denote a positive definitive fuction

where

Applying the formula yields

where and

From above equation we can get

and

Since

then we have

Note that

we have

According to (10)–(11), we get .

where

Next,

Therefore,

Construct a compact bounded subset U:

and will be given in the later. In the set , choosing small enough such that

Here are positive constants defined in equations (20), (21), (22), (23), (24), (25), respectively. Next, six domains are given in the following:

We need prove that . It is clear that is equivalent to

Case 1. If , due to

we get

where

According to (9), (12) and (13), we have that .

Case 2. If ,

where

By (9) and (14) we obtain that .

Case 3. If , due to

we have

where

In view of (9), (15) and (16), we get that .

Case 4. If

where

Together with (17), we have that

Case 5. If ,

where

By (18) we get that

Case 6. If,

where

From (19) we derive that for all .

Clearly, is small enough such that

We construct the following equivalent model to facilitate computer simulation:

where are independent random variables,is the time taken divided by the step size.

In system (2), let . = 4, . = 1.5, .. = 0.1, .. = 0.5, .. = 0.1, .. = 0.46,

In system (2) . With the changes of and , the diseases and will be extinct or persistent.

In each figure below, every figure has two subfigures. The first subfigure represents the development trend of, respectively. The second subfigure is the probability density of . From Theorem 3 we know that the diseases will be extinct when, the diseases will be persistent.

In our simulations, we only consider the influence with the changes in white noise on the disease. When the values of white noises are large than a certain value, the disease will be extinct. When the white noise is less than a certain value, the disease will be persistent. The figures are consistent with the theorem in our paper.

In Fig. 1, we let . By calculating we obtain that, then the conditions of Theorem 3 hold. So and are extinct.

In Fig. 2, we let . By calculating we gain that, which satisfy condition (i) of Theorem 4. We can obtain that is persistent (see Fig. 2(e)) and is extinct (see Fig. 2(f)).

In Fig. 3, we . By calculating we get that, then condition (iii) of Theorem 4 holds. So are persistent.

In Fig. 4, we simulate the influence of different noise intensities on system (2). It is found that as the intensity of white noise increases, the number of infections will decrease.

In addition, as time goes on, it shows periodic outbreaks and the duration of the outbreak is shortened.

Diseases are always affected by various noises in the environment, then the changes of environmental noise can lead to changes in diseases. According to Theorems 3 and 4, the conditions of extinction and persistence in mean about system (2) have been established. These theorems are in fact a development of the papers by Cai [2], Liu [16] and Meng [21]. Furthermore, we used a new stochastic method to investigate the extinction and persistence, which is different from the previous works [2, 6, 7, 16, 18, 21]. The results that obtained in the present work can be applied to stochastic model of proportional disturbance. The obtained theory is a positive and effective guidance for cross infection. Many diseases, such as diphtheria, typhoid and influenza, are cured, the susceptible class can have permanent immunity. This feature can be well reflected in this model.

The model can introduce telephone noises such as continuous time Markov chain [5, 17, 27]. We only study one susceptible person, and we can study multiple susceptible persons. We also can investigate a susceptible person infected with more than three dis- eases [32]. We can explore the periodic solution of the epidemic model [23]. The impact of white noise on not only mortality rate but also infection rates also will be considered. The methods also can be used in stochastic food chain models [26, 31]. In our future work, we will solve these problems.

This paper provides a modeling framework based on stochastic differential equations to explore the long-term dynamics of epidemic cross-infection with SIR epidemiological laws. Since the interaction between the disease and the environment is full of stochasticity, it is of great practical significance to explore the mechanism of environmental stochasticity on the dynamics of infection. For this reason, we assumed that each component of the population is subject to environmental stochasticity, which is positively correlated with the density of each component of the population. In addition, in view of the fact that immunization is widely used in the control of epidemic diseases and has achieved miraculous achievements repeatedly, we also considered the impact of immunization on the spread of diseases. With the help of stochastic analysis tools and auxiliary systems, we studied the properties of the global positive solution of the proposed system. Furthermore, we provided our main theoretical results including the extinction, persistence and the existence of a unique stationary distribution of the proposed model. The results show that: (i) When , both types of diseases will be extinct with probability one. Since the intensities of environmental stochasticities , are negatively correlated with the conditions for extinction , respectively, which indicates that environmental stochasticity is not conducive to the survival of the diseases. (ii) When will be persistent in the mean, while will be extinct; when will be persistent in the mean, while will be extinct; when will be persistent in the mean. Similar to the item (ii), this result suggests that small stochasticity is beneficial to the survival of the diseases. (iii) When and the parameters meet some other con- straints (see Theorem 5 for detail), the stochastic system has a unique ergodic stationary distribution. Ergodicity means that the statistical properties of the stochastic system will not change over time, which allows us to estimate the contour of the stationary distribution by simulating a trajectory of the solution. In this scenario, the results also indicate that the small stochasticity is necessary for the existence of the stationary distribution, i.e., small stochasticity contributes to the survival of the diseases.

This work is just our preliminary exploration of how stochasticity affects the dy- namics of disease transmission. In order to have a more comprehensive understanding of the interaction between environmental stochasticity and the diseases, the following explorations are needed: (i) The type of stochasticity considered in this article is white noise, and in complex actual environments, there are other types of stochasticity such as telegraph noise [5, 17]. It has practical significance to investigate how these different types of noise synergistically affect the spread of diseases. (ii) This article only considers the situation of one type of susceptibles. Next, we can explore more complex scenarios such as by including more than three types of diseases [32]. (iii) This article assumes that stochasticity is positively correlated with the density of each component. Next, we can also consider the situation where stochasticity mainly perturbs the infection rate, which will induce a different stochastic system with degenerate diffusion terms, and its theoretical analysis is also more challenging.