Abstract
A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value . That is, when and together with an additional condition, the disease is extinct with probability one, and when , the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when , the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.
1. Introduction
Our real life is full of randomness and stochasticity. Therefore, using stochastic dynamical models can gain more real benefits. Particularly, stochastic dynamical models can provide us with some additional degrees of realism in comparison to their deterministic counterparts. There are different possible approaches which result in different effects on the epidemic dynamical systems to include random perturbations in the models. In particular, the following three approaches are seen most often. The first one is parameters perturbation; the second one is the environmental noise that is proportional to the variables; and the last one is the robustness of the positive equilibrium of the deterministic models.
In recent years, various types of stochastic epidemic dynamical models are established and investigated widely. The main research subjects include the existence and uniqueness of positive solution with any positive initial value in probability mean, the persistence and extinction of the disease in probability mean, the asymptotical behaviors around the disease-free equilibrium and the endemic equilibrium of the deterministic models, and the existence of the stationary distribution as well as ergodicity. Many important results have been established in many literatures, for example, [1–16] and the references cited therein. Particularly, for stochastic SI type epidemic models, in [6], Gray et al. constructed a stochastic SIS epidemic model with constant population size where the authors not only obtained the existence of the unique global positive solution with any positive initial value, but also established the threshold value conditions; that is, the disease dies out or persists. Furthermore, in the case of the persistence, the authors also showed the existence of a stationary distribution and finally computed the mean value and variance of the stationary distribution.
However, from articles [1–16] and the references cited therein, we see that there are still many important problems which are not studied completely and impactfully. For example, see the following.(1)The stochastic epidemic models with general nonlinear incidence are not investigated. Up to now, only some special cases of nonlinear incidence, for example, saturated incidence rate, are considered. But, we all know that the nonlinear incidence rate in the theory of mathematical epidemiology is very important.(2)For the stochastic epidemic models with the standard incidence, up to now, we do not find any interesting researches.(3)The conditions obtained on the existence of unique stationary distribution are very rigorous. Whether there is a unique stationary distribution only when the model is permanent in the mean with probability one is still an open problem.
Motivated by the above work, in this paper, we consider the following deterministic SIS epidemic model with nonlinear incidence rate and disease-induced mortality:In model (1), and denote the susceptible and infectious individuals, denotes the recruitment rate of the susceptible, is the natural death rate of and , is the disease-related death rate, the transmission of the infection is governed by a nonlinear incidence rate , where denotes the transmission coefficient between compartments and , is a continuously differentiable function of and , and denotes the per capita disease contact rate.
Now, we assume that the random effects of the environment make the transmission coefficient of disease in deterministic model (1) generate random disturbance. That is, , where is a one-dimensional standard Brownian motion defined on some probability space. Thus, model (1) will become into the following stochastic SIS epidemic model with nonlinear incidence rate:
In this paper, we investigate the dynamical behaviors of model (2). By using the Lyapunov function method, Itô’s formula, and the theory of stochastic analysis [17, 18], we will establish a series of new interesting criteria on the extinction of the disease, permanence in the mean of the model with probability one. The stochastic oscillation of the disease about a positive number in the case where there is not disease-related death is also obtained. Further, we study the positive recurrence and the existence of stationary distribution for model (2), and a new criterion is established. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail. For some special cases of nonlinear incidence , for example, (standard incidence) and , many idiographic criteria on the extinction, permanence, and stationary distribution are established. Lastly, some affirmative answers for the open problems which are proposed in this paper also are given by the numerical examples (the numerical simulation method can be found in [19]).
The organization of this paper is as follows. In Section 2, the preliminaries are given, and some useful lemmas are introduced. In Section 3, the sufficient conditions are established which ensure that the disease dies out with probability one. In Section 4, we establish the sufficient conditions which ensure that the disease in model (2) is permanent in the mean with probability one, and when there is not disease-related death the disease oscillates stochastically about a positive number. In Section 5, the existence on the unique stationary distribution of model (2) is proved. In Section 6, the numerical simulations are carried out to illustrate some open problems. Lastly, a brief discussion is given in the end to conclude this work.
2. Preliminaries
Denote , , and . Throughout this paper, we assume that model (2) is defined on a complete probability space with a filtration satisfying the usual conditions; that is, is right continuous and contains all -null sets.
In model (2), and denote the susceptible and infected fractions of the population, respectively, and is the total size of the population among whom the disease is spreading; the parameters , , , and are given as in model (1); the transmission of the infection is governed by a nonlinear incidence rate ; denotes one-dimensional standard Brownian motion defined on the above probability space; and represents the intensity of the Brownian motion . Throughout this paper, we always assume the following.() is two-order continuously differentiable for any , , and . For each fixed , is increasing for and for each fixed , is decreasing for . for any and , and , where .
Particularly, when , then assumption becomes in the following form: () and are continuously differentiable for and , is increasing for , and is decreasing for .
Remark 1. From , by simple calculating, we can obtain that for any and , , and for any , .
Remark 2. When (standard incidence), where , (Beddington-DeAngelis incidence) with constants and , and with constant , then is satisfied.
Now, we give the following result for function .
Lemma 3. For any constants , let . Then,
The proof of Lemma 3 is simple. In fact, from , we have Hence, conclusion (3) holds. Define the functionsUsing the L’Hospital principle, from , we haveThis shows that and are continuous for . Therefore, conclusion (4) also is true.
Next, on the existence of global positive solutions and the ultimate boundedness of solutions for model (2) with probability one, we have the result as follows.
Lemma 4. For any initial value , model (2) has a unique solution defined on satisfying for all with probability one. Furthermore, when then , and when then , where and .
Lemma 4 can be proved by using the method which is given in [6]. We hence omit it here.
3. Extinction of the Disease
Define the constantsWe have that is the basic reproduction number of deterministic model (1). On the extinction of the disease in probability for model (2) we have the following result.
Theorem 5. Assume that one of the following conditions holds:(a) and ;(b).Then disease in model (2) is extinct with probability one. That is, for any initial value , solution of model (2) has a.s.
Proof. By Lemma 4 we have a.s. for all and For any there is such that for all . Hence, for any ,With Itô’s formula (see [17, 18]), we haveHence, for any , Define a functionWhen , is monotone increasing for , and when , is monotone increasing for and monotone decreasing for .
If condition (a) holds, then when , from (9), we directly haveWhen , since , we can choose such that and . From (9) we also have inequality (13). Hence, when , By the large number theorem for martingales (see [17] or Lemma A.1 given in [9]), we obtainFrom the arbitrariness of and , we further obtainIf condition (b) holds, then, since , has maximum value at , and for any , we have which impliesWith the large number theorem for martingales and arbitrariness of , we obtainFrom (16) and (19) we finally have a.s. This completes the proof.
Now, we give a further discussion for conditions (a) and (b) of Theorem 5 by using the intensity of stochastic perturbation and basic reproduction number of deterministic model (1).
When , then, for any , , and it is easy to prove that one of the conditions (a) and (b) of Theorem 5 holds. Therefore, for any , the conclusions of Theorem 5 hold. Let . From we haveDenoteSince , we easily prove that when one of the conditions (a) and (b) of Theorem 5 holds. Therefore, for any , the conclusions of Theorem 5 hold. When , we have and . Hence, condition (a) in Theorem 5 does not hold. We only can obtain that for any the conclusions of Theorem 5 hold. Summarizing the above discussions we have the following result as a corollary of Theorem 5.
Corollary 6. Assume that one of the following conditions holds:(a) and ;(b) and ;(c) and .Then disease in model (2) is extinct with probability one.
Corollary 7. Let (standard incidence). Assume that one of the following conditions holds:(a) and ;(b).Then disease in model (2) is extinct with probability one.
Corollary 8. Let . Assume that holds and one of the following conditions holds:(a) and ;(b).Then disease in model (2) is extinct with probability one.
Remark 9. It is easy to see that in Theorem 5 the conditions and are not included. Therefore, an interesting conjecture for model (2) is proposed; that is, if the above condition holds, then the disease still dies out with probability one. In Section 6, we will give an affirmative answer by using the numerical simulations; see Example 1.
Remark 10. In the above discussions, we see that case has not been considered. An interesting open problem is whether when the disease in model (2) also is extinct with probability one. A numerical example is given in Section 6; see Example 2.
4. Permanence of the Disease
On the permanence of the disease in the mean with probability one for model (2), we establish the following results.
Theorem 11. If , then disease in model (2) is permanent in the mean with probability one. That is, there is a constant such that, for any initial value , solution of model (2) satisfies
Proof. From , we choose a small enough constant such thatBy Lemma 4, it is clear that, for any initial value , solution of model (2) satisfies and for above there is such that a.s. for all . Denote the set . Since , we obtain for any From (10), for any ,Since for and is continuously differentiable, exists for any , and set is convex and connected, by the Lagrange mean value theorem when we have where . Let constantsFrom Lemma 3 we have , . For any , we haveFrom (25) and Remark 1 we further havewhereBy the large number theorem for martingales and Lemma 4, a.s. Therefore, from Lemma 5.2 given in [16], we finally obtain a.s. This completes the proof.
Remark 12. From (20), we have that is equivalent to . Therefore, Theorem 11 also can be rewritten by using intensity of stochastic perturbation in the following form: if , then disease in model (2) is permanent in the mean with probability one.
Remark 13. Combining Corollary 6 and Remark 12 we can obtain that when , number is a threshold value. When , the disease in model (2) is permanent in the mean and when , the disease is extinct with probability one. However, when , then the alike results are not established. Therefore, it yet is an interesting open problem.
Theorem 14. Susceptible in model (2) also is permanent in the mean with probability one. That is, there is a constant such that, for any initial value , solution of model (2) satisfies
Proof. By Lemma 4 we easily see that, for any initial value , solution of model (2) satisfies and for any small enough constant there is such that for all . Hence, by Lemma 3, when we have , where . Integrating the first equation of model (2) we obtain for any Therefore, with the large number theorem for martingales, we finally haveThis completes the proof.
As consequences of Theorems 11 and 14, we have the following corollaries.
Corollary 15. Let (standard incidence). If , then model (2) is permanent in the mean with probability one.
Corollary 16. Let . Assume that holds and ; then model (2) is permanent in the mean with probability one.
We further have the result on the weak permanence of model (2) in probability.
Corollary 17. Assume that . Then there is a constant such that, for any initial value , solution of model (2) satisfies
Now, we discuss special case: for model (2); that is, there is not disease-related death in model (2). We can establish the following more precise results on the weak permanence of the disease in probability compared to the conclusion given in Corollary 17.
Theorem 18. Let in model (2). If , then, for any initial value , solution of model (2) satisfieswhere satisfies the equation
Proof. From Lemma 4, we know that . Without loss of generality, we assume that for all . From (10), for any ,Define a function . Then, for any ,where function . With condition we have andHence, has a positive root in which isSince is monotone decreasing for , , andthere is a unique such that and
When and , since function has maximum value at and , there is a unique , such that . From and we have . Hence, .
From the above discussion we obtain that is strictly increasing on , is strictly decreasing on , and is strictly decreasing on .
When , similarly to the above discussion, we can obtain that is strictly decreasing on and is strictly decreasing on .
Now, we firstly prove that (35) is true. If it is not true, then there is an enough small such that , where Hence, for every , there is a constant such thatWith the above discussion we know that for all . From (39) we further obtain for any From the large number theorem for martingales, we have , which implies as . This leads to a contradiction with (43).
Next, we prove that (36) holds. If it is not true, then there is an enough small such that , where . Hence, for every , there is such thatWith the above discussion we have for all . Together with (39), we further obtain for any With the large number theorem for martingales, we have , which implies as . This leads to a contradiction with (45). This completes the proof.
Remark 19. Theorem 18 indicates that if and , then any solution of model (2) with initial value oscillates about a positive number . Therefore, an interesting open problem is whether there is a more less positive than number such that any solution of model (2) with initial value , a.s. In Section 6, we will give an affirmative answer by using the numerical simulations; see Example 3.
From Theorem 18, we easily see that number will arise from the change when the noise intensity changes. Therefore, it is very interesting and important to discuss how number changes along with the change of . We have the following result.
Theorem 20. Assume that in model (2) and . Let number be given in Theorem 18 and . Then one has the following.(a) as the function of is defined for (b) is monotone decreasing for .(c), where is the endemic equilibrium of deterministic model (1).(d)If , then , and if , then , where satisfies
Proof. Sinceby the inverse function theorem we obtain that as the function of is defined for . Fromwe can obtain that when . Therefore, as a function of is defined for .
Computing the derivative of with respect to , we haveSincewe have . From the definition of , we easily see that is monotone decreasing for . From (49) and , we obtain that exists and is continuous for . Since , we have . Therefore, . It follows that is monotone decreasing as increases. Thus, both and exist. Let and . We haveHence, . This shows that . Let be the endemic equilibrium of deterministic model (1); then we have . Hence, . This shows that .
On the other hand, we haveIf , then from (54) we obtain . Hence,This shows that . If , then we have from (54)which impliesTherefore, we have , and satisfiesThis completes the proof.
Conclusion (b) of Theorem 20 shows that when in model (2), number monotonically decreases when increases in , and when , has a maximum value by Conclusion (c). Therefore, when . If , then when , has a minimum value and if then when , has a minimum value by Conclusion (d).
It is clear that when in model (2) then from (20). On the other hand, from Conclusion (c) of Corollary 7, we see that if then when , where is given in (21), we have a.s. for any solution of model (2) with initial value , which implies that . Therefore, when , we can propose an interesting open problem: whether there is a critical value such that when we have the fact that is monotonically decreasing and and when we have .
Remark 21. When , then from (56) we obtainnamely,where is the endemic equilibrium of deterministic model (1). Hence,Consequently, .
Remark 22. When , we easily validate that Theorems 20 and 24 degenerate into Theorems 5.1 and 5.4 which are given in [19], respectively. Therefore, Theorems 18 and 20 are the considerable extension of Theorems 5.1 and 5.4 in general nonlinear incidence cases, respectively.
Remark 23. For the case in model (2), an interesting and important open problem is when whether we also can establish similar results as Theorems 18 and 20. Furthermore, as an improvement of the results obtained in Corollary 17 we also propose another open problem: only when we also can establish the permanence of the disease with probability one; that is, there is a constant such that, for any solution of model (2) with initial value , one has In Section 6, we will give an affirmative answer by using the numerical simulations; see Example 3.
5. Stationary Distribution
From Theorems 11 and 14 we obtain that when model (2) is permanent in the mean with probability one. However, when model (2) also has a stationary distribution. We have an affirmative answer as follows.
Theorem 24. If , then model (2) is positive recurrent and has a unique stationary distribution.
Proof. Here, the method given in the proof of Theorem 5.1 in [17] is improved and developed. By Lemma 4 and Remark 9 we only need to give the proof in region , where . Let be any solution of model (1) with a.s. for all . Let be a large enough constant, and letWhen , then either or . The diffusion matrix for model (56) isFor any we have .
Choose a Lyapunov function as follows:whereand is a constant. Computing , by Remark 1, we haveApplying the Lagrange mean value theorem, we havewhere andBy Lemma 3, we have ; . We hence haveComputing , by Remark 1, we haveComputing , we havewhere, by Lemma 3, . From the above calculations, we obtain that for any Sinceand when is small enough, it follows thatwe finally obtain that when is large enoughFrom Theorem 2.2, given in [10], we know that model (2) has a unique stationary distribution such thatThis completes the proof.
Remark 25. Comparing Theorem 24 with Theorem 6.2 given in [19], we see that Theorem 6.2 is extended and improved to the general stochastic SIS epidemic model (2).
Remark 26. Since is equivalent to , we also have that if , then model (2) is positive recurrent and has a unique stationary distribution.
Particularly, for some special cases of nonlinear incidence , we have the following idiographic results on the stationary distribution as the consequences of Theorem 24.
Corollary 27. Let (standard incidence). If , then model (2) is positive recurrent and has a unique stationary distribution.
Corollary 28. Let . Assume that holds and ; then model (2) is positive recurrent and has a unique stationary distribution.
Combining Corollary 6, Theorem 11, Remark 12, Theorem 24, and Remark 26, we can finally establish the following summarization result by using intensity of stochastic perturbation and basic reproduction number of deterministic model (1).
Corollary 29. (a) Let . Then for any the disease in model (2) is extinct with probability one.
(b) Let . Then for any model (2) is permanent in the mean with probability one and has a unique stationary distribution, and for any the disease in model (2) is extinct with probability one.
(c) Let . Then for any model (2) is permanent in the mean with probability one and has a unique stationary distribution, and for any , where is given in (20), the disease in model (2) is extinct with probability one.
6. Numerical Simulations
In this section we analyze the stochastic behavior of model (2) by means of the numerical simulations in order to make readers understand our results more better. The numerical simulation method can be found in [19]. Throughout the following numerical simulations, we choose , where is a constant. The corresponding discretization system of model (2) is given as follows:where are the Gaussian random variables which follow standard normal distribution .
Example 1. In model (2) we choose , , , , , and .
By computing we have , , , and which is the case of Remark 9. From the numerical simulations, we see that the disease will die out (see Figure 1). An affirmative answer is given for the open problem proposed in Remark 9.
(a)
(b)
Example 2. In model (2), choose , , , , and .
By computing we have . From the numerical simulations given in Figure 2 we know that the disease will die out. Therefore, an affirmative answer is given for the open problem proposed in Remark 10.
(a)
(b)
Example 3. In model (2) choose , , , , , and .
We have , , and . The numerical simulations are found in Figure 3. We can see that solution of model (2) oscillates up and down at , which further show that the conclusions of Theorems 14 and 18 are true. At the same time, this example also shows that the disease in model (2) is permanent with probability one. Therefore, an affirmative answer is given for the open problems proposed in Remarks 19 and 23.
(a)
(b)
7. Discussion
In this paper we investigated a class of stochastic SIS epidemic models with nonlinear incidence rate, which include the standard incidence, Beddington-DeAngelis incidence, and nonlinear incidence . A series of criteria in the probability mean on the extinction of the disease, the persistence and permanence in the mean of the disease, and the existence of the stationary distribution are established. Furthermore, the numerical examples are carried out to illustrate the proposed open problems in this paper.
It is easily seen that the research given in [6] for the stochastic SIS epidemic model with bilinear incidence is extended to the model with general nonlinear incidence and disease-induced mortality. Particularly, we see that stochastic SIS epidemic model with standard incidence is investigated for the first time.
The researches given in this paper show that stochastic model (2) has more rich dynamical properties than the corresponding deterministic model (1). Particularly, stochastic model (2) has no endemic equilibrium. Thus, this can bring more difficulty for us to investigate model (2), but, on the other hand, this also makes model (2) have more rich researchful subjects than deterministic model (1). We can discuss not only the extinction, persistence, and permanence in the mean of disease in probability, but also the existence and uniqueness of stationary distribution, the asymptotical behaviors of solutions of stochastic model (2) around the equilibrium of deterministic model (1), and so forth.
In addition, we easily see that when intensity of the stochastic perturbation, then . This shows that when we still can have . Therefore, there is a very interesting and important phenomenon; that is, for deterministic model (1) the disease is permanent, but for the corresponding stochastic model (2) the disease is extinct with probability one; see Conclusion of Corollary 29.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This research is supported by the Doctorial Subjects Foundation of The Ministry of Education of China (Grant no. 2013651110001) and the National Natural Science Foundation of China (Grants nos. 11271312, 11401512, and 11261056).