Abstract
A five-dimensional nonautonomous schistosomiasis model which include latent period is proposed and studied. By constructing several auxiliary functions and using some skills, we obtain some sufficient conditions for the extinction and permanence (uniform persistence) of infectious population of the model. New threshold values of integral form are obtained. For the corresponding autonomous schistosomiasis model, our results are consistent with the past results. For the periodic and almost periodic cases, some corollaries for the extinction and permanence of the disease are established. In order to illustrate our theoretical analysis, some numerical simulations are presented.
1. Introduction
Schistosomiasis is a chronic, parasitic disease caused by blood flukes (trematode worms) of the genus Schistosoma. More than 207 million people, 85% of them living in Africa, are infected worldwide, with an estimated 700 million people at risk in 74 endemic countries. Schistosomiasis is prevalent in tropical and subtropical areas, especially in poor communities without access to safe drinking water and adequate sanitation [1]. In the developing countries, schistosomiasis is frequently a serious health problem [2].
The first mathematical models for schistosomiasis were those developed by Macdonald in [3] and Hairston in [4]. Since then, a number of mathematical models have been proposed. Barbour [5] studied the prevalence of infection in the definitive host population and the snails, Chiyaka and Garira [6] were concerned with the combined effect of the inclusion the dynamics of the miracidia and cercariae populations and a density dependent infection term, Bhunu et al. [7] developed a mathematical model of the transmission dynamics of HIV/AIDS in the presence of schistosomiasis in the human-snail hosts, Yang and Xiao [8] presented a dynamic model of Schistosoma japonicum transmission that incorporated effects of the prepatent periods of the different stages of Schistosoma into Barbour’s model, Qi and Cui [9] established a diffusive system of partial differential equations considering the diffusion of human and snail hosts, and Qi et al. [10] developed a schistosomiasis model of the prevalence of infection in the Rattus norvegicus and Oncomelania snail, in which the authors considered a schistosomiasis model on the Rattus norvegicus in Qianzhou and Zimuzhou, two islets in the center of Yangtze River near Nanjing, China. There have been no human residents or other livestock on the islets; thus the mammalian Rattus norvegicus is the only definitive host while snail serves as the unique intermediate host; the authors set up the compartmental model as follows:Here, , represent the number of susceptible and infected Rattus norvegicus, respectively. , , are the number of susceptible, infected and preshedding, and infected and shedding Oncomelania snail, respectively. The other parameters interpretation of model (1) is shown in Parameters Section. All the parameters in the model are assumed to be nonnegative constants.
According to the concept of the next generation of matrix [11], the authors obtain the basic reproduction number [12] of system (1) as follows: And they prove that the infection will go extinct if and schistosomiasis will be prevalent if .
Nonautonomous phenomena, however, are dominating in real epidemic systems. Many diseases show seasonal behavior and taking into account the seasonal influence in epidemic models is very necessary. The nonautonomous phenomenon come from various sources, such as the variation of transmission rate and fluctuations in death and birth rates. Particularly, these coefficients of models usually vary with time when long-time dynamical behaviors are studied for an epidemic system. More and more realistic and interesting nonautonomous models can be found in papers [13–24] and the references cited therein.
Motivated by the ideas of the above aspects, system (1) is modified to take into account factors mentioned above where all coefficients vary with time to give a more appropriate result and better understanding of the prevalence of schistosomiasis. Thus, we refer to the following nonautonomous system for the study of schistosomiasis: with initial value
In this paper, our purpose is to obtain the weaker conditions for the permanence and extinction of schistosomiasis. We will establish the sufficient conditions for the permanence and extinction of the disease and give some new threshold values of the integral form. When system (3) reduces to periodic or almost periodic case, the basic reproduction number is obtained (see Corollaries 18 and 19 given in Section 5). Our results show that the threshold value acts as a sharp threshold for the permanence and extinction of the disease.
This paper is organized as follows. In Section 2 we present preliminary setting and propositions which are used to analyze the long-time behavior of system (3) in the following sections. In Sections 3 and 4, we prove our main theorems on the extinction and permanence of infectious population of system (3). In Section 5, we derive some corollaries for the extinction and permanence of infectious population of system (3) for some special cases. In Section 6, we provide numerical examples to illustrate the validity of our analytical results.
2. Notations and Preliminaries
At first, we put the following assumptions for system (3).(H1)Functions , , , , , , , , and are positive, bounded, and continuous on .(H2)There exist constants such that
For any continuous periodic function with period , we denote by the average value of where .
For any continuous almost periodic function , we denote by the average value of where .
Remark 1. When system (3) degenerates into -periodic system, that is to say, all the coefficients of system (3) are all nonnegative and nonzero, continuous periodic functions with period , then (H2) is equivalent to the following cases:
Remark 2. When system (3) degenerates into almost periodic system, that is to say, all the coefficients in it are all nonnegative and nonzero, continuous almost periodic functions, then (H2) is equivalent to the following cases: In what follows, we denote . We also define that are the solutions of the following equations with initial value , , , , respectively: Obviously, from system (3) and the comparison theorem, we have Furthermore, we have the following result.
Lemma 3. Suppose that assumptions (H1) and (H2) hold. Then the following results hold.(a)There exist constants and , which are independent of the choice of initial value , such that (b)There exist constants and , which are independent of the choice of initial value , such that (c)Each fixed solution of (8), of (9), of (10), and of (11) is bounded and globally uniformly attractive on .(d)When (8) is -periodic, then (8) has a unique nonnegative -periodic solution which is globally uniformly attractive. And (9)–(11) have similar results.(e)When (8) is almost periodic, then (8) has a unique nonnegative almost periodic solution which is globally uniformly attractive. And (9)–(11) have similar results.(f)If for all and , then for any solution of (8), we have where (g)If for all and , then, for any solution of (9), we have where (h)If for all and , then, for any solution of (10), we have where(i)If for all and , then, for any solution of (11), we have where
Lemma 4. Suppose that assumptions (H1) and (H2) hold. Then the solution of system (3) with initial value (3) exists and is uniformly bounded. Furthermore, for any , we haveFor any , and we define where is any solution of system (3).
We use the following two results in order to investigate the long-time behavior of system (3).
Lemma 5. If there exist positive constants and such that for all , then there exists such that either for all or for all .
Proof. Suppose that there does not exist such that for all or for all hold. Then there necessarily exists such that and Hence we have Substituting (27) into (28), we have From Lemma 4, we have , which is a contradiction.
Lemma 6. If there exist positive constants , and such that for all , then there exists such that for every , we have either or , .
Proof. From Lemma 5, we know that there exists such that either for all or for all .
For the first case, if for all , we will show that there exists such that either for all or for all . Suppose that there does not exist such that for all or for all hold. Then there necessarily exists such that and Hence we have Substituting (30) into (31), we have From Lemma 4, we have , which is a contradiction.
For the second case, if for all , we will prove that there exists such that either for all or for all . Suppose that there does not exist such that for all or for all hold. Then there necessarily exists such that and . Hence we have Substituting (33) into (34), we have From Lemma 4, we have , which is a contradiction.
If we let , then the results of the lemma hold.
3. Extinction of Infectious Population
In this section, we obtain sufficient conditions for the extinction of the infectious population of system (3). The definition of the extinction is as follows.
Definition 7. We say that the infective of system (3) is extinct if
Similarly, we can define the extinction of . Next, one of the main results of this paper is investigated.
Theorem 8. If there exist positive constants , and such that and for all , then and
Proof. Since there exist such that holds for all , from Lemma 5, we only have to consider the following two cases:(i) for all .(ii) for all .First we consider case (i). Since we have for all , it follows from the fifth equation of system (3) that for all . Hence we have for all . Now it follows from (37) that there exist constants and such that for all . From (41) and (42) we have Then the second equation of the corresponding limit system (3) is . From (i) and (H2) we can easily see that .
Next we consider case (ii). Because there exist such that for all , then from the proof of Lemma 6, we know that there exists such that either for all or for all . Thus we also have two cases to discuss.
On the one hand, if holds for all , that is, holds for all , then from the second equation of system (3) we obtain for all . Hence, we get for all . From (38) we see that there exist constants and such that for all . From (44) and (45), we have .
On the other hand, if for all hold, then we have for all . From the fourth equation of system (3), we have for all . Hence we have From (39), we see that there exist constants and such that for all . From (47) and (48), we have . Thus the fifth equation of the corresponding limit system (3) is . From (i) and (H2), we can easily get that . In a similar way, we can also prove .
Remark 9. We assume that all coefficients of system (3) are given by identically constant functions; then system (3) becomes an autonomous system. Conditions (37)–(39) are as follows: These inequalities show that , which implies ; here is given in (2). This shows that the result is consistent with the literature [10].
Remark 10. It is obvious that if there exist positive constants , and such that and for all , then and
4. Permanence of Infectious Population
In this section, we obtain sufficient conditions for the permanence of infectious population of system (3).
The definition of the permanence is as follows.
Definition 11. We say that the infective of system (3) is permanent if there exist positive constants and , which are independent of the choice of initial value satisfying (3), such that Similarly, we can give the definition of the permanence of .
Let the function and let be some fixed solution of (8) with initial value be some fixed solution of (10) with initial value .
We have the following theorem.
Theorem 12. If there is a constant such that then the infective and of system (3) are permanent.
Proof. Since , for all , then, from system (3), we have Let us consider the following system: Firstly, we prove that the number is independent of the choice of and . In fact, (c) of Lemma 3 implies that, for any sufficiently small (), any solution of (8) with initial value , and any solution of (10) with initial value , there exists such that as Hence, For , since the inequality holds for all and from Taylor’s theorem of binary function, we obtain By the arbitrariness of , we finally obtainThis shows that is independent of the choice of and . From inequalities (15) and (16), we can set largely enough such that for all . From (53), for sufficiently small , there exists such that for all . We define From (60)–(63), we see that, for positive constants , there exist small and such thathold for all , where . From (H2), we can choose such that both the following two equations hold for all . First we claim that . In fact, if it is not true, then there exists such that for all . Suppose that for all . Then, from (60) and (69) we have for all . Thus, from (67), we have , which contradicts with (b) of Lemma 3. Therefore we see that there exist such that . Suppose that there exists such that . Then, we see that there necessarily exists such that and for all . It is easy to see that there exists an integer that satisfies . Then, from (67), we havewhich is a contradiction. Therefore, we see that for all . In a similar way, from (68), we can show that there exists such that for all .
For , we define a differentiable function . When , from (69)–(73), we have Since the inequality holds for all and from (65) and (66), we have Integrating the above inequality from to , we have for all . By (62), we obtain . This contradicts with the boundedness of , , and . From this contradiction, we finally conclude .
Next, we prove where is a constant given in the following lines. For convenience sake, we let be the least common multiple of , , and . If we define for , there exists such that for all . Then, from inequalities (64) and (67)-(68) and (H2), we see that there exist constants , here we choose is an integral multiple of , and such that for all , , and is an integral multiple of . Let be a positive integer satisfying where . Since we have proved , there are only two possibilities as follows:(i)There exists such that for all .(ii) oscillates about for large . In case (i), we have . In case (ii), then there necessarily exist two constants such that Suppose that . Then, from the third equation of system (55), we have Hence, we obtain for all . Suppose that . Then, from (87), we have for all . Now, we are in a position to show that for all . Suppose that for all . Then, from (80), we have which is a contradiction. Therefore, there exists such that . Then, similar to the proof that , we can show that for all . Similarly, from (75), we can show that there exists such that for all . Thus we have for all . From (87), we have for all . Thus, from (65), (91), and (92), we have for all . Hence, from (83), we obtain for all . Thus, from (66), (91), (92), and (95), we have for all . Hence, from (84), we obtain We claim that for all . If it is not true, there exists a such that and for all . Then there exists such that . Let . From (74), (75), (91), and (92), the derivative of along solutions of (55) satisfies for all . Integrating the above inequality from to , we have Thus, , which contradicts with (91).
Therefore, is valid for all , which implies . According to the comparison theorem, we have . Since , the infective of system (3) is permanent.
Furthermore, from system (3), there exists such that . Since , the infective of system (3) is also permanent. This completes the proof.
Remark 13. For the corresponding autonomous system of system (3), condition (53) can become as follows: This shows that , which implies ; here is given in (2). This means that the result is consistent with the literature [10].
Remark 14. It is easy to prove that if , then the infective of system (3) is permanent.
5. Some Corollaries
As consequences of Theorems 8 and 12, we have a series of corollaries as follows.
Corollary 15. Suppose that , for all and , . If there exist positive constants , , , and such that and for all , then the two infectious populations and of system (3) are extinct.
Corollary 16. If there is a constant such that then the infective of system (3) is permanent.
Corollary 17. If there exists a constant such that , where Here, is some fixed solution of (8) with initial value is some fixed solution of (10) with initial value . Then the infective of system (3) is permanent.
Corollary 18. Suppose system (3) is -periodic; then the infective of system (3) is permanent provided that Here, is the globally uniformly attractive nonnegative -periodic solution of (8) and is the globally uniformly attractive nonnegative -periodic solution of (10).
Corollary 19. Suppose system (3) is almost periodic; then the infective of system (3) is permanent provided that Here, is the globally uniformly attractive nonnegative almost periodic solution of (8) and is the globally uniformly attractive nonnegative almost periodic solution of (10).
Corollary 20. Suppose that , for all and , . If there exist positive constants , and such that and for all , then the two infectious populations and of system (3) are extinct.
Here, is some fixed solution of (9) with initial value is some fixed solution of (11) with initial value
Corollary 21. Suppose system (3) is -periodic; we suppose that , for all and , . Then if there exist positive constants , and such that and for all , then the two infectious populations and of system (3) are extinct. Here, is the globally uniformly attractive nonnegative -periodic solution of (9) and is the globally uniformly attractive nonnegative -periodic solution of (11).
Corollary 22. Suppose system (3) is almost periodic; also suppose that , for all and , . Then if there exist positive constants , and such that and for all , then the two infectious populations and of system (3) are extinct. Here is the globally uniformly attractive nonnegative almost periodic solution of (9) and is the globally uniformly attractive nonnegative almost periodic solution of (11).
6. Numerical Simulation
Numerical verification of the results is necessary for completeness of the analytical study. In this section, we will present some numerical simulations to substantiate and discuss our analytical findings of system (3) by the means of the software Matlab.
In order to testify the validity of our results, in system (3), fix , , , , , , , , . It is easy to verify that (i) and (H2) hold. From (8)–(11), we have Then system (3) becomes periodic with period . We choose , and ; then we have Obviously, it is easy to see that for all from Figure 1. Furthermore, we can get that Hence, from Corollary 21, we get that the disease will be extinct (see Figure 2). Increasing the infective rate to , from Corollary 18, we can get , which means the condition of Corollary 18 is satisfied. The conclusion of Corollary 18 is verified (see Figure 3).
(a)
(b)
(a)
(b)
Parameters
| : | Per capita birth rate of Rattus norvegicus |
| : | Per capita natural death rate of Rattus norvegicus |
| : | Per capita disease induced death rate of Rattus norvegicus |
| : | Per capita contact transmission rate from infected snails to susceptible Rattus norvegicus |
| : | Per capita birth rate of snail host |
| : | Per capita natural death rate of snail host |
| : | Per capita disease induced death rate of snail host |
| : | Per capita contact transmission rate from infected Rattus norvegicus to susceptible snails |
| : | Per capita transition rate from infected and preshedding snails to shedding snails. |
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The research has been supported by the Natural Science Foundation of China (nos. 11261004 and 11561004), the Natural Science Foundation of Jiangxi Province (20151BAB201016), and the Supporting Development for Local Colleges and Universities Foundation of China-Applied Mathematics Innovative Team Building.