Abstract
A mathematical model on schistosomiasis governed by periodic differential equations with a time delay was studied. By discussing boundedness of the solutions of this model and construction of a monotonic sequence, the existence of positive periodic solution was shown. The conditions under which the model admits a periodic solution and the conditions under which the zero solution is globally stable are given, respectively. Some numerical analyses show the conditional coexistence of locally stable zero solution and periodic solutions and that it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis.
1. Introduction
Schistosomiasis, a disease caused by a wormlike parasite, is one of the most prevalent parasitic diseases in the tropical and subtropical regions of the developing world. In China, despite remarkable achievements in schistosomiasis control over the past five decades, the disease still remains a major public health concern. It mainly prevails in 381 counties (cities, districts) of 12 provinces, autonomous regions, and municipalities along and to the south of the Yangtze River valley and caused a number of above 100 million victims (Chen, 2008 [1]).
The mathematical models in schistosomiasis appeared in the 1960s (Macdonald, 1965 [2], Hairston, 1965 [3]). Since then, a number of mathematical models of the transmission dynamics of schistosomes have been developed (Williams et al., 2002 [4], Liang et al., 2005 [5], and references therein).
Schistosomiasis is a serious infectious and parasitic disease transmitted through the medium of water. Male and female helminthes must mate in a host (e.g., humans, ducks, etc.). Thereafter, some of fertilized eggs leave the host in its feces. Upon contact with fresh water, it hatches and attempts to penetrate a snail. Once a snail is infected, a large number of larvae are produced and swim freely in search of a host for reproduction. It might penetrate the skin of a host or be ingested with water or food grown in the water (Lucas, 1983 [6], Hoppensteadt, and Peskin, 1992 [7]). According to the process above Lucas, 1983 [6], provided a modified version of MacDonald’s model: where (also see Wu, 2005 [8]) is the mean number of worms upon each host and is the number of infected snails. The constants and are death probabilities of worms and snails, and is the number (fixed) of snails. The indicates the ratio of infection caused in final host population by a snail per unit time, while the means the ratio of infection in snail population caused by a worm per unit time. The represents the probability of a snail infected once per time unit. If we suppose that (Hoppensteadt and Peskin 1992 [7]) and consider that there is a delay due to the development period of helminthes, then there is the following system: In fact, as reported by Lu et al., 2005 [9], the number of snails and the death ratio of snails are related to time; for example, which is generated by fitting the data reported in Lu et al., 2005 [9]. Therefore, the system (2) might be modified by It is easy to check that system (4) does not belong to the situations studied in the literature (Tang and Kuang, 1997 [10], Tang and Zhou, 2006 [11], and Fan and Zou, 2004 [12]), though the systems therein are more general.
The aim of this paper is to study the existence of periodic solution of the system (4). The periodic solutions of the models of schistosomiasis relate to the periodic phenomenon in epidemiology and control of schistosomiasis. As an application, we will discuss what conditions enable the zero solution of the model stable. We will also perform numerical simulations, which indicate that the conditions of existence of periodic solution can be further improved. To begin with, we study the boundedness of solutions of model (4). And then we will prove the existence of periodic solutions of this model and discuss the stability of zero solution.
2. Preliminaries
The following assumptions apply to the whole paper concerning model (4).(H)The constants , , and are positive, and , are periodic continuous positive functions with period of :
The following results will be used in next sections.
Lemma 1. If (H) holds, then the solution of periodic system (4) with initial value is positive; that is, , , and for , the interval of existence of .
Proof. Let be the solution of (4) and (6). By the first equation of (4), we have
So, as long as .
By the continuity of solutions, there is a such that for . If we suppose that there exists a such that
where is a positive number. By (7), holds for . By the continuity of solutions again, there is a such that for . Therefore in interval (), there hold and
which implies that for where is the solution of differential equation and . This is a contradiction to (8). Thus the does not exist. The proof is completed.
Lemma 2. Suppose that (H) holds. Then the interval of existence of the solution of periodic systems (4) and (6) is .
Proof. By Lemma 1, the solution of (4) and (6) is positive. Suppose that is the maximal interval of existence of ; that is, or .
If is true, then there is such that
which imply that
It follows comparison principle that
holds, where is the solution of differential equation and . Obviously, (12) contradicts (10) due to in (H), which means that does not hold. Since the function is continuous, one gets that
It completes the proof.
Now we give a result of boundedness of solutions.
Theorem 3. Suppose that (H) holds. Let be a solution of (4) and (6); then there is a number , which is independent of , such that
Proof. We claim that . In fact, if it is false, then there are two cases.(i)One has a such that for . By the second equation of (4),
which implies that for, , the inequality , is true. This is a contradiction.(ii)The solution is oscillatory with respect to . Then there a sequence with as , such that
By the second equation of (4), we get
which is a contradiction. Thus the following inequality is true:
By (18), one has a large enough such that for . The first equation of (4) gives that
which implies that
It is obvious that (14) follows (18) and (20). The proof is completed.
3. Existence of Periodic Solution
Suppose that (H) holds. We will discuss the existence of -periodic solution of (4)–(6).
Lemma 4. Assume that there is a number , such that , where . Then the following differential equations admit a unique positive -periodic solution , satisfying
Proof. Since the second equation of (21)
is a linear equation with respect to and it admits a unique periodic solution denoted by . It is obvious that there holds
and that the following equation,
has a globally stable equilibrium . Therefore, by comparison principle, the following inequality,
is true for , where is the solution of (25) with . Noting that is a periodic function, we obtain
Substitute into the first equation of (21); we get
where
Thus there holds that
The conclusion of the lemma follows (27), (30). It completes the proof.
Let be the set of all continuous functions in with the distance norm ; then is a Banach space.
Let be a continuous -periodic function satisfying , , where . Denote the set of such functions by , clearly, . For any , the following differential equations, admit a unique -periodic solution: where Obviously, is a periodic function. By the proof of Theorem 3, one knows that , and so , .
Denote by the mapping from into defined by (32)–(34); that is, .
Lemma 5. The mapping is monotonic; that is, if , and for , then , for .
Proof. Let and . Substitute into the second equation of (21); one gets
Therefore,
By comparison principle, the solution of the following initial value problem,
for satisfies
Because of the periodicity of , one has for . By (33) we obtain that, for ,
which implies that , for . It completes the proof of Lemma 5.
By Lemma 5, we know that the mapping maps into . Based on the results above, we can give the existence of periodic solution.
Theorem 6. If (H) is satisfied and if there is a positive number , such that , then (4) admits a positive periodic solution.
Proof. For fixed satisfying , consider the following iterative sequence: From Lemmas 4-5, for any positive integer , one has and that the is an increasing sequence. Let defined by (40), . It is obvious that , and so for any we have , where is independent on . We claim that is equicontinuous on . In fact, for arbitrary and any , by (32)-(33), we get that the following equation holds for any , where is determined by (32) with and is given by the second equation of (33) with . By differential mean theorem, there is a between and such that Therefore where Since is a solution of the second equation of (31) with , by the proof of Theorem 3, has a bound of . Consequently , which is independent on . Therefore is equicontinuous on . By Arzela-Ascoli’s theorem, one has a subsequence such that . Obviously, for any , inequality holds. For all , one has as . On the other hand, the cone is normal, and in cone , there holds that if with , and , then (see Guo 2001 [13]). Thus as , it follows . Replace in (32) and (34) by the limit ; we get . Obviously is the periodic solution of (4). It completes the proof.
4. Stability of Zero Solution
Theorem 7. The zero solution of (4) is uniformly stable, if the following inequality holds:
Proof. Take ; we get that if . Because (45) holds, there are two positive numbers and small enough such that hold for all . It follows that . Consequently, if and (45). As , by the stability theorems (Hale and Verduyn Lunel, 1993 [14]), the conclusions of the theorem are true. The proof is completed.
Theorem 8. If and hold, then the zero solution of (4) is globally asymptotically stable.
Proof. Define
Now we calculate and estimate the upper-right derivative of along the solutions of model (4):
According to the assumption and , there exists such that
Integrating both sides of (50) from a sufficiently large number to , we get
which implies that is bounded on and
By Theorem 3, and are bounded on . It is obvious that and are bounded for . Therefore, and are uniformly continuous on . By Barbalat’s lemma (Lemmas and , Gopalsamy [15]), one can conclude that
It follows that the zero solution is globally asymptotically stable.
5. Numerical Simulations
In order to exemplify the results presented above to show some possible behaviors of the solutions of the model, we performed some simulations.
For the following model based on the fourth-order Runge-Kutta’s formula for a ordinary differential system, we applied the following scheme: where the step size is equal to , where ( is about 20–400 ) is the total number of iterative steps. In computation the virtual range of parameters was , , , and ; both of and were given by (3).
If one takes , , and , then the conditions of Theorem 6 are satisfied and a periodic solution follows (Figure 1(a)). If one takes , , and , then the condition (45) is satisfied and it can be found that the zero is uniformly stable (Figure 1(b)). Furthermore, the sufficient conditions given in Theorems 6 and 7 are far from being sharp. In some cases where the initial value is small enough, the solution tends to zero (solid squares and circles in Figure 2, , ), meanwhile in other cases where the initial value is larger both the solution tends to a periodic solution (dots and hollow squares in Figure 2, , ), where , , and . Further simulations indicate that the periodic solution, if it exists, is locally stable.
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Concerning control strategies, many methods have been applied in order to reduce the number of snails, that is, , and at the same time to enlarge the death ratio of snails, . If a control approach was applied such that and (dash lines in Figure 3), we obtain some results shown in Figure 4. A periodic solution is shown in Figures 4(a)-4(b) and its amplitude is reduced by half by the control strategy. A solution tending to zero is shown in Figures 4(c)-4(d) and the time infected snails go extinct is reduced by a quarter by the control strategy. A periodic solution is changed to a solution tending to zero by the control strategy (Figures 4(e)-4(f)). Therefore it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails.
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(b)
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In order to study the effects of the delay on the behavior of solutions, a simulation was performed in two cases: , , and (Figures 5(a)–5(c)) and , , and (Figures 5(d)–5(f)) with initial data of , . It can be seen that the amplitude and period of the periodic solution decrease, but no obvious changes were observed on the periodic solution when the delay is changed from 8 to 1. Furthermore when the delay is changed from 1 to 0.2, the similar changes could not be found.
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6. Discussion and Conclusion
In the preceding sections we modified MacDonald’s models in schistosomiasis. It may be more reasonable in natural conditions that we consider a periodic model and introduce a delay to simulate the development period of helminthes.
The existence of periodic solution shows that the infected snails and infecting worms in the host coexist in a periodic pattern. The stability of the zero solution shows that the population of infected snails and infecting worms in a host will extinct eventually. Although much to our expectation, the condition of existence of periodic solution is easier to be satisfied, rather than that of global stability of zero solution.
It is needed to add that the condition of Theorem 6 can be improved, for example, the periodic solution shown in Figure 4(a) with the parameters satisfying for all rather than the condition of Theorem 6. Also, the condition of Theorem 7 may be improved, for example, the stable zero solution shown in Figure 4(b) with the parameters not satisfying (45).
From the simulations and Theorem 7, while the practical meaning of the parameter values chosen in numerical simulations has not been clear yet, we can see that an increase in and decrease in , would enable (45) and thus lead to the stability of zero solution. In other words, to annihilate worms, it is required to increase the death probability of worms, while reducing the penetration of worms into snails and the disease transmission of snails to final hosts. This implication is in line with other published literature (Williams et al., 2002 [4], Liang et al., 2005 [5]).
In conclusion, we present the conditions under which the model admits a periodic solution and the conditions under which the zero solution is uniformly stable and globally stable, respectively. We show the conditional coexistence of locally stable zero solution and periodic solutions in numerical method and show that it is an effective treatment of simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are very grateful to the referees for careful reading and valuable comments which led to improvements of the original paper. This research was supported in part by Beijing Leading Academic Discipline Project of Beijing Municipal Education Commission (PHR201110506) and Capital Medical University Research Foundation (2013ZR10).