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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.

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 [

The mathematical models in schistosomiasis appeared in the 1960s (Macdonald, 1965 [

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 [

The aim of this paper is to study the existence of periodic solution of the system (

The following assumptions apply to the whole paper concerning model (

The constants

The following results will be used in next sections.

If (H) holds, then the solution

Let

By the continuity of solutions, there is a

Suppose that (H) holds. Then the interval of existence of the solution of periodic systems (

By Lemma

If

It follows comparison principle that

Now we give a result of boundedness of solutions.

Suppose that (H) holds. Let

We claim that

One has a

which implies that for,

The solution

By the second equation of (

By (

It is obvious that (

Suppose that (H) holds. We will discuss the existence of

Assume that there is a number

Since the second equation of (

Substitute

Let

Let

Denote by

The mapping

Let

Therefore,

By Lemma

If (H) is satisfied and if there is a positive number

For fixed

The zero solution of (

Take

If

Define

It follows that the zero solution is globally asymptotically stable.

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

If one takes

A periodic solution ((a)

Coexistence of a stable zero solution (solid squares and circles,

Concerning control strategies, many methods have been applied in order to reduce the number of snails, that is,

The graphs (solid lines) and a control strategy (dash line) of

Comparison between uncontrolled and the controlled snails: a periodic solution ((a-b),

In order to study the effects of the delay on the behavior of solutions, a simulation was performed in two cases:

The effects of delays on the behavior of solutions: (1)

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

From the simulations and Theorem

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.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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).