Existence of Periodic Solutions and Stability of Zero Solution of a Mathematical Model of Schistosomiasis

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.


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]).
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 () =  2 /( + 1) (Hoppensteadt and Peskin 1992 [7]) and consider that there is a delay  ≥ 0 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.

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.
Now we give a result of boundedness of solutions.
In fact, if it is false, then there are two cases.
(ii) The solution  0 () is oscillatory with respect to  0 .
Then there a sequence By the second equation of ( 4), we get which is a contradiction.Thus the following inequality is true: By ( 18), one has a  1 > 0 large enough such that () ≤  0 for  ≥  1 .The first equation of (4) gives that which implies that lim sup It is obvious that ( 14) follows ( 18) and (20).The proof is completed.

Existence of Periodic Solution
Suppose that (H) holds.We will discuss the existence of periodic solution of ( 4)- (6).
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  0  ≥  0 [ 0 (1 + ) +  2 ], then (4) admits a positive periodic solution.

Stability of Zero Solution
Theorem 7. The zero solution of (4) is uniformly stable, if the following inequality holds: Proof.Take (, ) = ( 2 +  2 )/2; we get that that − According to the assumption  >  0 and  <  0  0 , there exists  > 0 such that It follows that the zero solution is globally asymptotically stable.

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: in Figure 4(a) with the parameters satisfying  0  <  0 [ 0 (1 + ) +  2 ] for all  > 0 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.