Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 765498, 10 pages http://dx.doi.org/10.1155/2014/765498
Research Article Existence of Periodic Solutions and Stability of Zero Solution of a Mathematical Model of Schistosomiasis Lin Li1,2 and Zhicheng Liu1,2 1 2
School of Biomedical Engineering, Capital Medical University, Beijing 100069, China Beijing Key Laboratory of Fundamental Research on Biomechanics in Clinical Application, Capital Medical University, Beijing 100069, China
Correspondence should be addressed to Zhicheng Liu; zcl
[email protected] Received 29 July 2013; Revised 23 November 2013; Accepted 28 November 2013; Published 13 February 2014 Academic Editor: Yongkun Li Copyright Β© 2014 L. Li and Z. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 MacDonaldβs model: ππ = βππΏ + πΆ (π) π΅ (π β π) , ππ‘ ππ π΄ = βππ + π , ππ‘ π
(1)
where (also see Wu, 2005 [8]) π is the mean number of worms upon each host and π is the number of infected snails. The constants π and πΏ are death probabilities of worms and snails, and π is the number (fixed) of snails. The π΄ indicates the ratio of infection caused in final host population by a snail per unit time, while the π΅ means the ratio of infection in snail population caused by a worm per unit time. The πΆ(π) represents the probability of a snail infected once per time unit. If we suppose that πΆ(π) = π2 /(π + 1) (Hoppensteadt and Peskin 1992 [7]) and consider that there is a delay π β₯ 0
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due to the development period of helminthes, then there is the following system: ππ₯ (π‘) π΄ = βππ₯ (π‘) + π¦ (π‘) , ππ‘ π ππ¦ (π‘) π₯2 (π‘ β π) . = βπΏπ¦ (π‘) + π΅ (π β π¦ (π‘)) ππ‘ 1 + π₯ (π‘ β π)
π₯ (π) = π₯0 (π) > 0,
ππ‘ 2ππ‘ β 2.897 cos 6 6
ππ‘ 2ππ‘ + 4.036 sin β 0.441 sin , 6 6 2ππ‘ ππ‘ β 0.006 cos πΏ (π‘) = 0.048 β 0.030 cos 12 12 β 0.047 sin
which is generated by fitting the data reported in Lu et al., 2005 [9]. Therefore, the system (2) might be modified by ππ₯ (π‘) π΄ π¦ (π‘) , = βππ₯ (π‘) + ππ‘ π (π‘) ππ¦ (π‘) π₯2 (π‘ β π) = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) . ππ‘ 1 + π₯ (π‘ β π)
(4)
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 π΄, π΅, and π are positive, and πΏ(π‘), π(π‘) are periodic continuous positive functions with period of π: πΏ0 = min πΏ (π‘) > 0, π‘β[0,π]
π0 = min π (π‘) > 0. π‘β[0,π]
The following results will be used in next sections.
Proof. Let (π₯(π‘), π¦(π‘)) be the solution of (4) and (6). By the first equation of (4), we have π‘
0
ππ‘ 2ππ‘ + 0.021 sin , 12 12
(5)
(6)
is positive; that is, π₯(π‘) β₯ 0, π¦(π‘) β₯ 0, and π₯(π‘) + π¦(π‘) > 0 for π‘ β πΌ0 , the interval of existence of (π₯(π‘), π¦(π‘)).
π₯ (π‘) = π₯0 (0) πβππ‘ + β« (3)
π β [βπ, 0] ,
π¦ (0) = π¦0 > 0
(2)
In fact, as reported by Lu et al., 2005 [9], the number of snails π and the death ratio of snails πΏ are related to time; for example, π (π‘) = 6.060 + 0.750 cos
Lemma 1. If (H) holds, then the solution (π₯(π‘), π¦(π‘)) of periodic system (4) with initial value
π΄ π¦ (π) πβπ(π‘βπ) ππ. π (π)
(7)
So, π₯(π‘) > 0 as long as π¦(π‘) β₯ 0. By the continuity of solutions, there is a π‘1 > 0 such that π¦(π‘) > 0 for π‘ β [0, π‘1 ). If we suppose that there exists a π‘β > 0 such that π¦ (π‘) > 0,
π‘ β [0, π‘β ) ,
π¦ (π‘β ) = 0,
π‘ β (π‘β , π‘β + π1 ) ,
π¦ (π‘) < 0,
(8)
where π1 is a positive number. By (7), π₯(π‘) > 0 holds for π‘ β [βπ, π‘β ]. By the continuity of solutions again, there is a π2 > 0 such that π₯(π‘) > 0 for π‘ β [π‘β , π‘β + π2 ). Therefore in interval [π‘β , π‘β + π) (π = min(π1 , π2 )), there hold π¦(π‘) < 0 and ππ¦ (π‘) = βπΏ (π‘) π¦ (π‘) ππ‘ + π΅ (π (π‘) β π¦ (π‘))
π₯2 (π‘ β π) > βπΏ (π‘) π¦ (π‘) , 1 + π₯ (π‘ β π) π‘ β [π‘β , π‘β + π) , (9)
which implies that π¦(π‘) β₯ π’(π‘) for π‘ β (π‘β , π‘β + π) where π’(π‘) is the solution of differential equation π’σΈ (π‘) = βπΏ(π‘)π’(π‘) and π’(π‘β ) = 0. This is a contradiction to (8). Thus the π‘β 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 [0, +β). Proof. By Lemma 1, the solution (π₯(π‘), π¦(π‘)) of (4) and (6) is positive. Suppose that [0, π‘β ) is the maximal interval of existence of (π₯(π‘), π¦(π‘)); that is, limπ‘ β π‘β π₯(π‘) = +β or limπ‘ β π‘β π¦(π‘) = +β. If limπ‘ β π‘β π¦(π‘) = +β is true, then there is π‘0β < π‘β such that π¦ (π‘0β ) = π0 = max π (π‘) > 0, π‘β[0,π]
0
π¦ (π‘) > π ,
π‘β
(π‘0β , π‘β ) ,
(10)
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3
which imply that
By the second equation of (4), we get
ππ¦ (π‘) π₯2 (π‘ β π) = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) ππ‘ 1 + π₯ (π‘ β π) < βπΏ (π‘) π¦ (π‘) ,
π‘β
0= (11)
[π‘0β , π‘β ) .
π π¦ (π‘ ) ππ‘ 0 π
= βπΏ (π‘π ) π¦0 (π‘π ) + π΅ (π (π‘π ) β π¦0 (π‘π ))
It follows comparison principle that < βπΏ (π‘π ) π¦0 (π‘π ) < 0,
π‘
π¦ (π‘) β€ π’ (π‘) = π¦ (π‘0β ) exp (β β« πΏ (π ) ππ ) π‘0β
π‘
0
= π exp (β β« πΏ (π ) ππ ) , π‘0β
(π‘0β , π‘β )
holds, where π’(π‘) is the solution of differential equation π’σΈ (π‘) = βπΏ(π‘)π’(π‘) and π’(π‘0β ) = π¦(π‘0β ) = π0 . Obviously, (12) contradicts (10) due to πΏ(π‘) β₯ πΏ0 > 0 in (H), which means that limπ‘ β π‘β π¦(π‘) = +β does not hold. Since the function (π΄/π(π’))π¦(π’)ππ(π’) is continuous, one gets that π‘
lim π₯ (π‘) = limβ [πβππ‘ β«
π‘ β π‘β
π‘βπ‘
0
π΄ π¦ (π’) πππ’ ππ’] < +β. π (π’)
(13)
Theorem 3. Suppose that (H) holds. Let (π₯(π‘), π¦(π‘)) be a solution of (4) and (6); then there is a number π > 0, which is independent of (π₯(π‘), π¦(π‘)), such that lim supπ¦ (π‘) β€ π. π‘ β +β
(14)
Proof. We claim that lim supπ‘ β +β π¦(π‘) β€ π0 = max[0,π] π(π‘). In fact, if it is false, then there are two cases. 0
(i) One has a π0 > 0 such that π¦0 (π‘) > π for π‘ β₯ π0 . By the second equation of (4), ππ¦0 (π‘) π₯2 (π‘ β π) = βπΏ (π‘) π¦0 (π‘) + π΅ (π (π‘) β π¦0 (π‘)) ππ‘ 1 + π₯ (π‘ β π) (15) < βπΏ (π‘) π¦0 (π‘) ,
π‘ β₯ π0 ,
which implies that for, π‘ β₯ π0 , the inequality π¦0 (π‘) < π‘
π¦0 (π0 )πβ β«π πΏ(π )ππ β 0, (π‘ β +β) is true. This is a contradiction.
(ii) The solution π¦0 (π‘) is oscillatory with respect to π0 . Then there a sequence {π‘π } with π‘π β β as π β β, such that π¦0 (π‘π ) > π0 ,
π π¦ (π‘ ) = 0, ππ‘ 0 π
π = 1, 2, . . . .
lim sup π¦ (π‘) β€ π0 = max π (π‘) . [0,π]
π‘ β +β
(18)
By (18), one has a π1 > 0 large enough such that π¦(π‘) β€ π0 for π‘ β₯ π1 . The first equation of (4) gives that ππ₯ (π‘) π΄ π΄ 0 = βππ₯ (π‘) + π¦ (π‘) β€ βππ₯ (π‘) + π ππ‘ π (π‘) π (π‘) (19) π‘ β₯ π1 ,
which implies that
Now we give a result of boundedness of solutions.
π‘ β +β
π = 1, 2, . . . ,
which is a contradiction. Thus the following inequality is true:
π΄π0 β€ βππ₯ (π‘) + , π0
It completes the proof.
lim supπ₯ (π‘) β€ π,
1 + π₯ (π‘π β π)
(17) (12)
π‘β
π₯2 (π‘π β π)
(16)
lim sup π₯ (π‘) β€ π‘ β +β
π΄ 0 π0 . ππ0
(20)
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 πperiodic solution of (4)β(6). Lemma 4. Assume that there is a number πΌ > 0, such that π΄π΅π0 πΌ β₯ ππ0 [πΏ0 (1 + πΌ) + π΅πΌ2 ], where πΏ0 = maxπ‘β[0,π] πΏ(π‘). Then the following differential equations ππ₯ (π‘) π΄ = βππ₯ (π‘) + π¦ (π‘) , ππ‘ π (π‘) ππ¦ (π‘) πΌ2 = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) ππ‘ 1+πΌ
(21)
admit a unique positive π-periodic solution (π₯β (π‘), π¦β (π‘)), satisfying π₯β (π‘) β₯ πΌ,
π¦β (π‘) β₯
π΅π0 πΌ2 , πΏ0 (1 + πΌ) + π΅πΌ2 π‘ β [0, π] .
(22)
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Proof. Since the second equation of (21)
Thus there holds that 2
ππ¦ (π‘) πΌ = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) ππ‘ 1+πΌ = β (πΏ (π‘) +
π΅πΌ2 π΅πΌ2 ) π¦ (π‘) + π (π‘) 1+πΌ 1+πΌ
π₯β (π‘) β₯ (23)
π
πΌ2 β₯ βπΏ π¦ (π‘) + π΅ (π0 β π¦ (π‘)) , 1+πΌ and that the following equation,
(24)
0
ππ’ (π‘) πΌ2 = βπΏ0 π’ (π‘) + π΅ (π0 β π’ (π‘)) , ππ‘ 1+πΌ
π΅π0 πΌ2 , πΏ0 (1 + πΌ) + π΅πΌ2
(π‘ σ³¨β +β) ,
ππ¦ (π‘) π2 (π‘ β π) , = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) ππ‘ 1 + π (π‘ β π)
(26)
is true for π‘ β₯ 0, where π’ (π‘) is the solution of (25) with π’β (0) = π¦β (0). Noting that π¦β (π‘) is a periodic function, we obtain π΅π0 πΌ2 , π¦ (π‘) β₯ 0 πΏ (1 + πΌ) + π΅πΌ2
ππ₯ (π‘) π΄ π¦ (π‘) , = βππ₯ (π‘) + ππ‘ π (π‘)
π‘ β [0, π] .
π‘
π¦ (π‘) = π¦0 exp [β β« (πΏ (π) + 0
(27)
0
β₯ π₯β (0) πβππ‘ +
π΄ β π¦ (π) πβπ(π‘βπ) ππ π (π)
π΅π0 πΌ2 π΄ 1 = 0 0 π πΏ (1 + πΌ) + π΅πΌ2 π
(28)
π‘
π΄ π¦ (π) πβπ(π‘βπ) ππ’, π (π)
(33)
where π
π
0
π
π¦0 = ( β« exp [β β« (πΏ (π) +
π₯β (0)
Γ π
β«0 (π΄π¦β (π) πβπ(πβπ) /π (π)) ππ (π΄/π0 ) (π΅π0 πΌ2 / (πΏ0 (1 + πΌ) + π΅πΌ2 )) (1/π) (1 β πβππ ) 1 β πβππ
π΅π2 (π β π) ) ππ] 1 + β (π β π)
π΅π2 (π β π) π (π) ππ) 1 + π (π β π) β1
π΅π2 (π β π) ) ππ]) . Γ (1 β exp [β β« (πΏ (π) + 1 + π (π β π) 0 (34) π
1 β πβππ
π΅π0 πΌ2 1 π΄ . 0 0 π πΏ (1 + πΌ) + π΅πΌ2 π
(32)
π΅π2 (π β π) π (π) ππ, 1 + π (π β π)
π΄ 1 π¦ (π) πβπ(πβπ) ππ, π₯ = β« 1 β πβππ 0 π (π)
where
=
π΅π2 (π β π) ) ππ] 1 + π (π β π)
π
0
π΅π0 πΌ2 1 βππ‘ π΄ + (π₯ (0) β 0 0 )π , 2 π πΏ (1 + πΌ) + π΅πΌ π
β₯
π
0
β
=
0
π₯ (π‘) = π₯0 πβππ‘ + β«
2
π΅π0 πΌ 1 π΄ (1 β πβππ‘ ) π0 πΏ0 (1 + πΌ) + π΅πΌ2 π
π‘
Γ
Substitute π¦ = π¦ (π‘) into the first equation of (21); we get π‘
π‘
π΅π2 (π β π) ) ππ] 1 + π (π β π)
+ β« exp [β β« (πΏ (π) +
β
π₯β (π‘) = π₯β (0) πβππ‘ + β«
(31)
admit a unique π-periodic solution:
β
β
(30)
Let πΆπ (π
) = {π₯ β πΆ(π
) : π₯(π‘ + π) = π₯(π‘), π‘ β π
} be the set of all continuous functions in [0, π] with the distance norm βπ(π‘) β ](π‘)β = maxπ‘β[0,π] |π(π‘) β ](π‘)|; then πΆπ (π
) is a Banach space. Let π(π‘) be a continuous π-periodic function satisfying πΌ β€ π(π‘) β€ π½, π‘ β [0, π], where π½ = π΄π0 /π0 π. Denote the set of such functions by π·, clearly, π· β πΆ[0, π]. For any π β π·, the following differential equations,
(25)
has a globally stable equilibrium π’β = π΅π0 πΌ2 /(πΏ0 (1+πΌ)+π΅πΌ2 ). Therefore, by comparison principle, the following inequality, π¦β (π‘) β₯ π’β (π‘) σ³¨β
π‘ β [0, π] .
The conclusion of the lemma follows (27), (30). It completes the proof.
is a linear equation with respect to π¦(π‘) and β«0 (π΅πΌ2 /(1 + πΌ) + πΏ(π‘))ππ‘ > 0, it admits a unique periodic solution denoted by π¦β (π‘). It is obvious that there holds ππ¦ (π‘) πΌ2 = βπΏ (π‘) π¦ (π‘) + π΅ (π (π‘) β π¦ (π‘)) ππ‘ 1+πΌ
π΅π0 πΌ2 1 π΄ β₯πΌ 0 0 2 π πΏ (1 + πΌ) + π΅πΌ π
Obviously, π₯(π‘) β πΆπ (π
) is a periodic function. By the proof of Theorem 3, one knows that lim supπ‘ β +β π₯(π‘) β€ π΄ 0 π0 /ππ0 , (29)
and so π₯(π‘) β€ π½, π‘ β [0, π].
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Denote by π the mapping from π· into πΆπ (π
) defined by (32)β(34); that is, π : π· β πΆπ (π
), π₯ = ππ. Lemma 5. The mapping π is monotonic; that is, if π1 , π2 β π· and π1 (π‘) β€ π2 (π‘) for π‘ β [0, π], then (ππ2 )(π‘) β₯ (ππ1 )(π‘), for π‘ β [0, π]. Proof. Let π1 , π2 β π· and π1 β€ π2 . Substitute π1 , π2 into the second equation of (21); one gets ππ¦π (π‘) ππ‘
= βπΏ (π‘) π¦π (π‘) + π΅ (π (π‘) β π¦π (π‘)) Γ
ππ2 (π‘ β π) 1 + ππ (π‘ β π)
(35) ,
π = 1, 2.
which implies that (ππ2 )(π‘) β₯ (ππ1 )(π‘), for π‘ β [0, π]. It completes the proof of Lemma 5. By Lemma 5, we know that the mapping π maps π· into π·. 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 πΌ, such that π΄π΅π0 πΌ β₯ ππ0 [πΏ0 (1 + πΌ) + π΅πΌ2 ], then (4) admits a positive periodic solution. Proof. For fixed πΌ satisfying π΄π΅π0 πΌ β₯ ππ0 [πΏ0 (1 + πΌ) + π΅πΌ2 ], consider the following iterative sequence: π₯π = ππ₯πβ1 ,
Therefore, π (π¦2 (π‘) β π¦1 (π‘)) ππ‘ = βπΏ (π‘) (π¦2 (π‘) β π¦1 (π‘)) + π΅ (π (π‘) β π¦2 (π‘)) β π΅ (π (π‘) β π¦1 (π‘))
π22 (π‘ β π) 1 + π2 (π‘ β π)
π12 (π‘ β π) 1 + π1 (π‘ β π)
β₯ β (π¦2 (π‘) β π¦1 (π‘)) (πΏ (π‘) + π΅
π12 (π‘ β π) ). 1 + π1 (π‘ β π)
By comparison principle, the solution π’(π‘) of the following initial value problem,
(37)
π’ (0) = π¦2 (0) β π¦1 (0) ,
π2 (π β π) + πΏ (π)] ππ) σ³¨β 0, = π’ (0) exp (β β« [ 1 0 1 + π1 (π β π) π‘
(π‘ σ³¨β +β) . (38) Because of the periodicity of π¦2 β π¦1 , one has π¦2 (π‘) β₯ π¦1 (π‘) for π‘ β [0, π]. By (33) we obtain that, for π‘ β [0, π], π₯2 (π‘) β π₯1 (π‘)
π‘2
= (πβππ‘2 β πβππ‘1 ) π’0 + β« πβπ(π‘2 βπ ) π‘1
π΄ π¦ (π) ππ, π (π) π’
(41)
where π¦π’ is determined by (32) with β(β
) = π’πβ1 (β
) and π’0 is given by the second equation of (33) with π¦(β
) = π¦π’ (β
). By differential mean theorem, there is a π between π‘1 and π‘2 such that
+
π π‘ βπ(π‘βπ ) π΄ [β« π π¦ (π ) ππ ] (π‘2 β π‘1 ) ππ‘ 0 π (π ) π’ π‘=π
= βππβππ π’0 (π‘2 β π‘1 ) +[
π π΄ π΄ π¦π’ (π) + β« πβπ(π‘βπ ) π¦ (π ) ππ ] (π‘2 β π‘1 ) . π (π) π (π ) π’ 0 (42)
Therefore
= (π₯20 β π₯10 ) πβππ‘ + β«
π‘
0
π΄ (π¦ (π) β π¦1 (π)) πβπ(π‘βπ) ππ’ π (π) 2
π π΄ πβππ‘ = (π¦2 (π) β π¦1 (π)) πβπ(πβπ) ππ β« βππ 1βπ 0 π (π)
0
From Lemmas 4-5, for any positive integer π, one has π₯π β π· and that the {π₯π } is an increasing sequence. Let π·0 = {π₯π (π‘) | defined by (40), π = 1, 2, . . .}. It is obvious that π·0 β π· β πΆ[0, π], and so for any π, we have |π₯π (π‘)| β€ π½, where π½ is independent on π·0 . We claim that π·0 = {π₯π } is equicontinuous on π·. In fact, for arbitrary π > 0 and any π’π β π·0 , by (32)-(33), we get that the following equation holds for any π‘1 , π‘2 β [0, π],
= βππβππ π’0 (π‘2 β π‘1 )
π¦2 (π‘) β π¦1 (π‘) β₯ π’ (π‘)
π‘
(40)
π’π (π‘2 ) β π’π (π‘1 )
for π‘ β₯ 0 satisfies
+β«
(π = 1, 2, . . .) .
π’π (π‘2 ) β π’π (π‘1 ) (36)
π2 (π‘ β π) ππ’ (π‘) π’ (π‘) , = βπΏ (π‘) π’ (π‘) β π΅ 1 ππ‘ 1 + π1 (π‘ β π)
π₯0 = πΌ
π΄ (π¦ (π) β π¦1 (π)) πβπ(π‘βπ) ππ’ β₯ 0, π (π) 2
(39)
σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨π’π (π‘2 ) β π’π (π‘1 )σ΅¨σ΅¨σ΅¨ π΄ σ΅¨ σ΅¨ σ΅¨ σ΅¨ β€ σ΅¨σ΅¨σ΅¨βππβππ σ΅¨σ΅¨σ΅¨ π’0 σ΅¨σ΅¨σ΅¨(π‘2 β π‘1 )σ΅¨σ΅¨σ΅¨ + [ π¦ (π)] π (π) π’ σ΅¨σ΅¨ π σ΅¨σ΅¨ π΄ σ΅¨ σ΅¨ σ΅¨ σ΅¨ + β« σ΅¨σ΅¨σ΅¨ππβπ(πβπ ) π¦π’ (π )σ΅¨σ΅¨σ΅¨ ππ σ΅¨σ΅¨σ΅¨π‘2 β π‘1 σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ π (π ) 0 σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨ 0 β€ [ππ’ + π1 (1 + ππ)] σ΅¨σ΅¨π‘2 β π‘1 σ΅¨σ΅¨σ΅¨ ,
(43)
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0.5
10
0.4
8
0.3
6 0.2
4
0.1
2 0
0
20
40
60
0.0
80
0
20
40
(a)
60
80
(b)
Figure 1: A periodic solution ((a) π΄ = 1, π΅ = 2.4, and π = 0.2 satisfy the conditions of Theorem 6) and a solution tending to zero ((b) π΄ = 0.2, π΅ = 0.02, and π = 1.5 satisfy the conditions of Theorem 7), where square is π₯(π‘), line is π¦(π‘), π¦0 = 0, and π₯0 (π) = cos(ππ/10).
Guo 2001 [13]). Thus as π β +β, it follows 0 β€ π₯β β π’π β€ π₯β β π’ππ β 0. Replace π in (32) and (34) by the limit π₯β ; we get π¦β . Obviously (π₯β , π¦β ) is the periodic solution of (4). It completes the proof.
8
6 4
4. Stability of Zero Solution Theorem 7. The zero solution of (4) is uniformly stable, if the following inequality holds:
2 0
0
20
40
60
max [(
80
π‘β[0,π]
Figure 2: Coexistence of a stable zero solution (solid squares and circles, π¦0 = 0.2, π₯0 (π) = 0.2) and a periodic solutions (dots and hollow squares, π¦0 = 2, π₯0 (π) = 2) where π΄ = 0.6, π΅ = 0.4, and π = 0.8.
ππ | ππ‘ (4) π΄ π¦) + π¦ [ β πΏ (π‘) π¦ + π΅ (π (π‘) β π¦) π (π‘)
σ΅¨σ΅¨ π΄ σ΅¨σ΅¨ σ΅¨ σ΅¨ π¦π’ (π‘)σ΅¨σ΅¨σ΅¨ , π1 = max max σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ π‘β[0,π] σ΅¨σ΅¨ π (π‘) π₯0 = β€
π π΄ 1 π¦ (π ) ππ πβπ(πβπ ) β« 1 β πβππ 0 π (π )
Γ (44)
π1 π . 1 β πβππ
Since π¦π’ (π‘) is a solution of the second equation of (31) with β(β
) = π’πβ1 (β
), by the proof of Theorem 3, π¦π’ (π‘) has a bound of π0 . Consequently π1 β€ π΄π0 /π0 , which is independent on π’π . Therefore π·0 is equicontinuous on π·. By Arzela-Ascoliβs theorem, one has a subsequence {π’ππ } β π·0 such that π’ππ β π₯β . Obviously, for any π = 1, 2, . . ., inequality π₯β β₯ π’π holds. For all π‘ β [0, π], one has 0 β€ π₯β β π’π β€ π₯β β π’ππ as π > ππ . On the other hand, the cone π = {π | π β πΆπ (π
), π(π‘) β₯ 0} is normal, and in cone π, there holds that if π₯π , π¦π , π§π β π with π₯π β€ π§π β€ π¦π , π₯π β π₯, and π¦π β π₯, then π§π β π₯ (see
(45)
Proof. Take π(π₯, π¦) = (π₯2 + π¦2 )/2; we get that
= π₯ (βππ₯ +
where
2 π΄ + π΅π (π‘)) β 4ππΏ (π‘)] < 0. π (π‘)
= βππ₯2 β πΏ (π‘) π¦2 + β π΅π¦2
π₯2 (π‘ β π) ] 1 + π₯ (π‘ β π)
π₯2 (π‘ β π) π΄ π₯π¦ + π΅π (π‘) π¦ π (π‘) 1 + π₯ (π‘ β π)
π₯2 (π‘ β π) 1 + π₯ (π‘ β π)
β€ βππ₯2 β πΏ (π‘) π¦2 +
π΄ π₯π¦ + π΅π (π‘) π¦π₯ (π‘ β π) π (π‘)
β€ βππ₯2 β πΏ (π‘) π¦2 + [
π΄ + π΅π (π‘)] π₯π¦, π (π‘)
(46)
if π₯(π‘) β₯ π₯(π‘ β π). Because (45) holds, there are two positive numbers πΎ1 and πΎ2 small enough such that (π΄/π(π‘) + π΅π(π‘))2 β 4ππΏ(π‘)+4πΎ1 πΏ(π‘)+4πΎ2 (πβπΌ) < 0 hold for all π‘ β [0, π]. It follows
Journal of Applied Mathematics
7
0.12
12
0.10
10
0.08
8
0.06
6
0.04
4
0.02
2
0.00
0
5
10
15
0
20
0
5
10
(a)
15
20
(b)
Figure 3: The graphs (solid lines) and a control strategy (dash line) of πΏ(π‘) (a) and π(π‘) (b).
that βππ₯2 β (π΄/π(π‘) + π΅π(π‘))π₯π¦ β πΏ(π‘)π¦2 β€ β(πΎ1 π₯2 + πΎ2 π¦2 ). Consequently,
According to the assumption π > π΅π0 and π΄ < πΏ0 π0 , there exists π > 0 such that
ππ | β€ β (πΎ1 π₯2 + πΎ2 π¦2 ) ππ‘ (4)
σ΅¨ σ΅¨ π·+ π (π‘) β€ βπ (|π₯ (π‘)| + σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨) .
(47)
if π₯(π‘) β₯ π₯(π‘ β π) and (45). As π(π₯, π¦) = (π₯2 + π¦2 )/2, by the stability theorems (Hale and Verduyn Lunel, 1993 [14]), the conclusions of the theorem are true. The proof is completed.
Integrating both sides of (50) from a sufficiently large number πβ to π‘, we get π‘
σ΅¨ σ΅¨ π (π‘) + π β« [|π₯ (π )| + σ΅¨σ΅¨σ΅¨π¦ (π )σ΅¨σ΅¨σ΅¨] ππ β€ π (πβ ) < +β, πβ
Theorem 8. If π > π΅π0 and π΄ < πΏ0 π0 hold, then the zero solution of (4) is globally asymptotically stable.
σ΅¨ σ΅¨ π (π‘) = |π₯ (π‘)| + σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ + π΅π0 β«
π‘
π‘βπ
|π₯ (π )| ππ .
(48)
Now we calculate and estimate the upper-right derivative of π(π‘) along the solutions of model (4): π·+ π (π‘) = π₯Μ (π‘) sgn π₯ (π‘) + π¦Μ (π‘) sgn π¦ (π‘) + π΅π0 |π₯ (π‘)| β π΅π0 |π₯ (π‘ β π)|
β«
+β
πβ
|π₯ (π )| ππ < +β,
β«
(52)
By Theorem 3, |π₯(π‘)| and |π¦(π‘)| are bounded on [πβ , +β). It is Μ and π¦(π‘) Μ are bounded for π‘ β₯ πβ . Therefore, obvious that π₯(π‘) |π₯(π‘)| and |π¦(π‘)| are uniformly continuous on [πβ , +β). By Barbalatβs lemma (Lemmas 1.2.2 and 1.2.3, Gopalsamy [15]), one can conclude that σ΅¨ σ΅¨ lim σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ = 0.
(53)
5. Numerical Simulations
+ π΅π0 |π₯ (π‘)| β π΅π0 |π₯ (π‘ β π)| π΄ σ΅¨σ΅¨ σ΅¨ σ΅¨ σ΅¨ 0 σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ β πΏ (π‘) σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ + π΅π |π₯ (π‘ β π)| π0 σ΅¨
2 σ΅¨ π₯ (π‘ β π) σ΅¨ β π΅ σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ + π΅π0 |π₯ (π‘)| β π΅π0 |π₯ (π‘ β π)| 1 + π₯ (π‘ β π)
π΄ σ΅¨ σ΅¨ β πΏ0 ) σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ . π0
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π¦ (π )σ΅¨σ΅¨σ΅¨ ππ < +β.
It follows that the zero solution is globally asymptotically stable.
π₯2 (π‘ β π) sgn π¦ (π‘) 1 + π₯ (π‘ β π)
β€ β (π β π΅π0 ) |π₯ (π‘)| + (
+β
πβ
lim |π₯ (π‘)| = 0,
π΄ σ΅¨ σ΅¨ π¦ (π‘) sgn π₯ (π‘) β πΏ (π‘) σ΅¨σ΅¨σ΅¨π¦ (π‘)σ΅¨σ΅¨σ΅¨ π (π‘)
+ π΅ (π (π‘) β π¦ (π‘))
β€ βπ |π₯ (π‘)| +
(51)
(πβ ) for π‘ > πβ , which implies that π(π‘) is bounded on [πβ , +β) and
Proof. Define
= βπ |π₯ (π‘)| +
(50)
(49)
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 ππ₯ = π (π‘, π₯ (π‘) , π¦ (π‘)) , ππ‘ ππ¦ = π (π‘, π₯ (π‘) , π¦ (π‘) , π₯ (π‘ β π)) , ππ‘ π¦ (0) = π¦0 ,
π₯ (π ) = π₯0 (π ) ,
π β [βπ, 0] ,
(54)
8
Journal of Applied Mathematics 12
12
10
10
8
8
6
6
4
4
2
2
0
0
20
40
60
80
0
0
20
(a)
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5 0
20
40
60
80
0.0
0
20
(c)
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5 0
20
40
80
40
60
80
60
80
(d)
3.0
0.0
60
(b)
3.0
0.0
40
60
80
(e)
0.0
0
20
40 (f)
Figure 4: Comparison between uncontrolled and the controlled snails: a periodic solution ((a-b), π΄ = 1, π΅ = 2.4, and π = 2.5), a solution tending to zero ((c-d), π΄ = 2, π΅ = 0.2, and π = 3.8), and a changed case ((e-f), π΄ = 2, π΅ = 0.2, and π = 2.5), where square is π₯(π‘), dot is π¦(π‘), π¦0 = 0, and π₯0 (π) = cos(ππ/10).
based on the fourth-order Runge-Kuttaβs formula for a ordinary differential system, we applied the following scheme:
π₯π+1 = π₯π + π¦π+1
β (πΎ + 2πΎ2 + 2πΎ3 + πΎ4 ) , 6 1
β = π¦π + (π1 + 2π2 + 2π3 + π4 ) , 6 πΎ1 = π (π‘π , π₯π , π¦π ) ,
π1 = π (π‘π , π₯π , π¦π , π₯πβπ ) ,
π=
π , β
β β β πΎ2 = π (π‘π + , π₯π + πΎ1 , π¦π + π1 ) , 2 2 2 β β β π2 = π (π‘π + , π₯π + πΎ1 , π¦π + π1 , π₯πβπ ) , 2 2 2 β β β πΎ3 = π (π‘π + , π₯π + πΎ2 , π¦π + π2 ) , 2 2 2
Journal of Applied Mathematics
9
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0
0
20
40
60
80
0
20
(a)
40
60
80
0
12
12
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10
8
8
8
6
6
6
4
4
4
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2
2
0
0
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0
20
(d)
40 (e)
40
60
80
60
80
(c)
12
0
0
(b)
60
80
0
0
20
40 (f)
Figure 5: The effects of delays on the behavior of solutions: (1) π΄ = 0.4, π΅ = 0.98, and π = 0.5 (aβc) and (2) π΄ = 1, π΅ = 2.4, and π = 2.5 (dβf) with initial data of π¦0 = 0, π₯0 (π) = cos(ππ/10).
β β β π3 = π (π‘π + , π₯π + πΎ2 , π¦π + π2 , π₯πβπ ) , 2 2 2 πΎ4 = π (π‘π + β, π₯π + βπΎ3 , π¦π + βπ3 ) , π1 = π (π‘π + β, π₯π + βπΎ3 , π¦π + βπ3 , π₯πβπ ) , (55) where the step size β is equal to π/π1 , where ππ1 (π is about 20β400 ) is the total number of iterative steps. In computation the virtual range of parameters was π΄ β [0.1, 2], π΅ β [0.02, 3], π β [0.5, 4], and π β [0, 8]; both of π(π‘) and πΏ(π‘) were given by (3). If one takes π΄ = 1, π΅ = 2.4, and π = 0.2, then the conditions of Theorem 6 are satisfied and a periodic solution follows (Figure 1(a)). If one takes π΄ = 0.2, π΅ = 0.02, and π = 1.5, 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, π¦0 = 0.2, π₯0 (π) = 0.2), 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, π¦0 = 2, π₯0 (π) = 2), where π΄ = 0.6, π΅ = 0.4, and π = 0.8. Further simulations indicate that the periodic solution, if it exists, is locally stable. Concerning control strategies, many methods have been applied in order to reduce the number of snails, that is, π(π‘), and at the same time to enlarge the death ratio of snails, πΏ(π‘). If a control approach was applied such that π(π‘) β€ 6 and πΏ(π‘) β₯ 0.04 (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. In order to study the effects of the delay on the behavior of solutions, a simulation was performed in two cases: (1) π΄ = 0.4, π΅ = 0.98, and π = 0.5 (Figures 5(a)β5(c)) and (2) π΄ = 1, π΅ = 2.4, and π = 2.5 (Figures 5(d)β5(f)) with initial data of π¦0 = 0, π₯0 (π) = cos(ππ/10). It can be seen that the amplitude and period of the periodic solution π¦(π‘) decrease, but no obvious changes were observed on the periodic solution π₯(π‘) 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.
6. Discussion and Conclusion In the preceding sections we modified MacDonaldβs 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
10 in Figure 4(a) with the parameters satisfying π΄π΅π0 πΌ < ππ0 [πΏ0 (1 + πΌ) + π΅πΌ2 ] for all πΌ > 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 π and decrease in π΄, π΅ 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).
References [1] W. Chen, Mathematical models of schistosomiasis epidemic and the simulation of applications in practical [Ph.D. Dissertation], Fudan University, Shanghai, China, 2008 (Chinese). [2] G. Macdonald, βThe dynamics of helminth infections, with special reference to schistosomes,β Transactions of the Royal Society of Tropical Medicine and Hygiene, vol. 59, no. 5, pp. 489β 506, 1965. [3] N. G. Hairston, βOn the mathematical analysis of schistosome populations,β Bulletin of the World Health Organization, vol. 33, pp. 45β62, 1965. [4] G. M. Williams, A. C. Sleigh, Y. Li et al., βMathematical modelling of schistosomiasis japonica: comparison of control strategies in the Peopleβs Republic of China,β Acta Tropica, vol. 82, no. 2, pp. 253β262, 2002. [5] S. Liang, R. C. Spear, E. Seto, A. Hubbard, and D. Qiu, βA multigroup model of Schistosoma japonicum transmission dynamics and control: model calibration and control prediction,β Tropical Medicine and International Health, vol. 10, no. 3, pp. 263β278, 2005.
Journal of Applied Mathematics [6] W. F. Lucas, Modules in Applied Mathematics: Life Science Models, vol. 3, Springer, New York, NY, USA, 1983. [7] F. C. Hoppensteadt and C. S. Peskin, Mathematics in Medicine and the Life Science, Springer, New York, NY, USA, 1992. [8] K. Wu, βMathematical model and transmission dynamics of schistosomiasis and its application,β China Tropical Medicine, vol. 5, no. 4, pp. 837β844, 2005 (Chinese). [9] D. Lu, Q. Jiang, T. Wang et al., βStudy on the ecology of snails in Chengcun reservoir irrigation area of Jiangxian county,β Chininese Journal of Parasit Diseases Control, vol. 18, no. 1, pp. 52β55, 2005 (Chinese). [10] B. R. Tang and Y. Kuang, βExistence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems,β The Tohoku Mathematical Journal, vol. 49, no. 2, pp. 217β239, 1997. [11] X. H. Tang and Y. G. Zhou, βPeriodic solutions of a class of nonlinear functional differential equations and global attractivity,β Acta Mathematica Sinica, vol. 49, no. 4, pp. 899β908, 2006 (Chinese). [12] M. Fan and X. Zou, βGlobal asymptotic stability of a class of nonautonomous integro-differential systems and applications,β Nonlinear Analysis, vol. 57, no. 1, pp. 111β135, 2004. [13] D. Guo, Nonlinear Functional Analysis, Shandong Science and technology Publishing House, Jinan, China, 2001 (Chinese). [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to FunctionalDifferential Equations, vol. 99, Springer, New York, NY, USA, 1993. [15] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
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