Mean-Square Almost Periodic Solutions for Impulsive Stochastic Host-Macroparasite Equation on Time Scales

Many important human diseases, particularly in tropical and subtropical regions, arise from infection by macroparasites or metazoan organisms. These organisms tend to have much larger generation times and more complex life cycles than microparasites. In life cycles, there are two ormore obligatory host species together with the final host (humans).Macroparasitic infections are generally chronic in form and they are more a cause of morbidity than mortality and tend to be persistent in character in areas where they are endemic. The final hosts of parasites are usually humans (the hosts in which the parasite attains reproductive maturity) and they gain entry to the definitive host as a consequence of developmental changes which normally occur before the organism arrives at its preferred site and attains reproductivematurity. Because of this fact, the host-macroparasite system has been attracting the attention of many researchers (see [1–4]). A time delay, therefore, exists between entry to the definitive host and the point when the parasite begins the production of eggs or larvae for transmission to other hosts (see [5–7]). In [7], by means of a continuation theorem in coincidence degree theory, the authors considered the oscillation and global attractivity of the nonlinear delay host-macroparasite model with periodic coefficients. On the one hand, the theory of impulsive differential equations is now being recognized to not only be richer than the corresponding theory of differential equations without impulses, but also represent a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamical models and neural networks. Since many dynamical processes are characterized by the fact that, at certain moments of time, they undergo abrupt changes of state, with the development of the theory of impulsive differential equations (see [8, 9]), various models of impulsive differential equations have been proposed and studied extensively (see [10–14]). For example, authors of [14] considered the nonlinear impulsive delay host-macroparasite model with periodic coefficients:


Introduction
Many important human diseases, particularly in tropical and subtropical regions, arise from infection by macroparasites or metazoan organisms.These organisms tend to have much larger generation times and more complex life cycles than microparasites.In life cycles, there are two or more obligatory host species together with the final host (humans).Macroparasitic infections are generally chronic in form and they are more a cause of morbidity than mortality and tend to be persistent in character in areas where they are endemic.The final hosts of parasites are usually humans (the hosts in which the parasite attains reproductive maturity) and they gain entry to the definitive host as a consequence of developmental changes which normally occur before the organism arrives at its preferred site and attains reproductive maturity.Because of this fact, the host-macroparasite system has been attracting the attention of many researchers (see [1][2][3][4]).A time delay, therefore, exists between entry to the definitive host and the point when the parasite begins the production of eggs or larvae for transmission to other hosts (see [5][6][7]).In [7], by means of a continuation theorem in coincidence degree theory, the authors considered the oscillation and global attractivity of the nonlinear delay host-macroparasite model with periodic coefficients.
On the one hand, the theory of impulsive differential equations is now being recognized to not only be richer than the corresponding theory of differential equations without impulses, but also represent a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamical models and neural networks.Since many dynamical processes are characterized by the fact that, at certain moments of time, they undergo abrupt changes of state, with the development of the theory of impulsive differential equations (see [8,9]), various models of impulsive differential equations have been proposed and studied extensively (see [10][11][12][13][14]).For example, authors of [14] considered the nonlinear impulsive delay host-macroparasite model with periodic coefficients: where  is a positive integer, () = 1/(), () = ()/ +1 (), and () are -periodic functions.By use of the continuation theorem of coincidence degree, some sufficient 2 Discrete Dynamics in Nature and Society conditions are obtained for the global attractivity and oscillation of positive periodic solutions.
In fact, both continuous and discrete systems are very important in implementation and applications.But it is troublesome to study the dynamical properties for continuous and discrete systems, respectively.Therefore, it is significant to study that on time scales which can unify the continuous and discrete cases.In [15], the author considered the following host-macroparasite equation on time scales: By using the contraction principle and Gronwall-Bellman's inequality on time scale, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions.
On the other hand, as a matter of fact, population systems are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in population models are not absolute constants, and they may fluctuate around some average values.Based on these factors, more and more people began to be concerned about stochastic population systems (see [16][17][18][19][20]).Meanwhile, almost periodicity is universal than periodicity, and the mean-square almost periodicity is important in probability for investigating stochastic processes.To the best of our knowledge, there exist few results for mean-square almost periodic solutions for impulsive stochastic process models with delays; one can see some results in [21][22][23].However, there exists no result on the existence and uniqueness of mean-square almost periodic solutions for impulsive stochastic host-macroparasite equation on time scales.Motivated by the above, we consider the following impulsive stochastic host-macroparasite equation on time scales: where Let (Ω, F, P) be a complete probability space furnished with a complete family of right continuous increasing sub-algebras {F  } ≥0 satisfying F  ⊂ F. () is a standard Brownian motion over (Ω, F, P).Throughout this paper, we assume the following.
Our main purpose of this paper is to study the existence and exponential stability of mean-square almost periodic solutions to (3) by means of the Banach fixed point theorem and Gronwall-Bellman's inequality technique.

Preliminaries
In this section, we shall recall some basic definitions and lemmas which are used in what follows.
A time scale T is an arbitrary nonempty closed subset of the real numbers; the forward and backward jump operators ,  : T → T and the forward graininess  : T → R + are defined, respectively, by  A function  : T → R is right-dense continuous provided it is continuous at right-dense point in T and its leftside limits exist at left-dense points in T. If  is continuous at each right-dense point and each left-dense point, then  is said to be continuous function on T.
For  : T → R and  ∈ T  , we define the delta derivative of (),  Δ (), to be the number (if it exists) with the property that, for a given  > 0, there exists a neighborhood  of  such that for all  ∈ .
If  is continuous, then  is right-dense continuous, and if  is delta differentiable at , then  is continuous at .
Let  be right-dense continuous; if  Δ () = (), then we define the delta integral by Lemma 1 (see [24]).Assume ,  : T → R are delta differentiable at  ∈ T; then, The set of all regressive and rdcontinuous functions  : T → R will be denoted by R = R(T) = R(T, R).We define the set If  ∈ R, then the generalized exponential function   is defined by for all ,  ∈ T, with the cylinder transformation Let ,  : T → R be two regressive functions; we define Then, the generalized exponential function has the following properties.
Definition 8 (see [23]).One can say that a random process () with ( 0 ) =  0 is a solution of the impulsive stochastic dynamic system (19) on [ 0 , ] T if the following conditions hold: (i)  is adapted to the filtration F.
(ii) For all  ∈ [ 0 , ] T , we have almost surely where (, ) is the Cauchy matrix of the following system: Definition 9 (see [23]).One can say  : T →  For convenience, PC rd (T,  2 (R)) denotes the set of all piecewise continuous functions with respect to a sequence {  },  ∈ Z.For any integers  and , denote    =  + −   .
Definition 10 (see [26]).A time scale T is called an almost periodic time scale if Definition 11 (see [10]).For any  > 0, let Γ  ⊂ Π be a set of real numbers and {  } ⊂ T. One can say {   }, ,  ∈ Z, is equipotentially almost periodic on an almost periodic time scale T if, for  ∈ Γ  ⊂ Π, there exists at least one integer  such that Definition 12 (see [23]).Let T be an almost periodic time scale and assume that {  } ⊂ T satisfying the derived sequence {   }, ,  ∈ Z, is equipotentially almost periodic.One can call a stochastic process  ∈ PC rd (T,  2 (R)) mean-square almost periodic if (i) for any  > 0, there is a positive number  = () such that if the points   and   belong to the same interval of continuity and |  −  | < , then E‖(  )−(  )‖ 2 < ; (ii) for any  > 0, there is relative dense set Γ such that if  ∈ Γ, then E‖( + ) − ()‖ 2 < , for all  ∈ T, which satisfy the condition | −   | > ,  ∈ Z.
Definition 14.The equation ( 3) is said to be exponentially stable if, for all  > 0, there exist  = () > 0 and  > 0 such that if ‖  0 − V  0 ‖ ≤ , then, for all  ≥  0 , In order to study (3), we consider the linear system: Now let us consider the equations and their solutions Then, by [10], the Cauchy matrix of the linear equation ( 25) is and the solutions of (25) are in the following form: Similar to the proofs of Lemma 3.1 and Lemma 3.2 in [10], one can easily show the following two lemmas, respectively.Lemma 15.For system (3), let ( 1 )-( 3 ) hold.Then, for each  > 0, there exist  1 > 0, 0 <  1 < , a relative dense set  ⊂ Π of real numbers and Q, such that the following relations are fulfilled:  3) is said to be exponentially stable if, for all  > 0, there exist  = () > 0 and  > 0 such that if ‖  0 − V  0 ‖ ≤ , then, for all  ≥  0 ,

Existence of Almost Periodic Solution
In this section, we will study the existence and exponential stability of mean-square almost periodic solutions of (3) by using the Banach fixed point theorem and Gronwall-Bellman's inequality technique on time scales.
Theorem 18. Assume that ( 1 )-( 3 ) hold and there exist two positive constants ,  such that the following conditions hold.
Remark 19.According to the conditions of Theorem 18, we can find that the uniqueness and exponential stability of the mean-square almost periodic solution for the impulsive stochastic host-macroparasite equation on time scales are independent of the magnitude of delays but are dependent on the magnitude of noise and impulse.(58)

Conclusion
In this paper, we have investigated the host-macroparasite equation with impulsive and stochastic effects on time scales.By employing the Banach fixed point theorem, some stochastic analysis techniques, and Gronwall-Bellman's inequality technique on time scales, we have obtained some sufficient conditions ensuring the existence and exponential stability of mean-square almost periodic solutions for impulsive hostmacroparasite equation on time scales.These sufficient conditions can be easily checked by simple algebraic methods.
An example has been given to demonstrate the effectiveness of the presented results.The methods used in paper can be applied to study square-mean almost periodic problems of many other types of impulsive stochastic models on time scales.

Remark 20 .
It is the first time that the sufficient conditions for the existence and exponential stability of piecewise meansquare almost periodic solutions for the impulsive stochastic host-macroparasite equation on time scales are investigated.The obtained results are essentially new.Without considering impulsive and stochastic effects on (3), then (3) reduce to (2).
mean-square almost periodic in  uniformly for  ∈ R,   (, 0) = 0, and there exists constant   ) The sequences   , ]  are almost periodic in  and the sequences {  ()} are almost periodic in  uniformly for  ∈ PC rd (T,  2 (R)) and there exists constant  such that 2(R) is rd-piecewise continuous with respect to a sequence {  } ⊂ T which satisfies   <  +1 ,  ∈ Z, if () is continuous on [  ,  +1 ) T and rd-continuous on T \ {  }.Furthermore, [  ,  +1 ) T ,  ∈ Z, are called intervals of continuity of the function ().
. Next, we prove that the mapping  is a contraction mapping of  * .In fact, in view of ( 1 )-( 3 ), for any ,  ∈  * , we can easily obtain *