Dynamical Analysis of a Parasite-Host Model within Fluctuating Environment

A parasite-host model within fluctuating environment is proposed. Firstly, the positivity and boundedness of solutions of the model within deterministic environment are discussed, and, then, the asymptotical stability and global stability of equilibria of deterministic model are investigated. Secondly, we show that the stochastic model has a unique global positive solution; furthermore, we show that the stochastic model has a stationary distribution under certain conditions. Finally, we give some numerical simulations to illustrate our analytical results.


Introduction
Mathematical model plays an ever more important role in the study of epidemiology, which provides understanding of the underlying mechanisms that influence the spread of disease, and, in the process, it suggests control strategies.Classical epidemic model has been extended in many ways to understand mechanisms of disease transmissions; for example, to understand how parasites regulate host populations is an interesting topic in the study of infectious diseases.
It is now widely believed that diseases and parasites were responsible for a number of extinctions on islands and on large land masses (see [1] and the references therein).As a result, ecologists acknowledge the importance of disease and parasites in the dynamics of population.Recently, theory on the effects of parasites on host population dynamics has received much attention, such as how the parasite induced reduction of host fecundity and survival rates changes the host population dynamics and how such dynamics are applied to predict threats to biodiversity in general and endangered species in particular [2][3][4].In order to understand how six microparasites regulate Daphnia populations and drive the populations to extinction, Ebert et al. [5] proposed the following epidemiological microparasite model: where  and  are the densities of uninfected and infected hosts, respectively;  > 0 is the maximum per capita birth rate of uninfected hosts;  > 0 is the relative fecundity of an infected host;  > 0 measures the per capita densitydependent reduction in birth rate;  > 0 is the parasiteindependent host background mortality;  > 0 is the infection rate constant; and  > 0 is the parasite-induced excess death rate.Model (1) uses the classical mass action incidence and has the usual asymptotic behavior.However, this model is deterministic and does consider the effect of environmental noise.As is known to us, real life is full of randomness and stochasticity.So it is important to know whether or not the long-time behavior of the solution for deterministic dynamics system can be changed by stochastic perturbations.In this paper, following the idea of [6][7][8], we introduce random noise to model (1) as follows:
The rest of the paper is organized as follows.In Section 2, we show the positivity and the boundedness of solutions of deterministic model (1) with positive initial condition; the local and global stability of the equilibria of system ( 1) is also investigated in this section.In Section 3, we show that the stochastic model ( 2) has a unique global positive solution; furthermore, we show that the stochastic model (2) has a stationary distribution under certain conditions.In Section 4, we will give some numerical simulations to support the theoretical prediction.In Section 5, a brief discussion is given.

Deterministic Model
In this section, we first discuss some basic dynamical properties of the deterministic model (1), which is subjected to positive initial conditions: (3)

Positivity and Boundedness.
In this subsection, we study the positivity and boundedness of solutions of system (1) with initial condition (3).Using the analysis methods in [1], we can have the following results.Lemma 1.Let ((), ()) be the solution of system (1) with initial condition (3).Then ((), ()) is positive and ultimately bounded for all  ⩾ 0.

Mathematical Problems in Engineering
Based on the above analysis, we have the following.Theorem 4. For system (1), we have the following results:

Global Positive Solution.
In this subsection we show the solution of system ( 2) is global and nonnegative.As we have known, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [10].However, the coefficients of (2) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of ( 2) may explode at a finite time [10].In this subsection, using the Lyapunov analysis method (mentioned in [10]), we will show the solution of ( 2) is positive and global.
First, we prove () and () do not explode to infinity in a finite time.Let  0 > 0 be sufficiently large for (0) ∈ [1/ 0 ,  0 ] and (0) ∈ [1/ 0 ,  0 ].For each integer  ⩾  0 , define the stopping time where throughout this paper we set inf 0 = ∞ (0 denotes the empty set).Clearly,   is increasing as s., then   = ∞ and ((), ()) ∈ R 2 + a.s.for all  ⩾ 0. In other words, to complete the proof it is required to show that  ∞ = ∞ a.s.If this statement is false, then there are a pair of constants  > 0 and  ∈ (0, 1) such that Hence, there is an integer  1 >  0 such that and the nonnegativity of this function can be seen from Let  ⩾  0 and  > 0 be arbitrary.Applying Itô's formula, we obtain where  : Thus,  (, ) We can now integrate both sides of (28) from 0 to   ∧  and then take the expectations It follows from (30) that where 1 Ω  is the indicator function of Ω  .Letting  → ∞, we have and it is a contradiction; then we must have  ∞ = ∞.Therefore, it implies () and () will not explode in a finite time with probability one.

Stochastic Persistence.
In this subsection, we are intended to prove the stochastic persistence of the model system (2) under certain parametric restrictions.Stochastic persistence means that if we start from a positive initial condition, that is, from an interior point of the first quadrant, then solution trajectories of the stochastic model will always remain within the interior of the first quadrant and remain bounded at all future time.There are several concepts of stochastic persistence [11]; here we use the notion of stochastic persistence in mean.Before proving the main results of this section we define stochastic persistence in mean.Definition 6.The population of () is said to be strongly persistent in the mean if ⟨()⟩ * > 0, where The proof of strong persistence result for the stochastic model ( 2) is based upon the following lemma (see Lemma 4 in [11]).Integrating both sides from 0 to  and dividing by , we have ln [( () +  ()) / ( (0 From Lemma 7, we have the following inequality: It follows that ⟨(() + ())⟩ * = 0, whenever  −  +  2 1 /2 +  2  2 /2 ⩽ 0.

Existence of Stationary Distribution.
In this section, we prove the existence of stationary distribution of prey and predator populations.For this purpose we find the stationary distribution for solutions of system (2), which in turn implies the stability in stochastic sense.Before proving the main theorem related to the stationary distribution we state a useful lemma from [12] which will be useful to prove the theorem.
Let () be a homogeneous Markov process defined in the -dimensional Euclidean space, denoted by   , and described by the following system of stochastic differential equation: We assume there exists a bounded domain  ∈   with regular boundary Γ, having the following properties: (P1) In the domain  and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix  3 () is bounded away from zero.(P2) If  ∈   \, the mean time  at which a path emerging from  reaches the set  is finite, and sup ∈    < ∞ for every compact subset  ⊂   .(46) Applying Itô's formula, we can calculate Therefore where  lies in Int(R 2 + ).We can take  to be a neighborhood of the hyperbola with  ⊆  2 = Int(R 2 + ), where  is the compact closure of .So for  ∈  \  2 ,  < 0, which implies condition (P2) in Lemma 10 is satisfied.
Furthermore, there is for all (, ) ∈ ,  ∈ R 2 , which indicates that condition (P1) in Lemma 10 is also satisfied.Hence, we conclude that stochastic system (2) has a stationary distribution (⋅).

Numerical Simulation Results
In this section we provide numerical simulation results to substantiate the analytical findings for the deterministic and stochastic model reported in the previous sections.To find the approximate strong solution of system (2) in Itô sense with positive initial condition we use the Milstein's method having strong order of convergence  = 1 [13].This scheme is obtained from Euler-Maruyama scheme by incorporating a correction term for stochastic increment.
Example 12.We first choose  = 0.2,  = 0.2,  = 0.14,  = 0.2,  = 0.2, and  = 0.2.It is easy to compute that the deterministic model (1) has semisingular equilibrium  0 (1.5, 0).From Theorem 4,  0 (1.5, 0) is globally stable in this case.But, the white noise may make system (2) appear to be different phenomena.In detail, Figure 1 shows that the trajectory of () is fluctuating in a neighborhood of  0 = 1.5; however, the trajectory of () tends to 0 when  1 = 0.01,  2 = 0.02.The stationary distribution of population of () is also provided in Figure 1(b).From stationary distribution of the population of (), it is clear that the population of () is distributed normally around the mean value 1.5.
Figure 2 also shows that the trajectory of () is fluctuating in a neighborhood of  0 = 1.5, and the trajectory of () also tends to 0 when  1 = 0.1,  2 = 0.05, but we can see from Figures 1 and 2 that when the intensity of noise is amplified, the amplitude of fluctuation is also amplified.Furthermore, if we take  1 = 0.4,  2 = 0.2, then, from Theorem 9, ⟨(() + ())⟩ * = 0, it implies that the total population of () and () will tend to extinction (see Figure 3).Example 13.We choose  = 0.2,  = 0.2,  = 0.04,  = 0.2,  = 0.2, and  = 0.2.We compute that the deterministic model (1) has positive equilibrium  * (1.2, 0.5135).From Theorem 4,  * (1.2, 0.5135) is globally stable in this case.But the white noise may make system (2) appear to be different phenomena.In detail, Figure 4(a) shows that the trajectory of () and () is fluctuating in a neighborhood of (1.2, 0.5135) when  1 = 0.01,  2 = 0.02.The stationary distributions of population of () and () are also provided in Figure 4(b).From stationary distributions of two populations it is clear that they are distributed normally around the mean values 1.2 and 0.5153, respectively.4 and 5 that when the intensity of noise is amplified, the amplitude of fluctuation is also amplified.Furthermore, if we take  1 = 0.6,  2 = 0.3, then, from Theorem 9, ⟨(() + ())⟩ * = 0, it implies that the total population of () and () will tend to extinction (see Figure 6).

Discussion
In this paper, we investigated a parasite-host model within fluctuating environment.We first investigated the positivity and boundedness of the solution of deterministic model (1); we show that the solution of deterministic model (1) with the initial condition from the first quadrant is positive and bounded.Our results also show that when the parasiteindependent host background mortality is less than the maximum per capita birth rate of uninfected hosts, then the population of infected hosts maybe tend to 0. However, when the parasite-independent host background mortality is bigger than the maximum per capita birth rate of uninfected hosts, then the uninfected and infected hosts will coexist.
For the stochastic model (2), by using suitable Lyapunov functions, we show that the solution of stochastic model is positive and global, and the total population of the solutions will be persistent or tend to extinction under certain conditions.Further, we can conclude that the stochastic model (2) has a stationary distribution when the deterministic model (1) has a positive equilibrium.Our simulation results show that the solutions of stochastic model oscillate around the equilibria of deterministic model under certain conditions.That is to say, if the effects of environmental stochastic perturbations are smaller enough than the natural death rate, the solution of stochastic model will oscillate around the equilibria of deterministic model, and we observe that the amplitude of fluctuation of population distribution is amplified with the intensity of noise amplified.However, the numerical results suggest that the populations of uninfected and infected hosts become extinct after some initial large amplitude oscillation.