Dynamics of a Nonautonomous SIR Model with Time-Varying Impulsive Release and General Nonlinear Incidence Rate in a Polluted Environment

In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if R < 1 , whereas the disease persists if R∗ > 1 . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.


Introduction
It is well known that Poyang Lake located in the middle and lower reaches of the Yangtze River is the current largest freshwater lake in China.Its wetland ecosystem has a significant impact on the change of China's environment.The sufficient water resource and the superior natural environment nurture the abundant aquatic living resources of Poyang Lake.There are 136 kinds of fishes, 87 kinds of shells, 102 kinds of aquatic vascular plants and 266 kinds of identified plankton in Poyang Lake.The fishes in Poyang Lake take up 16.39% of the fresh water fish varieties in China, and 36.76% of the fish varieties of Yangtze River system.There are also first-level and second-level national protected precious rare aquatic animals such as white-flag dolphin, cowfish, chinese sturgeon, hilsa herring and so on in Poyang Lake, making it known as the treasury of fishery resources and the fish species genetic base with a significant position in the ecology system of the fish industry of Yangtze River reaches [1].
At present, the grand development of Poyang Lake ecological economy is under way in a large scale in province, which promotes the establishment of the ecological economy zone [2].However, the rapid economic development of Poyang Lake will have a negative influence on the living circumstances of fishes in the area.For the past few years, with the rapid development of modern industry and agriculture, a great quantity of toxicant and contaminants enter into Poyang Lake wetland ecosystem one after another.In order to use and regulate toxic substances wisely, we must assess the risk of the populations exposed to toxicant.Therefore, it is very important to investigate the effects of toxicants on populations and to find a theoretical threshold value, which determines permanence or extinction of fish population or community.
In recent years, many scholars have been conducted to investigate the effect of toxicant emitted into the environment from industrial, agricultural and household sources on biological species [3]- [19] by using mathematical models.For instance, Wang and Ma [18] investigated a nonautonomous SIS epidemic model with toxicant influence.They showed the existence and global attractiveness of periodic solutions and obtained the threshold between extinction and weak persistence of the infected class.Liu and Duan [19] considering the biological population infected with some kinds of diseases and hunted by human beings, and they formulate two SI pollution-epidemic models with continuous and impulsive external effects, respectively, and investigate the dynamics of such systems.But these previous models have invariably assumed that the exogenous input of toxicant is continuous or emitted in regular pulses.However, in the real life, it is often the case that toxicant is emitted in irregular pulses.In this paper, according to the above biological background, we investigate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and study dynamical behaviors of the model.
The organization of this paper is as follows.In the next section, we give some useful notations, definitions and preliminary lemmas which will be used to proof our main results.In Section 3, we mainly investigate a nonautonomous mathematical model with general nonlinear incidence rate and time-varying impulsive release, under some assumptions and the biological interpretation.In Section 4, we show that global attractivity of the disease-free periodic solution is determined by the threshold parameter * R .In Section 5, we give another expression of threshold parameter * R , and show that if * 1 R > , the disease is permanent.In the last section, we give a brief discussion and some numerical simulation results which conform the theoretical conclusions.

Notations, Definitions and Preliminary Lemmas
In this section, we introduce some notations, definitions and state some lemmas which will be useful in the subsequent sections.Let C denote the space of all bounded continuous functions.Given f C ∈ , we let If f is ω-periodic, then the average value of f on a time interval [ ] 0,ω can be defined as ( ) Before demonstrating the global attractivity of disease-free periodic solution of system (7), we need to introduce an important lemma.
Lemma 1. (see [20]) Consider the following nonautonomous linear differential equation: where ( ) a t and ( ) b t are continuous and positive ω-periodic functions.Then the system has a unique posi- tive ω-periodic solution ( ) * x t which is globally asymptotically stable.

Model Formulation and Preliminary
First of all, the total freshwater fish is divided into three groups: Susceptible fish (S), Infected fish (I) and Removed fish (R).Motivated by the above works and these literatures [21]- [29], now we investigate the properties of fish's dynamical behaviour of the model and human intervention in the polluted environment.The system is modeled by the following equations: , The model is derived with the following assumptions.

E h t C t −
represents the loss of population in the environment due to natural degradation.
• The coefficients to be nonnegative, continuous and bounded ω-periodic functions in the interval [ ) 0 , t +∞ .
• There exists a positive integer q such that k q k t t ω + = + for all k ∈  .The exogenous quantity of impulsive input of toxin into the environment is represented by k p at each fix time, and , , f t S I is a piecewise continuous, nonnegative, periodic function with period ω .The form of ( ) , , f t S I is as follows: In the following, we give some basic properties of the following subsystem of model (1), which are very important for deriving our main results.
where ( ) ) which is globally asymptotically stable, where ( ) Integrating and solving the first equation of system (2) between pulses for , , using the inductive method, we have Set ( ) f is the stroboscopic map.It is easy to see that system (6) has a unique positive equilibrium: f U is a straight line with slope less than 1, we obtain that * 0 E C is globally asymptotically stable.
It implies that the corresponding periodic solution of system (2) ( ) * E C t is globally asymptotically stable.Furthermore, according to Lemma 1, we can obtain that the system (2) has a unique positive ω -periodic solu- tion ( ) ( ) ( ) which is globally asymptotically stable.Therefore, the limit system of ( 1) is as follows: By Lemma 1, it is easy to see that system (7) has a unique disease-free periodic solution ( ) ( )

Global Attractivity of the Disease-Free Periodic Solution
To discuss the attractivity of the disease-free periodic solution of system (7), we firstly give the following hypothesis: (A) There exist positive, continuous, periodic functions Proof.Let ( ) ( ) ( ) ( ) , , S t I t R t be any solution of system (7).Since * 1 R < , we can choose a sufficiently small number 1 0 ε > such that From the second equation of system (7), we obtain that

S t t d t S t t C t S t t α ≤ Λ − −
By the comparison theorem, we can get that there exists a constant ( ) for all 1 t t ≥ .It follows from ( 9) and the second equation of system (7) that, for ( ] ( ) Then, we obtain that By using the similar method, we can infer that for ( ] Especially, when for any positive integer 1 l .It follows from ( 9) that ( ) ( ) From the (10) and (11), we get ( ) Therefore, for above mentioned 1 ε , there exist ( ) , we have for all 2 t t > .From the first and third equation of system ( 6) and ( 12), we have for where 1 ε is a sufficiently small number.Thus, we get ( ) By using the similar method, we can see that where 1 ε is an arbitrary small.Therefore, we also obtain that From ( 14) and ( 15), we can see that the disease-free periodic solution ( ) ( ) * , 0, 0 S t is global attractive.

Permanence of the Disease
In this section, we mainly obtain the sufficient conditions for the permanence of system (7).Therefore, we give the following hypotheses at first.(B) There exist positive, continuous, periodic functions with the periodic ω , such that ( ) S t be the solution of the following system: According to Lemma 1, we can obtain that the system has a unique positive ω-periodic solution ( ) S t which is globally asymptotically stable.
Theorem 2. If * 1 R > and system (7) satisfies the Hypotheses (A) and (B), then system ( Proof.Since * 1 R > , we can easily see that there exists a sufficiently small 0 In order to illustrate the conclusion, we firstly obtain the disease is uniformly weakly persistent, that is, there exists a positive constant 0 σ > , such that ( ) . By contradiction, we have that, for all given 0 η > , there exists a 3 0 t > such that ( ) < for all 3 t t > .In view of the Hypothesis (A) and the first equation of system (7), we get By comparison theorem, we have ( ) ( )

Z t S t →
as t → +∞ , where ( ) Z t is the solu- tion of the following comparison system: Therefore, for above mentioned η , there exists a * 0 n > , such that for all * 3 t n t ω > + .For above mentioned * 3 n t ω + , we have know that there exists a positive integer n 1 such that * 3 1 Then for all , by (17) and the second equation of system (6), we have Then we obtain that By using the similar method, we can get that for ( ] Therefore, for any positive integer 2 l , we have It follows from ( 19) and ( ) (2) If ξ ω > , then from the discussion in subcase (1), we have ( ) According to our above discussion, the choice of φ is independent of the positive solution of system (7), and we have proved that any solution of system (7)    Natural Science Foundation of Jiangxi Province (20151BAB201016), and the Science and Technology Plan Projects of Jiangxi Provincial Education Department (GJJ14673, GJJ150984, GJJ150995).The Supporting the Development for Local Colleges and Universities Foundation of China-Applied Mathematics Innovative Team Building.
the dose response parameter of the susceptible, infected and removed populations.the concentration of population in the organism and in the environment at time t, respectively.() E kC t represents the organisms net uptake of population from the environment.t − represent the egestion and depuration rates of population int the organism, respectively.

Figure 1 .
Figure 1.This figure shows that moment paths of susceptible fish (S) and infected fish (I) as functions of time t.* 0.9455 1 R = < .The infected fish will die out.

Figure 2 .
Figure 2.This figure shows that moment paths of susceptible fish (S) and infected fish (I) as functions of time t.* 1.1133 1 R = > .The infected fish is uniformly persistent. ) satisfies ( ) →+∞ ≥ .It is easy to obtain that, there exist positive constants * S such that ( ) * lim inf t S t S →+∞ ≥ .There- fore, system (7) is permanent.