Stage-Structured Impulsive SI Model for Pest Management

An SI epidemic model with stage structure is investigated. In the model, impulsive biological control is taken, that is, we release infected pests to the field at a fixed time periodically. We get a sufficient condition for the global asymptotical stability of the pesteradication periodic solution (0,0, ̃ I(t)), and a condition for the permanence of the system. At last, a brief discussion shows that our results will be helpful for pest management.


Introduction
According to reports of Food and Agriculture Organization of the United Nations, the warfare between human and pests (such as locust, Aphis, cotton bollworm, etc.) has sustained for thousands of years.With the development of society and the progress of science and technology, man has adopted some advanced and modern weapons such as chemical pesticides, biological pesticides, remote sensing and measuring, computers, atomic energy, and so on.At last, some brilliant achievements have been obtained [1][2][3][4].However, the warfare is not over, and will continue.A great deal of, and a large variety of, pesticides were used to control pests.In all, pesticides are useful because they can quickly kill a significant portion of a pest population and sometimes provide the only feasible method for preventing economic loss.However, pesticide pollution is also recognized as a major health hazard to human beings and beneficial insects.Biological control is, man's use of a suitably chosen living organism, referred as the biological control agent, to control another.Biological control agents can be predators, pathogens or parasites of the organism to be controlled that either kill the harmful organism or interfere with its biological progress.There is a vast amount of literatures on the applications of microbial disease to suppress pests [1][2][3], but there are a few papers on the mathematical model of the dynamical of microbial disease in pest control [4][5][6][7].The use of bacteria, fungi, and viruses is potentially one of the most important approaches in pest control.For example, Asian tiger mosquito (Aedes albopictus) can transmit viruses which cause dengue fever, Ross River fever, or Japanese encephalitis.To avoid this and to control the spread of mosquito swarms, we could spray with Bt, which is a variety of the bacterium Bacillus thuringiensis (Bt).An advantage of using insect pathogens is that they are safer to man and are usually safer to beneficial insects.In this paper, our aim is to control pests.
Stage structure is an important notation in epidemiology and demography because there are some kinds of diseases which are only spread or have more opportunities to be spread in children, for example, measles, mumps, chickenpox, scarlet fever, and diphtheria, while other infectious diseases such as gonorrhea and syphilis are spread only in adults.There are literatures about single species with stage structure [26,27], while others are about competitive or predator-prey systems with stage structure [28][29][30][31].There are still other models with stage structure [27,[32][33][34][35][36].In this paper, we consider the case in which the disease spreads only in mature pests, while the immature pests cannot be infected, for the immature are protected by their eggshells.For the purpose of pests control, we take biological control measures, that is, we release infected pests impulsively to the field, then more of the pests will be infected and die.The infected pests may be cultivated artificially in laboratory.
Xiao and Chen [34,36] have studied SIS models with stage structure, but they did not consider the effects of impulse.To our knowledge, there are few papers about stagestructured models with impulse interruption (see [7]).The present paper constructs and analyzes a realistic model of impulsive biological control system for pest management.The paper is organized as follows.In Section 2, the main biological assumptions are formulated, and the model is constructed.In Section 3, by use of the Floquet theory for impulsive differential equations, the small-amplitude perturbation method and comparison techniques, we get the sufficient condition for the global asymptotic stability of the pest-eradication periodic solution and also the sufficient condition for the permanence of the system.Finally, a brief discussion is provided in the last section.

Model formulation
The basic stage-structured SI model is where S 2 (t) and I(t) denote the densities of susceptible and infected mature populations, respectively, and S 1 (t) denotes the density of an immature population which does not contract the disease.All coefficients are positive constants.The models are derived with the following assumptions.
(H 1 ) All newborns are assumed to be susceptible.We assume that at any time t > 0, birth into the immature population is proportional to the existing mature susceptible, with proportionality r, and the infected mature population does not have offsprings.
(H 2 ) The immature susceptible population has the natural death rate μ.The death rate of the mature susceptible population is proportional both to the existing mature population and to the square of it with proportionality constants μ and 1.The infected mature population has the natural death rate μ and an extra disease-related death rate α.
(H 3 ) The immature population enters into mature susceptible with proportionality constant δ.
(H 4 ) It is assumed that the incidence rate is double linear βS 2 (t)I(t), that is, the rate at which the susceptible mature becomes infected.
In this paper, with an additional assumption (H 5 ), we construct the following model, and the parameters are the same as those in system (2.1).
(H 5 ) We release an infected mature population with constant number p > 0 periodically with period τ > 0;

Qualitative analysis for system (2.2)
In this section, we will study the effect of impulsive releasing of infected pests to the original model (2.1).To prove our main results, we give the following definition and lemmas.
Lemma 3.3 (see [9]).Let the function m ∈ pC [R + ,R] satisfy the inequalities If S 1 (t) = S 2 (t) = 0, we have the following subsystem of system (2.2): Obviously, ), is a positive periodic solution of system (3.3).Therefore, the complete expression for the pest-eradication periodic solution of system (2.2) is obtained as (0,0, n ∈ Z + , so we derive the following. Lemma 3.4 (see [8]).For every solution of system (3.3) By Lemma 3.3, for t ∈ (nτ,(n + 1)τ], we have So U(t) is uniformly ultimately bounded by a constant, and there exists a constant M > 0 such that S 1 (t) ≤ M, S 2 (t) ≤ M, I(t) ≤ M for each solution (S 1 (t),S 2 (t),I(t)) of system (2.2) for all t large enough.The proof is complete.
In the following, we investigate the stability of the pest eradication periodic solution of system (2.2).R. Shi and L. Chen 5 Theorem 3.6.Let (S 1 (t),S 2 (t),I(t)) be any solution of system (2.2) with initial values S 1 (0 + ) > 0, S 2 (0 + ) > 0, and I(0 + ) > 0, then (0,0, I(t)) is locally asymptotically stable, provided that ) Proof.The local stability of pest-eradication periodic solution may be determined by considering the behaviors of a small amplitude perturbation of the solution.Define x(t) = S 1 (t), y(t) = S 2 (t), z(t) = I(t) − I(t), then the linearized system of system (2.2) reads as Let Φ(t) be the fundamental solution matrix of system (3.7), then Φ(t) must satisfy and Φ(0) = I, the identity matrix.We can easily see that one of the eigenvalues of matrix A is −(μ + α), and the other two eigenvalues are determined by the 2 × 2 matrix B, where Denote the eigenvalues of B as λ 1 , λ 2 , then we have Therefore, by the Floquet theorem, (0,0, I(t)) is locally asymptotically stable, provided that that is,
We will next find an m 1 > 0 such that S 1 (t) + S 2 (t) ≥ m 1 for t large enough.We will do it in the following two steps.
Step 1.Since R 0 > 1 and 0 Consider the Liapunov function We claim that V (t) < (μ + δ)m 2 = m 3 cannot hold for all t ≥ 0. Otherwise, we have that S 2 (t) < m 2 for all t ≥ 0.Then, from the third and the sixth equations of system (2.2), we get By Lemma 3.3, we know that there exists a t 1 > 0 such that I(t) ≤ u(t) + 1 /β, for all t ≥ t 1 , and here, u(t It is a contradiction since V (t) is ultimately bounded by Lemma 3.5.Therefore, there exists t 1 > 0 such that V (t 1 ) ≥ m 3 .
Step 2. If V (t) ≥ m 3 hold for all t ≥ t 1 , then we can select m 1 = m 2 , and our aim is obtained.We consider those solutions which leave the region Let T = (n 2 + n 3 )τ.We claim that there must be a ( By Lemmas 3.3 and 3.4, we have and by the comparison principle, we have When t ∈ [t * ,(n 1 + 1 + n 2 )τ], we have (3.33) R. Shi and L. Chen 9
Remark 3.9.From Theorem 3.8, we can see that if the coefficients satisfy the condition R 0 > 1, then, without control of releasing infected pests, or the releasing number p < (1 − e −aτ )(b/β), the pests will be permanent and they will do harm to the crops.Under this condition, we can increase the releasing amount until p > (e aτ − 1)(b/β), and by Theorem 3.7, we know that the pests will be doomed.

Discussion
In this paper, a stage-structured SI epidemic model is investigated.In the model, impulsive biological control is taken into consideration, and we analyzed the existence and stability of the pest-eradication periodic solution of the system.We also get a sufficient condition for the permanence of the system.In the real world, we can take impulsive biological control to constrain the pest population.There is still an interesting problem: if R 0 > 1 and 0 < p < b(1 − e −aτ )(b/β), then the system is permanent; if R 0 > 1 and p > (e aτ − 1)(b/β), then (0,0, I(t)) is globally asymptotically stable.But we have not considered the case that R 0 > 1 and b(1 − e −aτ )/β < p < (e aτ − 1)(b/β), how about the dynamical behaviors of the system?In this paper, our aim is to control the pest population, so we omit the above problems.But from a mathematical aspect, it is very interesting, and we leave it as a future work.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning 13) because f (x) = e x − 1 > 0, whenever x > 0 Theorem 3.8.If R 0 > 1 and 0 < p < b(1 − e −aτ )/β, then system (2.2) is permanent.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation