Processing math: 67%
Research article

Different retiming transformation technique to design optimized low power VLSI architecture

  • A different method for designing low power retime architecture is presented in this paper. The modified retiming transformation techniques approach to reduce the dynamic power consumption of the digital circuit, without compromising the output results. In this paper, retiming transformation is extended in two-ways to reduce the power consumption of the design. Graphical Circular Retiming and Tabular Shift Retiming are the two methods used to realize how the registers are mapped to reduce the glitching power. Proposed transformation technique delay value is placed in the form of metric and verified without sacrificing the functionality. Proposed transformation technique is applied to FIR filter to analyze the simulation and synthesis results as proofs of this concept. Experimental results are compared with the conventional retiming transformation technique with the same operating frequency, and with the significant reduction in dynamic power of FIR filter.

    Citation: Jalaja S, Vijaya Prakash A M. Different retiming transformation technique to design optimized low power VLSI architecture[J]. AIMS Electronics and Electrical Engineering, 2018, 2(4): 117-130. doi: 10.3934/ElectrEng.2018.4.117

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  • A different method for designing low power retime architecture is presented in this paper. The modified retiming transformation techniques approach to reduce the dynamic power consumption of the digital circuit, without compromising the output results. In this paper, retiming transformation is extended in two-ways to reduce the power consumption of the design. Graphical Circular Retiming and Tabular Shift Retiming are the two methods used to realize how the registers are mapped to reduce the glitching power. Proposed transformation technique delay value is placed in the form of metric and verified without sacrificing the functionality. Proposed transformation technique is applied to FIR filter to analyze the simulation and synthesis results as proofs of this concept. Experimental results are compared with the conventional retiming transformation technique with the same operating frequency, and with the significant reduction in dynamic power of FIR filter.


    As an insect control method, the sterile insect technique (SIT) has made a big success in pest management and disease vector control over the past few decades [1,2,3,4,5]. In most instances, sterile males are released into an area to compete with wild males. If a female mates with a sterile one, it is unable to produce offspring, then the insect population will be eventually controlled or even eliminated by the reproductive mate attrition. SIT was verified to work against mosquitoes by field trials in the 1970s and 1980s [6], and then developed rapidly for emerging approaches from SIT and new rearing techniques [7,8,9,10]. In addition to the classic SIT, the biological control methods already used in the laboratory or field also include the genetic approaches and the Wolbachia driven mosquito control technique.

    To investigate the impact and effectiveness of these biological control measures, various mathematical models have been formulated and analyzed. K. R. Fister et al. [11] proposed an optimal control framework to explore the effect of sterile mosquito releases on reducing the incidence of mosquito-borne diseases. S. M. White et al. explored the mechanism how an insect fitness cost affects different control policies by constructing a stage-structured mathematical model for the mosquito Aedes aegypti in [12]. J. Li et al. [13,14,15] investigated multiple policies of sterile mosquito release by formulating discrete and continuous dynamical systems for the interaction between two mosquito populations. While L. Cai et al. [16] studied the impact of the SIT on disease transmission by constructing dynamical systems incorporating constant, proportional and Holling-Ⅱ type release rates. Y. Dumont and J. M. Tchuenche proposed mathematical models of SIT to exploit the control of an epidemic of Chikungunya in [17], and pulsed periodic releases was especially studied. While in [8], M. Strugarek et al. investigated the application of SIT in reducing and eliminating wild mosquitoes by a simplified population model for Aedes, and several release modes were considered and necessary conditions to guarantee elimination in each case were obtained. There are also lots of works focusing on the dynamical analysis of mosquito population models with different characteristics [18,19,20].

    It was pointed out in [21] that the duration of the release process each time is relatively short and it usually takes multiple releases to make the mosquito population under control. Therefore, multiple pulsed releases may be a realistic assumption. The impulsive release has been studied in several cases [8,17,21,22,23,24,25,26,27]. The authors in [17] explained the pulsed release of sterile males by a mosquito-human epidemiological model for Chikungunya, and investigated the impact of periodic pulsed releases on the disease transmission. In [21], different two-dimensional models with periodic impulsive releases and state feedback impulsive releases are constructed, and the authors exploited the influence of different release strategies on the population development of wild mosquitoes. And P. A. Bliman et al.[22] also investigated impulsive release of sterile male mosquitoes, and they studied the periodic impulsive releases under open-loop control, state feedback impulsive releases under closed-loop control and a mixed release strategy that combines open-loop and closed-loop controls. While research in [8] established necessary conditions which can guarantee the eventual extinction of the wild mosquito population.

    For the Wolbachia driven mosquito control technique, J. Yu [23] introduced a model of differential equations with a time delay to study the suppression dynamics of wild mosquitoes intervened by the releasing of Wolbachia-infected males. Unlike many studies, the population suppression in this work tried to avoid releasing infected females, just released living infected males, and aimed for eliminating the whole population of mosquitoes. Then J. Yu et al. introduced the sexual lifespan of sterile mosquitoes and assumed that the interaction happens only when the sterile mosquitoes are still sexually active [24,25,26,27]. They investigated the impact of the sexual lifespan of sterile mosquitoes on mosquito population suppression based on delay differential equations and gave a lot of important results. J. Yu and B. Zheng [28] specially studied Wolbachia persistence by extra releases of Wolbachia-infected mosquitoes based on difference equations, and obtained a maximal maternal leakage rate threshold such that infected mosquitoes can persist. While M. Huang et al. [29] introduced a system of delay differential equations, including both the adult and larval stages of wild mosquitoes, interfered by Wolbachia infected males. They explored its global dynamics and determined a threshold level of infected male releasing which can ensure that the wild population is suppressed completely.

    Although rearing techniques are steadily updated, mosquito mass rearing is still one of the major obstacles preventing the application of the SIT against mosquitoes in large scale. Artificial rearing of sterile mosquitoes comes at an economic cost which cannot be neglected. Optimal control method can be used to balance the conflict between the control level of the wild mosquitoes and the economic cost.

    Optimal control problem for impulsive dynamical systems has its special peculiarities, and there is a technical difficulty compared to continuous dynamical systems due to the dependence of the state of variables on uncertain pulse effects. Many researchers have been making contributions to overcome the difficulty and providing available control methodologies. These methodologies have been used in the optimal impulsive management of population [11,13,30,31,32,33,34]. For example, in [33] the authors provided multiple kinds of optimal policies for an eco-epidemiological model with impulsive interferences. An pest management system incorporating impulsive release natural enemies is studied in [34], and three optimal release strategies are given.

    In this study, we did not consider the effect of the sexual lifespan and propose a two-sex mosquito population model with stage structure and impulsive releases of sterile males, and an Allee effect is also incorporated to describe the scarcity of the available mating area in the field. We firstly study the large-scale time control based on aims of the extinction of wild mosquitoes, and then investigate limited-time optimal control of wild mosquitoes to gain a suitable control strategy by selecting optimal release parameters.

    The structure of paper is as follows: In Section 2, we establish a hybrid dynamical system for the large-scale time control of wild mosquitoes with impulsive releases of sterile males, and then study its dynamical properties and exploit threshold conditions whether or not the wild mosquito population is eliminated. In Section 3, we take into account the control effect of mosquito population level and the economic input, and raise three limited-time optimal control problems for the impulsive release strategies. By using a time rescaling technique, the gradients of cost function with respect to all control parameters are obtained. Then numerical simulations are performed in Section 4 to determine the optimal values of the release timing and release amount. Finally, a brief conclusion is presented in Section 5.

    The model proposed in this section aims to investigate release tactics of sterilizing males which can effectively reduce and eventually eliminate wild mosquitoes in a field.

    According to the life habits of many species of mosquitoes (for example, Aedes genus), eggs produced by fertile females may keep unhatched for a long time to wait for rainy seasons when the natural breeding sites are available. So there may be large egg stocks in a given field which has to be taken into account when built our mathematic model. Besides, in many cases the available mating area in the field is relatively scarce and fertile females need overcome difficulties to get successful fertilization. Therefore, we add an Allee effect in the wild female population as some researchers have done. Based on the model assumption in [8], we propose the following population development model for wild mosquitoes with impulsive releases of sterile males

    {dWE(t)dt=βWF(1WEK)(ρ+μ1)WE,dWM(t)dt=θρWEμ2WM,dWF(t)dt=(1θ)ρWEbWMγ+WM+αGMμ3WF,dGM(t)dt=μ4GM(t),}   tkω,k=1,2,,WE(t+)=WE(t),WM(t+)=WM(t),WF(t+)=WF(t),GM(t+)=GM(t)+δ,}  t=kω, (2.1)

    with K>WE(0)>0,WM(0)>0,WF(0)>0 and GM(0)>0. WE(t),WM(t) and WF(t) represent the densities of eggs, fertile males and fertile females of wild mosquitoes at time t, while GM(t) is the density of sterilizing males released in the field. The logistic term βWF(1WEK) describes the "skip oviposition" behavior of fertile females for they are capable to avoid depositing eggs in an area which has supported too many larvae. β measures the effective fecundity and K is the environmental carrying capacity. μi(i=1,2,3,4) denote the death rates, ρ is the hatching rate and θ represents the sex ratio of wild mosquitoes. b stands for the insemination rate of emerging females, while parameters γ and α measure the strength of Allee effect which involves a female's mating likelihood and the mating competitiveness of sterile males, respectively. Besides, ω is the release period and δ is the amount released each time.

    In [8], the Allee effect caused by mating limitation is modeled by a negative exponential function 1exp(kWM), while in this work a rectangular hyperbola function bWMγ+WM+αGM is used. This form of Allee effect is also widely used to describe mating limitation in sexually reproducing organisms [14,15,16,35,36]. We hope to explore this kind of Allee effect in this work and to see if it will cause any changes.

    In the next subsection, we will investigate dynamical behaviors of system (2.1), and exploit the release tactics for the large-scale time control of wild mosquitoes aiming to wipe out wild mosquitoes from the field eventually.

    We firstly discuss the existence of the wild mosquito-free periodic solution of system (2.1), then determine conditions of its global stability. We will provide theoretical analysis and practical method for the selection of the release amount δ and the release period ω so that the wild mosquitoes can be wiped out from the field.

    Let X(t)=(WE(t),WM(t),WF(t),GM(t))T be any solution of system (2.1). Obviously, X(t) is piecewise continuous and X(kω+)=limε0+X(kω+ε) exists. Since the right hand side of system (2.1) is locally lipschitz continuous on R4+, system (2.1) has a unique solution [37,38].

    The positivity and boundedness of the solution of system (2.1) are firstly investigated.

    Proposition 1. Solutions of system (2.1) are always non-negative if the initial values are non-negative.

    Proof. Let (WE(t),WM(t),WF(t),GM(t)) be a solution of system (2.1) with non-negative initial conditions. From the forth equation of system (2.1), we can get dGM(t)/dt=0 if GM(t)=0, so we have GM(t)>0, t0 for GM(0)>0.

    Assume that there exists t>0 satisfying WE(t)<0. Denote τ1=inf{t:WE(t)<0}. From this, we have WE(τ1)=0 and WE(τ1)0. Substitute them into the first equation of system (2.1), then we obtain WE(τ1)=βWF(τ1)0.

    If WF(t) is non-negative for all t>0, then WF(τ1)0 and we get WE(τ1)0. It follows from WE(τ1)0 that WE(τ1)=0, then we have WE(τ1)=WE(τ1)=WF(τ1)=WF(τ1)=0. According to the definition of τ1, there must be a sufficiently small constant ϵ1>0 such that WE(τ1+ϵ1)<0 and WE(τ1+ϵ1)<0. Since WF(τ1+ϵ1)0, we have WE(τ1+ϵ1)=βWF(τ1+ϵ1)(1WE(τ1+ϵ1)K)(ρ+μ1)WE(τ1+ϵ1)>0, which leads to a contradiction. Therefore, there must exist t>0 such that WF(t)<0. Denote τ2=inf{t:WF(t)<0}, then WF(τ2)=0 and WF(τ2)0. We can easily obtain τ2<τ1 and WE(τ2)>0.

    In fact, if τ1=τ2, there is WE(τ1)=βWF(τ1)=βWF(τ2)=0, and as discussed above we can get a contradiction. If τ1<τ2, there is WF(τ1)>0, and then we have WE(τ1)=βWF(τ1)>0, which also leads to a contradiction.

    Further more, if WM(t) is non-negative for all t>0, then we obtain WM(τ2)0. By the third equation of system (2.1), we have

    WF(τ2)=(1θ)ρWE(τ2)bWM(τ2)γ+WM(τ2)+αGM(τ2)μ3WF(τ2)0.

    It follows from WF(τ2)0 that WF(τ2)=0, then we have WF(τ2)=WF(τ2)=WM(τ2)=0,WM(τ2)=θρWE(τ2)>0. According to the definition of τ2, there must be a sufficiently small constant ϵ2>0 such that WF(τ2+ϵ2)<0, WF(τ2+ϵ2)<0 and WM(τ2+ϵ2)>0. Since

    WF(τ2+ϵ2)=(1θ)ρWE(τ2+ϵ2)×bWM(τ2+ϵ2)γ+WM(τ2+ϵ2)+αGM(τ2+ϵ2)μ3WF(τ2+ϵ2)>0,

    which leads to a contradiction. Thus there must exist some t>0 such that WM(t)<0. Denote τ3=inf{t:WM(t)<0}, then we have WM(τ3)=0 and WM(τ3)0. According to the second equation of (2.1) yields WM(τ3)=θρWE(τ3)0. If τ1>τ3, then there is WE(τ3)>0 and WM(τ3)=θρWE(τ3)>0, which also leads to a contradiction. Thus we have τ1τ3.

    By the above discussion, we have τ2<τ1τ3 and WE(τ2)>0,WM(τ2)>0,WF(τ2)=0. However,

    WF(τ2)=(1θ)ρWE(τ2)bWM(τ2)γ+WM(τ2)+αGM(τ2)>0,

    which also contradicts the definition of τ2.

    To sum up, we have WE(t) is non-negative for all t>0. Then there are dWM(t)/dt=0 if WM(t)=0 and dWF(t)/dt0 if WF(t)=0, and all solutions of system (2.1) with non-negative initial conditions are always non-negative. The proof is completed.

    Denote

    Ω={(WE,WM,WF,GM)R4+:0WEK,0WMθρKμ2L1,0WF(1θ)ρKbμ3L2and0GMδ1exp(μ4ω)L3}. (2.2)

    Proposition 2. Ω is a forward invariant and globally attracting set of system (2.1) in R4+.

    Proof. Suppose X(t)=(WE(t),WM(t),WF(t),GM(t))T is a solution of system (2.1) with X(0)=(WE(0),WM(0),WF(0),GM(0))TR4+. Firstly, we prove that Ω is a forward invariant set. By the forth equation and the impulsive conditions of system (2.1), we get

    {dGM(t)dt=μ4GM(t),tkω,k=1,2,,GM(t+)=GM(t)+δ,t=kω. (2.3)

    It is obvious that the dynamics of sterilizing males is completely unaffected by that of the wild mosquitoes. System (2.3) admits a unique positive periodic solution

    {˜GM(t)=δexp(μ4(tkω))1exp(μ4ω),t(kω,(k+1)ω],k=0,1,,˜GM(0+)=δ/(1exp(μ4ω)), (2.4)

    which is globally asymptotically stable. Besides, we can easily get

    GM(t)=(GM(0)˜GM(0))exp(μ4t)+˜GM(t)

    and

    limt+GM(t)=˜GM(t). (2.5)

    If X(0)Ω, then we have GM(0)L3=˜GM(0) and GM(t)˜GM(t)˜GM(0)=L3,t0.

    Besides, from the first equation of system (2.1), we get dWEdt|X(t)ΩβL2(1WEK), thus we can easily obtain that 0WE(t)K,t0 for any X(0)Ω. According to the second equation of system (2.1), we have dWMdt|X(t)ΩθρKμ2WM=θρK(1WML2), thus there is 0WM(t)L2,t0 for any X(0)Ω. Similarly, by the third equation of system (2.1), we have dWFdt|X(t)Ω(1θ)ρKbμ3WF=(1θ)ρKb(1WML3), and then there is 0WF(t)L3,t0 for any X(0)Ω. Therefore, for any X(0)=(WE(0),WM(0),WF(0),GM(0))TΩ, there is

    X(t,0,X(0))=(WE(t),WM(t),WF(t),GM(t))TΩ,t0.

    That is to say, Ω is a forward invariant set of system (2.1) in R4+.

    In the following, we prove that Ω is globally attractive. For any initial point X(0)=(WE(0),WM(0),WF(0),GM(0))TΩ, we study the trajectory trend of X(t,0,X(0)) with the increase of time t. If WE(0)K, similar to the above discussion above the invariant set, we can get 0WE(t)K,t0. If WE(0)>K, from the first equation of system (2.1), we get dWEdt|WEK<(ρ+μ1)K<0, and there must exist a time t1>0 such that 0WE(t)K,tt1. When tt1, by the second equation of system (2.1) and WE(t)K, we have dWMdtθρKμ2WM. If WM(t1)L1, similar to the above discussion above the invariant set, we can get 0WM(t)L1,tt1. If WM(t1)>L1, we have dWMdt|tt1θρKμ2WM, then there must exist a time t2>t1 such that 0WM(t)L1,tt2. When tt2, by the third equation of system (2.1) and WE(t)K, we have dWFdt(1θ)ρKbμ3WF. If WF(t2)L2, similar to the above discussion above the invariant set, we can get 0WF(t)L2,tt2. If WF(t2)>L2, we have dWFdt|tt2(1θ)ρKbμ3WF, then there must exist a time t3>t2 such that 0WF(t)L2,tt3. Further more, since limt+GM(t)=˜GM(t) and 0˜GM(t)L3,t0, we deduce that Ω is a globally attracting set of system (2.1) in R4+. This completes the proof.

    Theorem 1. System (2.1) has a wild mosquito-free periodic solution (0,0,0,˜GM(t)), where

    ˜GM(t)=δexp(μ4(tkω))1exp(μ4ω),t(kω,(k+1)ω],k=0,1,

    and

    ˜GM(0+)=δ/(1exp(μ4ω)).

    Proof. According to the positive periodic solution of model (2.3) obtained in Proposition 2, we can easily get (0,0,0,˜GM(t)) is a periodic solution of system (2.1) which implies the eradication of wild mosquitoes. The proof is completed.

    We first prove the local stability of the ωperiod solution (0,0,0,˜GM(t)).

    Theorem 2. The wild mosquito-free periodic solution (0,0,0,˜GM(t)) of system (2.1) is locally asymptotically stable.

    Proof. In order to study the local stability of the wild mosquito-free periodic solution (0,0,0,˜GM(t)), we consider the following subsystem

    {dWE(t)dt=βWF(1WEK)(ρ+μ1)WE,dWM(t)dt=θρWEμ2WM,dWF(t)dt=(1θ)ρWEbWMγ+WM+α˜GMμ3WF. (2.6)

    Obviously, system (2.6) has a trivial equilibrium (0,0,0). Computing the Jacobian matrix of system(2.6) at (0,0,0), we get

    JO=((ρ+μ1)0βθρμ2000μ3)

    and it has three real and negative eigenvalues λi,i=1,2,3. Then the Floquet multipliers of the corresponding monodromy matrix eJOω are eλiω<1,i=1,2,3. It follows from the Floquet theorem that the trivial equilibrium (0,0,0) of system (2.6) is always locally stable, which also implies that the wild mosquito-free periodic solution (0,0,0,˜GM(t)) of system (2.1) is locally asymptotically stable. The proof is completed.

    In the following, we determine conditions under which the wild mosquito-free periodic solution (0,0,0,˜GM(t)) is also globally attractive.

    Theorem 3. The wild mosquito-free periodic solution (0,0,0,˜GM(t)) of system (2.1) is a global attractor provided one of the following conditions holds

    (i) (ρ+μ1)μ3β(1θ)ρb;

    (ii) (ρ+μ1)μ3β(1θ)ρ<b and 1α((b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ)<δexp(μ4ω)(1exp(μ4ω)).

    Proof. Since limt+GM(t)=˜GM(t), to discuss the global attractivity of (0,0,0,˜GM(t)), we only need to find conditions that can guarantee the global stability of (0,0,0) for system (2.6).

    According to Theorem 2, the trivial equilibrium (0,0,0) of system (2.6) is locally stable. We now exploit conditions under which it is globally attractive.

    By letting the sterilizing males ˜GM(t) be a constant, that is, ˜GM(t)GcstM0, we construct a comparison system as follows

    {dWE(t)dt=βWF(1WEK)(ρ+μ1)WE,dWM(t)dt=θρWEμ2WM,dWF(t)dt=(1θ)ρWEbWMγ+WM+αGcstMμ3WF. (2.7)

    We calculate the Jacobian matrix of system (2.7) as follows

    JE=((βWFK+ρ+μ1)0β(1WEK)θρμ20(1θ)ρWMγ+WM+αGcstM(1θ)ρWEb(γ+αGcstM)(γ+WM+αGcstM)2μ3).

    Obviously, all the off-diagonal elements are non-negative on the set Ω1={(WE,WM,WF)R3+:WEK}, which implies that system (2.7) is monotone on Ω1 in the sense of the monotone systems theory [39]. Besides, it has a trivial equilibrium O(0,0,0) which is locally stable. To verify the existence of positive steady state, we need to solve the following algebraic equations

    WE=βWF(1WEK)(ρ+μ1),WE=μ2WMθρ,WF=(1θ)ρμ3WEbWMγ+WM+αGcstM.

    By direct calculation, we get

    {WE=μ2θρWM,WF=(ρ+μ1)Kμ2θρKWMβ(1μ2θρKWM),μ2bθρK(WM)2+((ρ+μ1)μ3β(1θ)ρb)WM+(ρ+μ1)μ3β(1θ)ρ(γ+αGcstM)=0. (2.8)

    For simplicity, denote

    B1=μ2θρK,B2=(ρ+μ1)μ3β(1θ)ρ,

    then the number of positive steady states of system (2.7) equals the number of positive roots of the following equations with respect to x:

    {bB1x2+(B2b)x+B2(γ+αGcstM)=0,B1x<1. (2.9)

    For the quadratic equation in system (2.9), it is straightforward to show that it has no positive root if (i)B2b or (ii)B2<b and 1α((bB2)24bB1B2γ)<GcstM holds.

    If B2<b and 1α((bB2)24bB1B2γ)>GcstM, then the quadratic equation in system (2.9) has positive roots

    x±=bB2(bB2)24bB1B2(γ+αGcstM)2bB1,

    which must satisfy the second inequality in system (2.9). That is to say, system (2.7) must have two positive equilibria, one of which is locally stable.

    Due to the monotonicity of the system, the trivial equilibrium O(0,0,0) of system (2.7) is globally asymptotically stable if it is the unique steady state. According to the above analysis, when B2<b, there is a critical value GcritM=1α((bB2)24bB1B2γ) for GcstM and GcstM>GcritM can ensure that O(0,0,0) is a global attractor.

    Based on the analytical expression of ˜GM in system (2.4), we can easily get its upper and lower bounds

    ˜GLM=δexp(μ4ω)1exp(μ4ω)˜GM(t)δ1exp(μ4ω)=˜GUM.

    If ˜GLM=δexp(μ4ω)1exp(μ4ω)>GcritM, then we have ˜GM(t)>GcritM,t0. By the monotonicity and the relation between systems (2.6) and (2.7), we know that the trivial equilibrium O(0,0,0) is globally asymptotically stable for system (2.6) if it is globally asymptotically stable for system (2.7).

    Thus, the trivial equilibrium O(0,0,0) is globally asymptotically stable for system (2.6) if (i)B2b or (ii)B2<b and ˜GLM>GcritM holds, then wild mosquito-free periodic solution (0,0,0,˜GM(t)) of system (2.1) is globally asymptotically stable under the same conditions. The proof is completed.

    Remark 1. In Theorem 3, 1B2=β(1θ)ρ(ρ+μ1)μ3 involves with the fecundity of the wild mosquitoes in the field. According to the results in Theorem 3, if the fertility is weak enough, that is, b1B21, then the wild mosquitoes will eventually go extinct even without human intervention. If the fertility is relatively strong, for example, b1B2>1, we can also eliminate the wild mosquitoes in the long run by adjusting the intensity of releases of sterile males.

    In the following, we study the large-scale time control strategies for wild mosquitoes by theoretical analysis and give practical methods for selecting the release amount δ and release period ω so that the wild mosquitoes can be wiped out from the field.

    Based on the first kind of condition listed in Theorem 3, if the fertility in a given field is weak (β(1θ)ρ(ρ+μ1)μ31b), wild mosquitoes will always go extinct without any human interventions. So we mainly consider a more common case when wild mosquitoes has a relatively strong fertility and we need release sterilizing males reasonably into the field to wipe out the wild ones.

    According to the second kind of conditions listed in Theorem 3, we discuss the control strategies if (ρ+μ1)μ3β(1θ)ρ<b holds.

    Obviously, if (b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ, then the conditions listed in (ii) of Theorem 3 are valid and the wild mosquitoes will eventually go extinct without human intervention. If (b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ρ>γ, we denote N(δ,ω):=δexp(μ4ω)(1exp(μ4ω)) and consider the equation

    N(δ,ω)=δexp(μ4ω)(1exp(μ4ω))=1α((b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ). (2.10)

    For any given release period ω, N(δ,ω) is monotonically increasing with respect to δ. Besides, by simple calculation, we can get N(0,ω)=0 and N(+,ω)=+. Then there is a unique ˜δ0 satisfying N(δ,ω)=1α((b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ). Hence it follows from Theorem 3 that the wild mosquito-free periodic solution (0,0,0,˜GM(t)) of (2.1) is globally stable provided δ>˜δ holds.

    Similarly, for a given release amount δ, N(δ,ω) is monotonically decreasing with respect to ω. Since it is obvious to have N(δ,0)=+ and N(δ,+)=0, there exists a unique ˜ω0 such that N(δ,ω)=1α((b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ). Thus the wild mosquito-free periodic solution (0,0,0,˜GM(t)) of system (2.1) is globally stable provided ω<˜ω according to Theorem 3.

    In the following, we will investigate the release tactics for large-scale time control of wild mosquitoes by numerical simulations. Most model parameters are chosen from [8] and [17] (refer to Table 1). While the environmental carrying capacity K and the strength of Allee effect γ remain pending for their values often change with environments.

    Table 1.  Model parameter values from [8] and [17].
    Parameters Value interval Unit Parameters Value interval Unit
    β 7.46 - 14.85 day1 ρ 0.001 - 0.25 -
    μ1 0.023 - 0.046 day1 θ 0.51 -
    μ2 0.0770.139 day1 b 0 - 1 -
    α 01 - μ3 0.033 - 0.046 day1
    μ4 0.25 day1

     | Show Table
    DownLoad: CSV

    In this paper, we consider parameters as follows

    β=10,ρ=0.01,θ=0.51,μ1=0.03,μ2=0.1,μ3=0.04,μ3=0.04,μ4=0.25,α=1,b=0.7,K=5000. (2.11)

    For the wild mosquito extinction induced by a strong Allee effect, we select γ=1500 and there is (b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ρ=1240.1868<γ=1500. Then the wild mosquitoes eventually go extinct without sterile males deliveries (see Figure 1).

    Figure 1.  The global stability of the wild mosquito-free equilibrium of system (2.1) when the Allee effect is strong. Here, two different sets of initial values are selected.

    Keep other parameters the same as in Eq (2.11) but change γ=1500 to γ=200, then we have (b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ρ>γ. By direct calculation, we get

    1α((b(ρ+μ1)μ3β(1θ)ρ)24bμ2θρK(ρ+μ1)μ3β(1θ)ργ)=1040.1868.

    We first fix the release period ω=3 and through simple calculation we obtain that the unique positive root of N(δ,ω)=1040.1868 is ˜δ=1161.8887. To verify our theoretical results, we compare two release amounts δ=1200 and δ=800 in Figures 2 and 3, respectively. We can see that if the release amount δ=1200>˜δ, then the wild mosquito-free periodic solution of system (2.1) is globally stable (see Figure 2), while if the release amount δ=800<˜δ, then there exists a locally stable positive coexistence period solution in addition to the wild mosquito-free periodic solution (see Figure 3).

    Figure 2.  The global stability of the wild mosquito-free periodic solution of system (2.1) with δ=1200>˜δ. Here, three different sets of initial values are selected.
    Figure 3.  System (2.1) has two locally stable periodic solutions with δ=800<˜δ: a positive coexistence one and a wild mosquito-free one. The initial values are the same as those in Figure2.

    We then fix the release amount δ=1000 and get that the unique positive root of N(δ,ω)=1040.1868 is ˜ω=2.6946. Similarly, we compare two release periods ω=2.5 and ω=4 in Figures 4 and 5, respectively. We see that if the release period ω=2.5<˜ω, then the wild mosquito-free periodic solution of system (2.1) is globally stable (see Figure 4), while if the release period ω=4>˜ω, then a locally stable positive coexistence period solution coexists with the wild mosquito-free periodic solution (see Figure 5).

    Figure 4.  The global stability of the wild mosquito-free periodic solution of system (2.1) with ω=2.5<˜ω. The initial values are the same as those in Figure 2.
    Figure 5.  System (2.1) has two locally stable periodic solutions with ω=4>˜ω: a positive coexistence one and a wild mosquito-free one. The initial values are the same as those in Figure 2.

    In this section, we focus on the asymptotic behaviors of system (2.1), and verify the theoretical results by numerical simulations. From Figures 15, we can see that although the sequential impulsive releases of sterile mosquitoes can make the wild mosquitoes go extinct, there are disadvantages in terms of cost control. After the wild population entered the basin of attraction of the extinction solution, the extinction of wild mosquitoes is a foregone conclusion, and the releases of sterile mosquitoes only speed up the process. This is unreasonable from the perspective of cost control. Because the cost control is not involved in this part of the study, the conclusion of this part is more limited to the theoretical discussion, and ignores the rationality of practical application.

    As pointed out in Section 2, regardless of the state of wild mosquito population in the environment, blind sequential releasing sterile mosquitoes in large-scale time can ensure the extinction of wild mosquitoes, but it will cause unnecessary cost waste in practice, which is not desirable. Furthermore, the prevalence of mosquito borne diseases describes obviously seasonal and regional characteristics. To this end, in this section we investigate limited-time optimal control of wild mosquitoes incorporating both of the population control level of wild mosquitoes and the economic input, and study three different release strategies: optimal release amount for periodic releases, optimal release timing for a fixed release amount and mixed control with optimal release timing and amount each time.

    Firstly, we consider a relatively complicated scenario when both release timings and release amounts are chosen as control parameters. Assume that we need to make mosquito population under control in a predefined finite interval [0,T] and N1 releases of sterilizing males are planed. Suppose the release amount and release timing of the ith release are δi and ti[0,T], respectively, with i=1,2,,N1. Then a limited-time control system is proposed in the following

    {dWE(t)dt=βWF(1WEK)(ρ+μ1)WE,dWM(t)dt=θρWEμ2WM,dWF(t)dt=(1θ)ρWEbWMγ+WM+αGMμ3WF,dGM(t)dt=μ4GM(t),}  tti,t[0,T],WE(t+)=WE(t),WM(t+)=WM(t),WF(t+)=WF(t),GM(t+)=GM(t)+δi,}  t=ti,i=1,2,,N1 (3.1)

    with initial conditions

    WE(0)=W0E,WM(0)=W0M,WF(0)=W0F,GM(0)=G0M. (3.2)

    T, the length of control time, can be converted to be dependent on the transmission of mosquito borne infectious diseases in this situation. ti,i=1,2,,N1, the release timings, are assumed to satisfy 0=t0t1t2tN1tN=T. Denote Ti=titi1 which represents the time interval between the (i1)th and ith release. According to practical meaning, this time interval cannot be too long or too short. Thus we give constraints

    0τ1iTiτ2i,i=1,2,,N, (3.3)

    where τ1i and τ2i are given constants. Similarly, we also give constraint for the release amount δi

    0δ1iδiδ2i,i=1,2,,N1, (3.4)

    where δ1i and δ2i are given constants which represent the minimum and maximum amount allowed in the ith release. Denote Γ=(T1,T2,,TN)T and δ=(δ1,δ2,,δN1)T, where Ti and δi meet the stated constraints Eqs (3.3) and (3.4), respectively. Let Ψ1 and Ψ2 be sets of all ΓRN,δRN1 satisfying Eqs (3.3) and (3.4), respectively.

    Since the right hand side of system (3.1) are differentiable, system (3.1) with initial condition Eq (3.2) has a unique solution (WE(t),WM(t),WF(t),GM(t))T corresponding to each pair (Γ,δ)(Ψ1,Ψ2) [37,38].

    Based on our key considerations of the mosquito population control, we define a cost function as follows

    J(Γ,δ)=WM(T)+WF(T)+r0N1i=1δi. (3.5)

    Here, r0 stands for the per unit cost of rearing sterilizing mosquitoes.

    For the optimal control problem in this scenario, we can state it formally as follows.

    (P1) Subject to the dynamical system (3.1) with initial condition Eq (3.2), find a feasible parameter vector pair (Γ,δ) such that the cost function J(π,δ) is minimized over (Ψ1,Ψ2).

    Since the state of variables WE(t),WM(t),WF(t) and GM(t), functions of t,Γ and δ, depends on uncertain release timings and uncertain release amounts, the optimal control problem (P1) cannot be solved directly by general optimization techniques. In this paper, we apply a time rescaling method which has been used in several studies [33,34,40,41] to transform these uncertain pulse time points into fixed ones. By this method, the optimal problem (P1) is turned into an equivalent optimal parameter selection problem which is described as a series of ordinary differential equations with periodic initial conditions. And the new equivalent problem can be solved by utilizing a gradient-based optimization technique.

    To this end, let t=i1j=1Tj+Tis, t(i1j=1Tj,ij=1Tj], and denote

    {WiE(s)=WE(i1j=1Tj+Tis),WiM(s)=WM(i1j=1Tj+Tis),WiF(s)=WF(i1j=1Tj+Tis),GiM(s)=GM(i1j=1Tj+Tis). (3.6)

    Then system (3.1) with initial condition Eq (3.2) is converted into N subsystems

    {dWiE(s)ds=Fi1(s)=Ti[βWiF(1WiEK)(ρ+μ1)WiE],dWiM(s)ds=Fi2(s)=Ti[θρWiEμ2WiM],dWiF(s)ds=Fi3(s)=Ti[(1θ)ρWiEbWiMγ+WiM+αGiMμ3WiF],dGiM(s)ds=Fi4(s)=Ti[μ4GiM(t)],} s(0,1],i=1,,N,WiE(0)=Wi1E(1),WiM(0)=Wi1M(1),WiF(0)=Wi1F(1),GiM(0)=Gi1M(1)+δi,}  i=2,3,,N (3.7)

    with

    {W1E(0)=WE(0)=W0E,W1M(0)=WM(0)=W0M,W1F(0)=WF(0)=W0F,G1M(0)=GM(0)=G0M. (3.8)

    The cost function Eq (3.5) can also be redefined as follows

    J1(Γ,δ)=WNM(1)+WNF(1)+r0N1i=1δi. (3.9)

    Based on the above transformations, we can restate the problem (P1) as follows

    (P2) Subject to the dynamical system (3.7) with initial condition Eq (3.8), find a feasible parameter vector pair (Γ,δ) such that the cost function defined in Eq (3.9) is minimized over (Ψ1,Ψ2).

    Using Theorem 6.1 in [42], we define Hamiltonian functions Hi,i=1,2,,N in the following

    Hi(s,WiE(s),WiM(s),WiF(s),GiM(s),Γ,δ)=(λi1(s),λi2(s),λi3(s),λi4(s))(Fi1(s),Fi2(s),Fi3(s),Fi4(s))T. (3.10)

    Here λi(s)=(λi1(s),λi2(s),λi3(s),λi4(s)) are costate variables which are governed by the following costate equations

    {˙λi1(s)=HiWiE=Ti[λi1(βWiFK+ρ+μ1)θρλi2λi3(1θ)ρbWiMγ+WiM+αGiM],˙λi2(s)=HiWiM=Ti[μ2λi2λi3(1θ)ρbWiE(γ+αGiM)(γ+WiM+αGiM)2],˙λi3(s)=HiWiF=Ti[λi1β(1WiEK)+μ3λi3],˙λi4(s)=HiGiM=Ti[λi3(1θ)ρbαWiEWiM(γ+WiM+αGiM)2+μ4λi4] (3.11)

    with

    {λN1(1)=0,λN2(1)=1,λN3(1)=1,λN4(1)=0,λi1(1)=λi+11(0),λi2(1)=λi+12(0),λi3(1)=λi+13(0),λi4(1)=λi+14(0),i=1,2,,N1. (3.12)

    Define

    xi(s)=(WiE(s),WiM(s),WiF(s),GiM(s))T,

    and from system (3.7) there is

    xi(0)=xi1(1)+(0,0,0,δi1)T,i=2,,N.

    From Theorems 4.1 and 4.2 in [41], we have

    Proposition 3. The gradients of the cost functional J1 with respect to δ and Γ are given by

    J1δ=(r0N1i=1δi)δ+Ni=1(λi(0))T(xi(0)δ)+10Ni=1Hi(xi(s),δ,Γ,λi(s))δ

    and

    J1Γ=(r0N1i=1δi)Γ+Ni=1(λi(0))T(xi(0)Γ)+10Ni=1Hi(xi(s),δ,Γ,λi(s))Γ,

    respectively.

    By straightforward calculation, we get

    Theorem 4. The gradients of the cost function J1(Γ,δ) with respect to the release timing Ti and release amount δl are given by

    TiJ1(Γ,δ)=10NjHj(s,WiE(s),WiM(s),WiF(s),GiM(s),Γ,δ)Tids=10{λi1(s)[βWiF(1WiEK)(ρ+μ1)WiE]+λi2(s)[θρWiEμ2WiM]+λi3(s)[(1θ)ρWiEbWiMγ+WiM+αGiMμ3WiF]λi4(s)μ4GiM(t)}ds (3.13)

    for i=1,2,,N, and

    δlJ1(Γ,δ)=r0+N1i(λi+1(0)T)ϕi(xi(1),δi)δlds=r0+(λl+11(0),λl+12(0),λl+13(0),λl+14(0))(0,0,0,1)T=r0+λl+14(0) (3.14)

    for l=1,2,,N1, respectively.

    In this subsection, we study a relatively simple but common scenario when sterile male mosquitoes are released into the field periodically with a fixed release amount. To determine an optimal release amount for this mode, we suppose that sterile mosquitoes are periodically released with a constant release amount δd in the limited time [0,T] and N1 times of releases are totally planed, that is to say, the release period is ω=TN. Then the limited-time control system is proposed as follows

    {dWE(t)dt=βWF(1WEK)(ρ+μ1)WE,dWM(t)dt=θρWEμ2WM,dWF(t)dt=(1θ)ρWEbWMγ+WM+αGMμ3WF,dGM(t)dt=μ4GM(t),}  tiω,t[0,T],WE(t+)=WE(t),WM(t+)=WM(t),WF(t+)=WF(t),GM(t+)=GM(t)+δd,}  t=iω,i=1,2,,N1 (3.15)

    with initial conditions Eq (3.2).

    Just like we did in section 3.1, we also give a constraint for the release amount δd

    0δlowδdδup, (3.16)

    where δlow and δup are given constants which represent the minimum and maximum amount that are allowed in each release.

    In this scenario, we can define the cost function for the control problem (P1) as follows

    ˉJ(δd)=WM(T)+WF(T)+r0(N1)δd. (3.17)

    Here, δd is the only control parameter. That is, we need to determine a release amount δd such that ˉJ(δd) is minimized over [δlow,δup].

    Apply the time rescaling technique again and let t=(i1)ω+sω for i=1,2,,N, then the system (3.15) with initial condition Eq (3.2) is converted into the following N subsystems

    {dWiE(s)ds=Fi1(s)=ω[βWiF(1WiEK)(ρ+μ1)WiE],dWiM(s)ds=Fi2(s)=ω[θρWiEμ2WiM],dWiF(s)ds=Fi3(s)=ω[(1θ)ρWiEbWiMγ+WiM+αGiMμ3WiF],dGiM(s)ds=Fi4(s)=ω[μ4GiM(t)],} s(0,1],i=1,2,,N,WiE(0)=Wi1E(1),WiM(0)=Wi1M(1),WiF(0)=Wi1F(1),GiM(0)=Gi1M(1)+δd,}  i=2,3,,N (3.18)

    with the same initial conditions Eq (3.8).

    Then the cost function Eq (3.17) can be redefined as

    \begin{equation} \mathfrak{\bar{J}}_1(\delta_d) = W_M^N(1)+W_F^N(1)+r_0(N-1)\delta_d \end{equation} (3.19)

    while the optimal control problem can be restated as: determine a \delta_d such that \mathfrak{\bar{J}}_1(\delta_d) is minimized over [\delta_{low}, \delta_{up}] .

    Then by similar discussion, we get the following result.

    Theorem 5. The gradient of \mathfrak{\bar{J}}_1(\delta_d) with respect to the release amount \delta_d is

    \begin{equation} \nabla\mathfrak{\bar{J}}_1(\delta_d) = r_0(N-1)+ \sum\limits_{i = 1}^{N-1} \lambda_4^{i+1}(0). \end{equation} (3.20)

    In this subsection, the release timings are added as new control parameters on the basis of the preceding scenario. That is, sterile mosquitoes are released at irregular moments 0\leq t_1\leq t_2\leq\cdots\leq t_{N-1}\leq T with a same release amount \delta_d . To find a set of optimal release timings and an optimal release amount for this case, the following limited-time control system is proposed

    \begin{equation} \left\{\begin{array}{ll} \left. \begin{array}{l} \frac{dW_E(t)}{dt} = \beta W_F (1-\frac{W_E}{K})-(\rho +\mu_1)W_E,\\ \frac{ dW_M(t)}{dt} = \theta \rho W_E-\mu_2W_M,\\ \frac{ dW_F(t)}{dt} = (1-\theta)\rho W_E\frac{b W_M}{\gamma +W_M+\alpha G_M}-\mu_3W_F,\\ \frac{ dG_M(t)}{dt} = -\mu_4G_M(t),\\ \end{array}\right\}\ \ t\neq t_i, t\in [0,T],\\ \left.\begin{array}{l} W_E(t^+) = W_E(t), W_M(t^+) = W_M(t), \\ W_F(t^+) = W_F(t), G_M(t^+) = G_M(t)+\delta_d, \\ \end{array}\right\}\ \ i = 1,2,\cdots, N-1 \end{array}\right. \end{equation} (3.21)

    with the same initial conditions listed in Eq (3.2). Besides, release timings t_i, i = 1, 2, \cdots, N-1 and release amount \delta_d meet the stated constraints Eqs (3.3) and (3.16), respectively.

    We define the cost function of control problem (P1) as follows

    \begin{equation} \mathcal{\hat{J}}(\Gamma, \delta_d) = W_M(T)+W_F(T)+r_0(N-1)\delta_d \end{equation} (3.22)

    with \Gamma = (T_1, T_2, \cdots\cdots, T_N)^T, T_i = t_i-t_{i-1} .

    Use the time rescaling technique and let t = \sum_{j = 1}^{i-1}T_j+T_i s for i = 1, 2, \cdots, N , then the system (3.21) is turned into

    \begin{equation} \left\{\begin{array}{ll} \left. \begin{array}{l} \frac{d W_E^i(s)}{ds} = F_1^i(s) = T_i[\beta W_F^i (1-\frac{W_E^i}{K})-(\rho +\mu_1)W_E^i],\\ \frac{ d W_M^i(s)}{ds} = F_2^i(s) = T_i[ \theta \rho W_E^i-\mu_2W_M^i],\\ \frac{ d W_F^i(s)}{ds} = F_3^i(s) = T_i[ (1-\theta)\rho W_E^i\frac{b W_M^i}{\gamma +W_M^i+\alpha G_M^i}-\mu_3W_F^i],\\ \frac{ d G_M^i(s)}{ds} = F_4^i(s) = T_i[ -\mu_4G_M^i(t)],\\ \end{array}\right\}\ s\in (0,1],\\ \left.\begin{array}{l} W_E^i(0) = W_E^{i-1}(1), W_M^i(0) = W_M^{i-1}(1),\\ W_F^i(0) = W_F^{i-1}(1), G_M^i(0) = G_M^{i-1}(1)+ \delta_d, \\ \end{array}\right\}\ \ i = 2,3,\cdots, N \end{array}\right. \end{equation} (3.23)

    with the same initial conditions listed in Eq (3.8).

    Accordingly, the cost function Eq (3.22) is transformed into an equivalent form

    \begin{equation} \mathfrak{\hat{J}}_1(\Gamma,\delta_d) = W_M^N(1)+W_F^N(1)+r_0(N-1)\delta_d. \end{equation} (3.24)

    Then we obtain the following result.

    Theorem 6. The gradients of \mathfrak{\hat{J}}_1(\Gamma, \delta_d) with respect to the release timing T_k and release amount \delta_d are given by

    \begin{equation} \nabla_{T_k}\mathfrak{\hat{J}}_1(\Gamma,\delta_d) = \nabla_{T_k}\mathfrak{J}_1(\Gamma,\delta),\quad k = 1,2,\cdots,N \end{equation} (3.25)

    and

    \begin{equation} \nabla_{\delta_d}\mathfrak{\hat{J}}_1(\Gamma,\delta_d) = r_0(N-1)+ \sum\limits_{i = 1}^{N-1} \lambda_4^{i+1}(0), \end{equation} (3.26)

    respectively. Here \nabla_{T_k}\mathfrak{J}_1(\Gamma, \delta) is the same as Eq (3.13).

    To determine the optimal values of the release timings and release amounts for the three limited-time optimal control problems in the preceding section, a series of numerical simulations are performed in this section. These three optimal release strategies will also be compared in different ways.

    Before proceeding further, we need to introduce the calculation method of the cost function and its gradients with respect to the control parameters in detail. This method has been used in [34] and [41], and we will explain it in the following by the case presented in Section 3.1.

    (ⅰ) We firstly solve straightforward the differential equations (3.7) with initial conditions Eq (3.8) to obtain W_E^i(s), W_M^i(s), W_F^i(s), G_M^i(s), s\in [0, 1] for i = 1, 2, \cdots, N .

    (ⅱ) Using W_E^i(s), W_M^i(s), W_F^i(s), G_M^i(s) obtained in last step, we first solve backwards the costate equations (3.11) with boundary conditions Eq (3.12) and obtain the costate variables \lambda_1^i(s), \lambda_2^i(s), \lambda_3^i(s) and \lambda_4^i(s) for i = N . Then we obtain the costate variables for i = N-1 in the same way, and continue until we obtain the costate variables for i = 1 .

    (ⅲ) Based on the expression in Eq (3.9), we compute the cost function \mathfrak{J}_1(\Gamma, \delta) by using W_M^N(s) and W_F^N(s) and release amounts \delta_i .

    (ⅳ) Applying W_E^i(s), W_M^i(s), W_F^i(s), G_M^i(s), \lambda_1^i(s), \lambda_2^i(s), \lambda_3^i(s) and \lambda_4^i(s) obtained in step (i) and (ii) , we calculate \nabla_{T_i}\mathfrak{J}_1(\Gamma, \delta) for i = 1, 2, \cdots, N and \nabla_{\delta_l}\mathfrak{J}_1(\Gamma, \delta) for l = 1, 2, \cdots, N-1 .

    Keep the parameter values in Eq (2.11) and consider the Allee effect in a relatively high level with \gamma = 200 . Besides, suppose that the average cost of rearing per unit of sterilizing mosquitoes r_0 = 0.01 . When doing numerical simulation, we measure the time in days and take 20 days as the total control time, that is to say, T = 20 , and 4 releases of sterilizing males are planed, then these 20 days should be divided into N = 5 parts according to different rules.

    In the following, we will study three different optimal strategies in impulsive control by numerical simulations. Surely there is no guarantee that the optimal solution we find numerically is unique, so we just present some optimal ones with special initial release periods and amounts. Specifically, with the help of nonlinear optimization-Matlab function library, we will find the optimal parameters by Matlab according to the objective function and the correlation gradient calculated in the above steps.

    Example 1. Optimal release amount for periodic releases

    It is obvious that the release period is \omega = \frac{T}{N} = 4 . Starting with a initial release amount \delta_d = 600 , if no optimal control is taken and only simple impulsive releases are employed, we can obtain that after five periods the cost value is \mathfrak{J}_0 = 371.9008 and the total fertile wild mosquito population is W_M(T)+W_F(T) = 347.9008 at T = 20 .

    Under the constraint 0\leq \delta_d \leq 1000 , we solve the corresponding optimal problem numerically by using the algorithm listed above in Matlab. We get an optimal release amount \delta^*_d = 559.54 and the corresponding cost value \mathfrak{J}^* = 371.8468 , while the total fertile wild mosquito population is W_M^*(T)+W_F^*(T) = 349.4652 at time T = 20 . We plot the time series diagrams of fertile wild mosquitoes for this kind of optimal control, non-control and simple impulsive control in Figure 6(a). By these three curves, we find that although the simple impulsive control has an obvious superiority in reducing fertile mosquito population (in most time of the control process, the total fertile wild mosquito population of the optimal one is a little higher than that of the simple impulsive control one), it cost more sterile mosquitoes to achieve such an effect (see Table 2).

    Figure 6.  (a) Comparisons of total fertile wild mosquitoes population under three cases; (b) Influence of the release amount on the objective function value and the total fertile wild mosquito population at time T.
    Table 2.  Comparison of the optimal amount control and the simple impulsive control.
    W_M(T) W_F(T) Total release Cost value
    Optimal control 213.3239 136.1413 2238.16 371.8468
    Impulsive control 213.2209 134.6799 2400 371.9008

     | Show Table
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    In addition, we investigate the influence of release amount on the cost function and the number of the fertile wild mosquito population at time T (see Figure 6(b)), and find that when the release amount varies in the interval 0\leq \delta_d \leq 1000 , the cost function \mathfrak{J}(\delta_d) admits a minimum point, which also verifies the optimum result we obtained above. Furthermore, we notice that the increase of the release amount leads to the population reduction of fertile wild mosquitoes at the terminal time T . However, the wild mosquito cannot be eliminated regardless of the releasing amount of sterile mosquitoes each time.

    Example 2. Optimal release timing for a fixed release amount

    To be consistent with Example 1, we choose the same initial release amount \delta_d = 600 and select T_1 = 2.5, T_2 = T_3 = T_4 = 4, T_5 = 5.5 as the initial release intervals. In order to determine optimal time intervals T_i and optimal releasing amount \delta_d which can minimize the cost function \mathfrak{J} , we consider constraint conditions

    \begin{equation} 0\leq T_i\leq 10, i = 1,2,\cdots,5, \quad\sum\limits_1^5 T_i = 20 \end{equation} (4.1)

    and 0\leq \delta_d\leq 1000 .

    Then by using the Matlab program, we solve this optimal problem numerically and obtain the following optimal release intervals

    \begin{equation} T^*_1 = 2.4793, T^*_2 = 3.9990, T^*_3 = 4.0044, T^*_4 = 4.0057, T^*_5 = 5.5117 \end{equation} (4.2)

    and an optimal release amount

    \begin{equation} \delta^*_d = 574.16. \end{equation} (4.3)

    In addition, we get the minimum cost value \mathfrak{J}^* = 371.0832 and the total fertile wild mosquito population W_M(T)+W_F(T) = 348.1168 at T = 20 .

    We plot the time series diagrams of fertile wild mosquitoes for the optimal release timing control, optimal release amount control and non-control in Figure 7(a). From these curves, we find that the optimal release timing control has a better control effect with a relatively low cost function value. Besides, for every \delta_d \in [0, 1000] , we solve the corresponding optimal time intervals under the restriction Eq (4.1) and then calculate the value of cost function and the amount of total fertile wild mosquitoes at T = 20 . We find that, see Figure 7(b), when the release amount varies in the interval 0\leq \delta_d \leq 1000 , the cost function \mathfrak{J}(\Gamma, \delta_d) also admits a minimum point, which agrees with the optimal values we have obtained. Similarly, the increase of the release amount reduces the fertile wild mosquito population, but the wild mosquitoes cannot be eliminated even if the release amount reaches the upper constrained bound and the cost value is very high.

    Figure 7.  (a) Comparisons of total fertile wild mosquitoes population under different biological controls; (b) Influence of the release amount on the objective function value and the total fertile wild mosquito population at time T.

    Example 3. Optimal release timing and release amounts

    Keep the initial release intervals T_1 = 2.5, T_2 = T_3 = T_4 = 4, T_5 = 5.5 and choose the initial release amounts \delta_1 = \delta_2 = \delta_3 = \delta_4 = 600 , we deal with the optimal problem with constraints Eq (4.1) and 0\leq \delta_i\leq 1000, i = 1, 2, 3, 4 . Solving this optimal problem numerically in Matlab, we obtain the optimal release amounts

    \begin{equation} \delta^*_1 = 584.84, \delta^*_2 = 584.84, \delta^*_3 = 584.84, \delta^*_4 = 584.82 \end{equation} (4.4)

    and the optimal release intervals

    \begin{equation} T^*_1 = 2.4517, T^*_2 = 3.9976, T^*_3 = 4.0102, T^*_4 = 4.0133, T^*_5 = 5.5272. \end{equation} (4.5)

    This release strategy is showed in Figure 8(b). Besides, we obtain the minimum cost value \mathfrak{J}^* = 371.0803 and the total fertile wild mosquito population W_M(T)+W_F(T) = 347.6869 at T = 20 . The time series diagrams of the number of total fertile wild mosquitoes under four types of control modes are plotted in Figure 8(a) from which we can see that mixed optimal control produces the best control effect.

    Figure 8.  (a) Comparisons of total fertile wild mosquitoes population under different biological controls; (b) Release strategy of the mixed optimal control.

    Finally, we compare these three optimal release strategies (refer to Table 3 and Figure 9). We find that the optimal release timing control is superior to the optimal release amount control, while the mixed control produces the best integrated control effect since the lowest fertile wild mosquito level is reached at the minimal cost function value. From Figure 9(b), we also see that the mixed optimal control releases the most sterile mosquitoes in the whole control process, but its cost function value is not large because the least wild mosquitoes stayed in the field at the terminal time. Although it can be seen from Figure 9(b) that the largest accumulated amount of sterile mosquitoes are released during the whole control process in the mixed optimal control, its cost function value is the smallest due to the smallest final population size of wild mosquitoes in the field.

    Table 3.  Comparison of different release strategies.
    Optimal control parameters \mathfrak{J}^* W_M^*(T)+W_M^*(T)
    Amount control \delta^*_d=559.54 371.8468 349.4652
    T^*_1=2.4793 , T^*_2= 3.9990 ,
    Timing control T^*_3= 4.0044 , T^*_4=4.0057 , 371.0832 348.1168
    T^*_5= 5.5117 , \delta^*_d= 574.16
    T^*_1=2.4517 , T^*_2= 3.9976 ,
    T^*_3= 4.0102 , T^*_4=4.0133 ,
    Mixed control T^*_5= 5.5272 , \delta^*_1= 584.84 371.0803 347.6869
    \delta^*_2= 584.84 , \delta^*_3= 584.84
    \delta^*_4= 584.82

     | Show Table
    DownLoad: CSV
    Figure 9.  (a) Comparisons of three release strategies: the red, blue and green segments are for amount control, timing control and mixed control, respectively; (b) Comparisons of total release amounts of sterile mosquitoes for three optimal control methods.

    The SIT has been a hot topic in the research field of mosquito-borne infectious disease control in recent years, and lots of field trails have been conducted all over the world and large numbers of researchers have been devoting themselves in this area and have already achieved many progresses. Mathematical model, as an important tool, plays a significant role in the research process. However, most of these models in previous studies are constructed by continuous or discrete dynamical systems, which cannot describe the release process accurately. Most works focused on the asymptotic behavior of the system in infinite time. However, the control of mosquitoes in most cases should be a shorter-term action.

    Release of sterile mosquitoes has been used to reduce or eliminate the wild mosquito population in order to control vector-borne infectious diseases. In this paper, we proposed and studied a stage-structured two-sex mosquito population model with an Allee effect and impulsive releases of sterile males. By adjusting different types of control parameters, both large-scale time control and limited-time optimal control of wild mosquitoes were investigated.

    We firstly studied the large-scale time control aiming to wipe out wild mosquitoes. By using the monotone system theory and the comparison theorem, we showed the existence, uniqueness and globally stability of the wild mosquito-free periodic solution. For fixed release period \omega^* (or release amount \delta^* ), we established threshold value for release amount \tilde{\delta} (or release period \tilde{\omega} ) which determines the extinction or persistence of the wild mosquito population.

    Then for the limited-time optimal control of wild mosquitoes, we took into account both of the population control level of wild mosquitoes and the economic cost, and investigated three different release tactics: optimal release amount for periodic releases, optimal release timing for a fixed release amount and a combination of optimal release timing and release amounts. A time rescaling technique was applied to overcome the technical difficulty that the state of variables depends on uncertain pulse effects. We obtained the optimal release amounts and release timings numerically for each release strategy. Numerical simulations indicate that the optimal release timing control is a more effective strategy than the optimal release amount control. However, simultaneous optimal selection of release amount and release timing leads to the best control performance.

    In the limited-time control, we construct cost functions by referring to the pest control in agriculture and only focus on terminal control but ignore process control. And in our future work, we will consider both terminal control and process control in the limited time control, so as to ensure that the number of wild mosquitoes cannot be too large during the control process.

    This work is supported by the National Natural Science Foundation of China (12071407, 11901502 and 11871415), Training plan for young backbone teachers in Henan Province (2019GGJS157), Foundation of Henan Educational Committee under Contract (21A110022), Program for Science & Technology Innovation Talents in Universities of Henan Province (21HASTIT026), Scientific and Technological Key Projects of Henan Province (212102110025), Nanhu Scholars Program of XYNU and Nanhu Scholars Program for Young Scholars of XYNU.

    The authors declare that they have no competing interests.

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