SUPPRESSING TACTICS BY OPTIMAL STERILE RELEASE PROGRAM AGAINST WILD MOSQUITOES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The sterile insect technique (SIT) offers a promising strategy for reducing wild mosquitoes in nature. But many factors cause incomplete sterility in a sterile release program. This article develops a mosquitoes model involving pulsed releasing sterile mosquitoes with gender structure and partial residual fecundity. Firstly, we devote to eliminate the wild mosquitoes by chronical period control. Next the impulsive control tactics with alterable pulse times and releasing amounts in finite time restraining wild mosquitoes are established. To acquire optimal strategy, the corresponding impulsive optimal problem is translated into continuous parameter selection problem by means of the transformation of time-scaling and time translation. Lastly, simulations based on gradient descent and genetic algorithm are given. Results indicate that pulsed releasing sterile mosquitoes greatly suppress the wild mosquitoes. Moreover, hybrid control is superior to the amount control and periodic impulsive control.


INTRODUCTION
Sterile insect technique (SIT), in which radiation-sterilized males are released into the field to mate with wild females thereby preventing them from producing viable offspring [1,2], as successfully restrained field insect species [3]. Recently, SIT has been used against mosquitoes and the corresponding mathematical model are formulated to simulate the evolution of mosquitoes [4,5,6,7,8]. In these models, the released sterile males are suppose to be 100% sterile. Practically, complete sterilization cannot be achieved for some objectiv reasons (technical problems, such as lower irradiation doses, environmental conditions or others) [9]. So it is essential to study partial sterility in the control process.
Inspired by a Aedes albopictus model with the partial residual fecundity [1,10,11], we firstly study the perennial and periodic impulsive releasing sterile males strategy. Then due to the high cost of breeding and experimenting with mosquitoes in captivity, we research the short-term and optimal impulsive releasing tactics. Denoting by M(t) the wild male, by F(t) female, and by S(t) sterile male, P.A. Bliman in [1] proposed SIT models as following where µ M , µ F and µ S represent mortalities of M(t), F(t) and S(t), respectively. r is the sex ratio; ρ is the average number of eggs laid per female mosquito per day; Λ is the sustained release of sterile males; β is the competitive effect between the mosquitoes; ε is the residual fertility with 0 ≤ ε < 1 and ε = 0 meaning the complete sterility; γ is the competitiveness index of sterile males with 0 < γ ≤ 1 while γ = 1 meaning sterile males have the equally competitive with wild males.
Considering the basic number of offspring for both female and male populations, i.e. the average number of wild offspring that a mosquito produce per day, we obtain The outline of the paper is organized as follows. In Section 2, the wild mosquitoes control model with periodic pulse releasing sterile males are established and analysed; In Section 3, the wild mosquitoes control model with alterable pulse times and releasing amounts are built and the corresponding impulsive optimal problem are constituted and translated into continuous parameter selection problem; In Section 4, simulations and optimal control strategies based on gradient descent and genetic algorithm are given; Finally, a simple conclusion is list.

MODEL WITH PERIODIC PULSE CONTROLS
In the above model (1), converting continuous releases into pulsed releases leads to a new interaction model for mosquito populations in terms of sex structure: (3) Suppose that τ and Λ denote the period and amount of releasing sterile male mosquitoes, respectively.
Let X(t) = (M(t), F(t), S(t)) T is an arbitrary solution of (3). Then X(t) is continuous between every two adjacent pulses and X(iτ + ) = lim ε→0 X(iτ + ε)(ε > 0) exists. Thus, for the smooth characteristics of the functions in the first three equations, the existence and uniqueness of the solution of (3) are guaranteed. Referring to [1,7], we infer to the positivity and boundedness of the solution of system (3), as well as the existence and global asymptotic stability of bounded periodic solutions of the system (3).

Theorem 2.1
The solution X(t) of the system (3) with non-negative initial conditions is always non-negative, and there exists a constant K > 0 such that each component of X(t) is less than K for t ≥ 0.

OPTIMAL MOSQUITOES CONTROL BY IMPULSIVE RELEASING STERILE MALES
The stable wild mosquito population-eradication periodic solution means that the sterile males can suppress effectively wild mosquitos under impulsive controls. However, we have not considered the cost of various controls which is an important problem in practice. Hence, we devote this section to minimizing the amount of wild mosquitos at the terminal time with minimum control cost against fertile mosquitoes by impulsive releasing sterile mosquitoes. The optimal control problem can be solved by the time-scale conversion technology, that is, the time range is divided into several sub-intervals [12]. Then, the original switching time point is converted into the pre-fixed switching time point of the extended system. Therefore, the optimal problem is the same as the optimal parameter selection [13].
Assume that the amount of sterile males Λ i is released at t i in finite time [0, T ] for i = 1, 2, · · · n − 1. Then we optimize the releasing time and amount of sterile insects which is called hybrid control strategy. Taking the releasing interval and the amount as controlled variables to be controlled, then a finite-time control system is proposed as follows with initial conditions time t i and amount of the sterile insect Λ i satisfies: where a i , b i , c i and d i are given and non-negative constants.
Using the transformation of time scaling and time translation [8], the problem (P1) can be transformed into an equivalent optimal parameter selection problem, which can then be solved based on the gradient. Define an indicator function of I: Then construct the transformation from t ∈ [0, T ] to s ∈ [0, n] [8,14] as follows which maps the releasing moment 0, to the time point s = 0, 1, · · · , n. So system (5) reduces to with initial conditions (6).
However, this problem is still difficult to solve. So we will introduce time translation transformation. For i = 1, . . . , n, define Then, system (10) and (11) as well as initial condition (6) are reduced to Then cost function (12) and problem (P1) becomes respectively P1) Given the system (14) with the initial condition (15), find the parameter vector pair (τ, Λ) meeting (7) and (8) such that the cost function (16) is the corresponding costate determined by the following costate equationṡ with boundary conditions Hence, according to the system (14), for i = 2, . . . , n we get
Specially, if sterile insects are released n times at regular intervals τ on [0, T ], then nτ = T .
Thus the released amounts of sterile insects Λ i meeting constraint (8) are optimally selected. So we get a amount control model: with initial conditions (6). Our optimal control problem can now be formulated: (P2) Given a system (21), under the initial condition (6), find a vector of release parameters for sterile insects Λ fulfilling (8) such that the cost function Similarly, after the time scaling transform and time translation, system (21) reduces to and (15). And the ultimate cost function is According the definition of Hamiltonian function in (17), then the costate equations with boundry conditions and the gradient of (24) on Λ k are obtained.

COMPUTE AND SIMULATION
The genetic algorithm (GA) is a search technique used in computing to find exact or approximate solutions to optimization and search problems. Nowadays the toolbox of genetic algorithm is used in Matlab to solve the continuous optimization problems. In this section, we will apply gradient descent (GD) and genetic algorithm (GA) to solve the optimal control strategy for suppressing wild mosquitoes.
Some parameters derive from reference [2] in which sterile mosquitoes were released by an open-release field trial in Guangzhou with the highest dengue transmission rate in China. In our simulation, taking days as the unit of time, the total optimal control time is 63 days, that is, T = 63. Assume that the sterile mosquitoes are released N − 1 = 8 times in variable amounts.
Take the production and release cost of a sterile mosquito as c = 0.0002. Other parameter values shown in Table 1 derive from [1]. Then, assuming that sterile mosquitoes are free,

Numerical simulation of gradient descent (GD).
Referring to the algorithm in [8,16], the optimization strategies were solved numerically using gradient descent method on the Matlab.  which are described by Figure 1

Simulation 2. Optimal release amount at fixed moment
We take the same initial conditions as the optimized control above, then obtain the optimal release amount Λ * = (1. which is displayed in Figure 1  Comparison of two optimal control strategies shows that the hybrid control is more advantageous. In addition, it is found that wild mosquito populations continue to decline as the amount of releases increases. However, even if the release reaches the upper limit, the wild mosquitoes cannot be eliminated either in the short term.

Numerical simulation of genetic algorithm(GA).
Choosing the same initial conditions as the above simulation, we use the genetic algorithm to solve two optimal control strategies. Then for the hybrid control, optimal release time interval and amount respectively are τ * = (1, 1, 1, 10, 10, 10, 10, 10, 10) and Λ * = (1. Further, for the amount control, optimal release amount is The corresponding results of both algorithms are list in the Table 2. Our numerical simulations show that impulsive controls radically reduce the level of wild mosquitos. Futhermore, the results of both optimization algorithms are similar and the result of hybrid control is superior to the amount control and periodic impulsive control in terms of the minimal objective function, the lower releasing amount and the lesser wild mosquitos at the terminal time. Even so, there is no significant difference among various impulsive control measures in which sterile mosquitoes are released eight times.

CONCLUSION
SIT is an environmentally friendly and species-specific control tool for suppressing Aedes albopictus. According to Theorem 3, wild mosquitoes will become extinct after chronical SIT control. But it is not realistic so that the impulsive control tactics with alterable pulse times and releasing amounts in finite time restraining wild mosquitoes are established. The corresponding impulsive optimal problem are constituted and translated into continuous parameter selection problem by means of the transformation of time-scaling and time translation. Then simulations and optimal control strategies based on gradient descent and genetic algorithm are given. Simulations indicate that hybrid control is superior to the amount control and periodic impulsive control in terms of the objective function, the total release amount and the amount of wild mosquitos at the terminal time.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.