#### Abstract

Sterile insect technique has been successfully applied in the control of agricultural pests; however, it has a limited ability to control mosquitoes. A promising alternative approach is the Trojan Y Chromosome strategy, which works by manipulating the sex ratio of a population through the introduction of YY supermales that guarantee male offspring. To take the advantages of both approaches, a combined Trojan Y chromosome strategy and sterile insect technique (TYC-SIT) strategy considering intraspecies competition is modeled. The pure TYC method is compared with the pure SIT method by cancelling one-state variable. The dynamical analysis leads to results on both local and global stabilities of this combined TYC-SIT model. Optimal control analysis is also implemented to investigate the optimal mechanisms for extinction of mosquitoes. In particular, the numerical results affirm that the combined TYC-SIT enables near elimination of mosquitoes and works better than the pure TYC or pure SIT method. These conclusions have great significance for species controls with an XX-XY sex determinism system or ZZ-ZW system.

#### 1. Introduction

Mosquito-borne diseases are transmitted by mosquitoes infected with viruses, such as Zika virus, yellow fever virus, West Nile fever virus, and dengue fever virus [1, 2]. The spread of mosquito-borne virus in humans is mainly through the bite of mosquitoes infected with the virus such as Anopheles sinensis, Anopheles anthropophagus, Aedes aegypti, Aedes albopictus, and Culex mosquitoes [3–5]. Frequent outbreaks of mosquito-borne diseases in tropical and subtropical regions have caused large numbers of deaths and economic losses. Chemical treatments such as pesticides have been implemented for many years to eradicate mosquitoes. Heavy spraying of insecticides has effectively suppressed mosquito density. However, environmental problems caused by excessive use of pesticides [6], insecticide resistance [7, 8], and combined with the lack of vaccines [9, 10], have called for alternative environment-friendly and sustainable approaches [11], such as radiation-based sterile insect technique (SIT) and Trojan Y chromosome strategy (TYC).

SIT works by releasing radiation-sterilized males to an existing population, to mate with wild females so that they have no viable offspring [11, 12]. SIT has been successfully applied in the control of several agricultural pests such as invasive fruit flies, lepidopteran, and Hemiptera [13–17]; however, it has a limited ability to control mosquitoes due to the three issues. (1) It requires introducing large quantities of sterile mosquitoes. (2) The health of sterile male mosquitoes is significantly impaired, thus leading to the decline in mating competitiveness. (3) Mosquitoes are very prolific; the population may be supported even with a low mating ratio of normal females to normal males [18–21].

A promising approach that has been put forward for eliminating mosquitoes is TYC, in which supermales (containing two Y chromosomes) are released into the field to mate with wild females, resulting in a sharp sex imbalance in the offspring [22–25]. Figure 1 illustrates that an equal proportion of females and males are produced in the wild; however, the offspring is guaranteed to be male when a natural female mates with a YY supermale. The gradual reduction in females and increase in males may lead to eventual extinction of the targeted population after several generations [26]. The advantages of TYC strategy are as follows. (1) It only requires to recombine sex chromosomes of mosquitoes instead of changing their genes. (2) Only a fixed number of modified mosquitoes are required. (3) The intensity of this strategy can be controlled.

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However, it is expensive to design and produce YY supermales, that is why TYC is not widely used in controlling mosquitoes or agricultural pests [27, 28]. In this study, a combined TYC and SIT model is proposed to take advantages of both approaches. Intraspecies competition is especially considered, which is omitted in many literature. By setting different values to the release rate of radiation-based sterile males and the release rate of supermales with YY chromosome structure, we can switch to a pure SIT model or a pure TYC model. To the best of our knowledge, it is the first mathematical investigation to compare the combined SIT-TYC to TYC or SIT.

#### 2. Materials and Methods

##### 2.1. Mathematical Modelling

The population of mosquitoes in the wild tends to be numerous because their reproduction rates are high and matings occur constantly; therefore, continuous models of ordinary differential equations (ODEs) can be established to describe the population dynamics of mosquitoes [29]. Parameters will be used in this work and are firstly explained in Table 1.

The combined TYC-SIT model, in which both YY supermales and radiation-based sterile males are introduced, is described by a system of four ODEs for state variables: a wild-type XX female , a wild-type XY male , a sterile male , and a YY supermale . Half female and half male offspring are produced in the wild; however, only males can be produced if a wild female mates with a YY supermale, and no viable offspring can be produced if mates with a sterile male. Thus, the set of equations that describes the system iswhere define the number of individuals in each associated class, and

The intraspecies competition for female mates caused by the introduction of YY supermales and radiation-based sterile males is considered and modeled as

If , that is, , it implies that there is no mating pressure on wild males. Obviously,

The larger the value of is, the more competitive wild males are. Similarly, the larger the value of is, the more competitive YY supermales are.

##### 2.2. Equilibrium and Stability Analysis

The stability of the TYC-SIT model (1) is now investigated. There is one equilibrium, , on the boundary. It is clear is global stable. To get the positive interior equilibrium explicitly, i.e., , it is equivalent to solve the following equations:

By solving equations (5), , and can be obtained as follows:

And is the solution of

Theorem 1. *Let . The interior equilibrium is locally asymptotically stable if*

*Proof. *The Jacobian matrix of model (1) about is given aswhereThe corresponding characteristic equation iswhereAccording to Routh–Hurwitz criterion, is locally asymptotically stable if

Theorem 2. *In the case of , the trivial equilibrium of TYC-SIT model (1) is globally asymptotically stable if .*

*Proof. *Consider the Lyapunov function . Note that due to the positivity of its solutions and is radially unbounded. It is left to show that for all . Taking the derivative of about yieldsTherefore, we only need to show to get global stability, and this requires , which completes the proof.

##### 2.3. Optimal Control Analysis

The goal in this section is to investigate the mechanisms in TYC-SIT system to lead to an optimal level of female mosquitoes density. We assume that the influx of sterile males and YY supermales and are not known a priori and enter them into the system as time-dependent controls and . In fact, male mosquitoes do not bite human and only female mosquitoes bite and spread diseases [30–32], which implies that we do not have to kill all mosquitoes, eliminating females is enough instead. Furthermore, we also hope the production of sterile males and YY supermales is minimized. Herein, the following objective function is chosen:where subject to the governing equation (1).

Optimal strategies are derived from the objective function, where the female population is minimized and also the introduction of both radiation-based sterile males and supermales with YY chromosome structure are minimized. We search for the optimal controls within the set , which is given by

The goal is to find the optimal controls such that

Theorem 3. *There exists the optimal controls and for the minization problem (17).*

*Proof. *It is obvious that is concave in the argument and ; all state variables are bounded by the carring capacity , and the control variables and are also assumed bounded due to the cost of releasing sterile males and YY supermales; therefore, the existence of for minimization problem (16) is guaranteed based on [33–35].

Theorem 4. *An optimal control of system (1) that minimizes the objective function is characterized by*

*Proof. *Here, Pontryagin’s minimum principle is used to derive the necessary conditions on this problem. The Hamiltonian in this problem isThe Hamiltonian is used to find the adjoint functions :To find the optimal , minimize pointwise, that is,Note that cancels with 2 which comes from the square of the controls and . Furthermore, the problem is indeed minimization asHence, the optimal solutions areA compact way of writing the optimal control isNow, the proof is completed.

#### 3. Numerical Simulations

The numerical simulations are investigated by MATLAB R2019b with the values of initial conditions and parameters shown in Table 2. The ode15s solver was used to get numerical solutions of the combined TYC-SIT system. The TOMLAB Base Module and TOMLAB/SNOPT are also used to solve the optimal control problems of our dynamic systems.

The combined TYC-SIT system is modeled, and the initial conditions and parameters in Table 2 are utilized to observe the relative population decline of females in response to the addition of the radiation-based sterile males and YY supermales.

Under the condition of , that is, sterile males or YY supermales are added to the population only once at time and no additional will be introduced, it is observed that no matter how large the initial introduction of sterile males or YY supermales are, the system of the combined TYC-SIT cannot achieve extinction and instead leads to an equilibrium state. As shown in Figure 2, the population did decline for some time with large enough (i.e., purple/green star line) influx of sterile males or YY supermales; however, the population recovers soon and reaches the equilibrium state at approximately 172, which can be also calculated from equations (5) and (6). Extinction can occur with continuous introducing modified males; two examples are provided in Figure 3.

To further validate the effectiveness of the proposed TYC-SIT model, comparative analysis among the pure TYC, the pure SIT, and the combined TYC-SIT is carried out.

If we solely continue adding sterile males (i.e., ) or solely continue adding YY supermales (i.e., ) after the initial introduction of the both, how many modified males would be introduced has a great impact on the population decline of mosquitoes. With a relatively low influx of modified males, purely introducing sterile males or YY supermales cannot drive the population to extinction; however, the effectiveness of the continuous introduction of YY supermales is better because the population drops much more, see part (a) in Figure 4. As or is increased to 10, the initial decline rate with is faster than , but then the situation is reversed after the threshold point in Figure 4 part (b). As we can see in part (c) and (d) in Figure 4, the population decline rate with either or is similar if the influx of modified males is large enough.

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The left panel of Figure 5 shows the female density change varies with time under different influx of sterile males, and the right panel shows the female density change varies with time under different influx of YY supermales. Identical parameters are used to compare the effectiveness of SIT and TYC. TYC is much more effective with low influx of modified males. Furthermore, combined TYC and SIT requires less input of modified males, as shown in Figure 6.

Considering the cost of the production of sterile males and YY supermales, optimal strategies and its corresponding optimal states have also been numerically simulated to compare the effects of the combined TYC-SIT strategy with the pure SIT and the pure TYC strategy, see Figures 7–9. The population can be driven to extinction with an optimal time controls or by using either SIT , or TYC , or the combined TYC-SIT approach.

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Figure 9 shows the optimal eradication strategy of the combined TYC-STT model requires to introduce YY supermales at 20% of the initial female population (or 5% of the carrying capacity) for a short time to bring the population of females below some threshold, then gradually drops to a relatively low level after introducing sterile males at time , and finally turns off until the entire population vanishes. However, both pure TYC and pure SIT require to release at 30% of the initial female population, as shown in Figures 7 and 8. In overall, the combined TYC-SIT approach works better in eradicating mosquitoes, which reduces the whole releasing rate by 33%.

#### 4. Conclusions and Discussion

In this study, we established a mathematical model of the combined TYC and SIT approaches. The population was divided by the following four state variables: wild female mosquitoes , wild male mosquitoes , radiation-based sterile males , and YY supermales . Six parameters, the birth coefficient , the death coefficient , the logistic term *L*, the carrying capacity K, the influx of sterile males , and the influx of sterile males , are included. The intraspecies competition for female mates caused by the introduction of modified male mosquitoes is especially considered, which is omitted in many studies. By setting or , pure SIT is compared with pure TYC strategy. If only a single method is considered, the TYC strategy is better than SIT with low influx of modified males, and the situation is reversed with high influx of modified males. The dynamical analysis and optimal control analysis of the combined TYC-SIT model in this study are important to understand the efficiency of the combined strategy to eliminate mosquitoes. Optimal control analysis also provides the optimal release rates of YY supermales and sterile males. In particular, the comparative studies confirm that the combined TYC-SIT approach can indeed eliminate mosquitoes and perform better than pure TYC or pure SIT.

The combined TYC-SIT approach is a safe biological control method because it has the advantage of being species-specific and does not harm other nontarget species. In addition, older generations would not transfer genetically engineered genes to their offspring. The intensity of this method is also controllable because the number of radiation-based sterile males or supermales with YY chromosome structure to be released to the targeted population can be determined by the executor. Unlike other strategies, the combined TYC-SIT does not rely on eliminating all wild matings to affect the total population. Instead, it depends on the number of wild females dwindling over a period of several generation cycles. The results of this research are of great significance for the biological control of mosquitoes.

Besides mosquitoes, the combined TYC-SIT strategy can also be applied to eradicate species with an XX-XY sex determinism system or ZZ-ZW system. Our results will help experts devise suitable management strategies and measures to control invasive species or agricultural pests.

#### Abbreviations

SIT: | Sterile insect technique |

TYC: | Trojan Y chromosome strategy |

ODEs: | Ordinary differential equations. |

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Authors’ Contributions

J.L., C.K., W.S., and M.G conceptualized the study; J.L. carried out formal analysis; J.L. and C.K. investigated the study; J.L. prepared and wrote the original draft; C.K., M.G., and W.S. reviewed and edited; J.L., C.K., and W.S. administrated the project; J.L. and M.G. carried out funding acquisition All authors have read and agreed to the published version of the manuscript.

#### Acknowledgments

The authors thank Dr. Chuan He for his support throughout this work. This research was funded by the Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province (Grant no. MSSB-2021-09), and Talent Initiation Program of Chengdu University (Grant no. 2081921002).