# Combined Trojan Y Chromosome Strategy and Sterile Insect Technique to Eliminate Mosquitoes: Modelling and Analysis

Authors: Jingjing Lyu; Musong Gu; Sheng Wang; Kun Cheng
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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.Figure 1
Mating pedigrees of TYC. (a) Mating of a wild-type XX femaleFXX and a wild-type XY male MXY. (b) Mating of a wild-type XX female FXX and a YY supermale MYY. Note. We called YY male as supermale.
(a)(b)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.Table 1
List of parameters. All parameters are nonnegative.
ParameterDescriptionβBirth coefficient proportional to the viability of progenyδDeath coefficient proportional to the death from predators and illnessLLogistic term (to limit the size of the population)KCarrying capacity of the ecosystemμ1Constant influx of radiation-based sterile malesμ2Constant influx of YY supermalesThe 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 femalef, a wild-type XY male m, a sterile male s1, and a YY supermale s2. 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 is(1)dfdt=12fmβLmm+s1+s2−δf,dmdt=12fmβLmm+s1+s2+fs2βLs2m+s1+s2−δm,ds1dt=μ1−δs1,ds2dt=μ2−δs2,where f,m,s1,ands2 define the number of individuals in each associated class, and(2)L=1−f+m+s1+s2K.The intraspecies competition for female mates caused by the introduction of YY supermales and radiation-based sterile males is considered and modeled as(3)mm+s1+s2,s2m+s1+s2.Ifs1=s2=0, that is, m/m+s1+s2=1, it implies that there is no mating pressure on wild males. Obviously,(4)0≤mm+s1+s2≤1,0≤s2m+s1+s2≤1.The larger the value ofm/m+s1+s2 is, the more competitive wild males are. Similarly, the larger the value of s2/m+s1+s2 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, 0,0,μ1/δ,μ2/δ, on the boundary. It is clear 0,0,μ1/δ,μ2/δ is global stable. To get the positive interior equilibrium explicitly, i.e., f∗,m∗,s1∗,s2∗, it is equivalent to solve the following equations:(5)12f∗m∗βL∗m∗m∗+s1∗+s2∗−δf∗=0,12f∗m∗βL∗m∗m∗+s1∗+s2∗+fs2∗βL∗s2∗m∗+s1∗+s2∗−δm∗=0,μ1−δs1∗=0,μ2−δs2∗=0,L∗=1−f∗+m∗+s1∗+s2∗K.By solving equations (5), f∗,s1∗, and s2∗ can be obtained as follows:(6)f∗=K−m∗+s1∗+s2∗−2δKm∗+s1∗+s2∗m∗2β,s1∗=μ1δ,s2∗=μ2δ.Andm∗ is the solution of(7)2βm∗5+βs1+s2−Km∗4+2Kδ+βs22m∗3+2δKs1+s2+2βs22s1+s2−1m∗2+4δK∗s22m∗+4Kδs22s1+s2=0.Theorem 1.
Letμ>0. The interior equilibrium f∗,m∗,s1∗,s2∗ is locally asymptotically stable if(8)R1>0,R3>0,R4>0,R1R2R3>R32+R12R4.Proof.
The Jacobian matrix of model (1) about f∗,m∗,s1∗,s2∗ is given as(9)J=J11J12J13J14J21J22J23J2400J330000J44,where(10)J11=βKm∗2L∗−βf∗m∗22Km∗+s1∗+s2∗−δ,J12=2KL∗−m∗βf∗m∗2Km∗+s1∗+s2∗−βf∗m∗2L∗2∗m∗+s1∗+s2∗2,J13=−βf∗m∗2L∗2m∗+s1∗+s2∗2−βf∗m∗22Km∗+s1∗+s2∗,J14=−βf∗m∗2L∗2m∗+s1∗+s2∗2−βf∗m∗22Km∗+s1∗+s2∗,J21=βL∗m∗2+2s2∗22m∗+s1∗+s2∗−βf∗m∗2+2s2∗22Km∗+s1∗+s2∗,J22=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2+βf∗2Km∗L∗−m∗2−2s2∗22Km∗+s1∗+s2∗−δ,J23=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2−b∗fm∗2+2s2∗22Km∗+s1∗+s2∗,J24=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2+βf∗4Ks2∗L∗−m∗2−2s2∗2Km∗+s1∗+s2∗,J33=−δ,J44=−δ.
The corresponding characteristic equation is(11)λ4+R1λ3+R2λ2+R3λ+R4=0,where(12)R1=−J11+J22+J33+J44,R2=J11J22+J11J33+J11J44+J22J33+J22J44+J33J44−J12J21,R3=J33+J44J12J21−J11+J22J33J44,R4=J11J22J33J44−J12J21J33J44.
According to Routh–Hurwitz criterion,f∗,m∗,s1∗,s2∗ is locally asymptotically stable if(13)R1>0,R3>0,R4>0,R1R2R3>R32+R12R4.Theorem 2.
In the case ofμ1=μ2=0, the trivial equilibrium 0,0,0,0 of TYC-SIT model (1) is globally asymptotically stable if β<δ/K.Proof.
Consider the Lyapunov functionVf,m,s1,s2=f+m+s1+s2. Note that Vf,m,s1,s2≥0 due to the positivity of its solutions and Vf,m,s1,s2 is radially unbounded. It is left to show that V′f,m,s1,s2<0 for all f,m,s1,s2≠0,0,0,0. Taking the derivative of Vf,m,s1,s2 about t yields(14)V′f,m,s1,s2=dVdt=dfdt+dmdt+ds1dt+ds2dt=fmβLmm+s1+s2+fs2βLs2m+s1+s2−δf−δm−δs1−δs2≤fmβ+fs2β−δf+m+s1+s2sinceL≤1,mm+s1+s2≤1ands2m+s1+s2≤1≤fmβ+fs2β−δm+s2=fβm+s2−δm+s2≤Kβm+s2−δm+s2≤Kβ−δm+s2.
Therefore, we only need to showKβ−δ<0 to get global stability, and this requires β<δ/K, 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 supermalesμ1 and μ2 are not known a priori and enter them into the system as time-dependent controls μ1t and μ2t. 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:(15)Objμ1t,μ2t=∫0T12f2+12μ1t2+12μ2t2dt,where f,m,s1,ands2 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 setU, which is given by(16)U=μ1,μ2μ1,μ2are measurable,0≤μ1t,0≤μ2t,t∈0,150.The goal is to find the optimal controlsμ1∗t,μ2∗t such that(17)Objμ1∗t,μ2∗t=minμ1t,μ2t∫0T12f2+12μ1t2+12μ2t2dt.Theorem 3.
There exists the optimal controlsμ1∗t and μ2∗t for the minization problem (17).Proof.
It is obvious thatObjμ1∗t,μ2∗t is concave in the argument μ1 and μ2; all state variables f,m,s1,ands2 are bounded by the carring capacity K, and the control variables μ1 and μ2 are also assumed bounded due to the cost of releasing sterile males and YY supermales; therefore, the existence of μ1∗t,μ2∗t for minimization problem (16) is guaranteed based on [33–35].Theorem 4.
An optimal controlμ1∗t,μ2∗t∈U of system (1) that minimizes the objective function Obj is characterized by(18)μ1∗t,μ2∗t=maxt0,−λ3,maxt0,−λ4.Proof.
Here, Pontryagin’s minimum principle is used to derive the necessary conditions on this problem. The HamiltonianH in this problem is(19)H=12f2+12μ1t2+12μ2t2+λ1f′+λ2m′+λ3s1′+λ4s2′.
The Hamiltonian is used to find the adjoint functionsλi,i=1,2,3,4:(20)λ1′=−∂H∂f=λ1δ+βm2f−KL2Km+s1+s2−λ2βLm2+2s222m+s1+s2+βfm2+2s222Km+s1+s2−f,λ2′=−∂H∂m=λ1βfm2L2m+s1+s22+βfmm−2KL2Km+s1+s2+λ2δ−βfmLm+s1+s2+βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2,λ3′=−∂H∂s1=λ1βfm2L2m+s1+s22+βfm22Km+s1+s2+λ2βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2+λ3δ,λ4′=−∂H∂s2=λ1βfm2L2m+s1+s22+βfm22Km+s1+s2+λ2−2βfs2Lm+s1+s2+βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2+λ4δ.
To find the optimalμ1∗t,μ2∗t, minimize H pointwise, that is,(21)∂H∂μ1=μ1+λ3,∂H∂μ2=μ2+λ4.
Note that1/2 cancels with 2 which comes from the square of the controls μ1 and μ2. Furthermore, the problem is indeed minimization as(22)∂2H∂μ12=1>0,∂2H∂μ22=1>0.
Hence, the optimal solutions are(23)∂H∂μ1>0⇒μ1∗t=0,∂H∂μ1=0⇒0≤μ1∗t=−λ3,∂H∂μ2>0⇒μ2∗t=0,∂H∂μ2=0⇒0≤μ2∗t=−λ4.
A compact way of writing the optimal controlμ1∗t,μ2∗t is(24)μ1∗t=maxt0,−λ3,μ2∗t=maxt0,−λ4.
Now, the proof is completed.


## 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.Table 1
List of parameters. All parameters are nonnegative.
ParameterDescriptionβBirth coefficient proportional to the viability of progenyδDeath coefficient proportional to the death from predators and illnessLLogistic term (to limit the size of the population)KCarrying capacity of the ecosystemμ1Constant influx of radiation-based sterile malesμ2Constant influx of YY supermalesThe 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 femalef, a wild-type XY male m, a sterile male s1, and a YY supermale s2. 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 is(1)dfdt=12fmβLmm+s1+s2−δf,dmdt=12fmβLmm+s1+s2+fs2βLs2m+s1+s2−δm,ds1dt=μ1−δs1,ds2dt=μ2−δs2,where f,m,s1,ands2 define the number of individuals in each associated class, and(2)L=1−f+m+s1+s2K.The intraspecies competition for female mates caused by the introduction of YY supermales and radiation-based sterile males is considered and modeled as(3)mm+s1+s2,s2m+s1+s2.Ifs1=s2=0, that is, m/m+s1+s2=1, it implies that there is no mating pressure on wild males. Obviously,(4)0≤mm+s1+s2≤1,0≤s2m+s1+s2≤1.The larger the value ofm/m+s1+s2 is, the more competitive wild males are. Similarly, the larger the value of s2/m+s1+s2 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, 0,0,μ1/δ,μ2/δ, on the boundary. It is clear 0,0,μ1/δ,μ2/δ is global stable. To get the positive interior equilibrium explicitly, i.e., f∗,m∗,s1∗,s2∗, it is equivalent to solve the following equations:(5)12f∗m∗βL∗m∗m∗+s1∗+s2∗−δf∗=0,12f∗m∗βL∗m∗m∗+s1∗+s2∗+fs2∗βL∗s2∗m∗+s1∗+s2∗−δm∗=0,μ1−δs1∗=0,μ2−δs2∗=0,L∗=1−f∗+m∗+s1∗+s2∗K.By solving equations (5), f∗,s1∗, and s2∗ can be obtained as follows:(6)f∗=K−m∗+s1∗+s2∗−2δKm∗+s1∗+s2∗m∗2β,s1∗=μ1δ,s2∗=μ2δ.Andm∗ is the solution of(7)2βm∗5+βs1+s2−Km∗4+2Kδ+βs22m∗3+2δKs1+s2+2βs22s1+s2−1m∗2+4δK∗s22m∗+4Kδs22s1+s2=0.Theorem 1.
Letμ>0. The interior equilibrium f∗,m∗,s1∗,s2∗ is locally asymptotically stable if(8)R1>0,R3>0,R4>0,R1R2R3>R32+R12R4.Proof.
The Jacobian matrix of model (1) about f∗,m∗,s1∗,s2∗ is given as(9)J=J11J12J13J14J21J22J23J2400J330000J44,where(10)J11=βKm∗2L∗−βf∗m∗22Km∗+s1∗+s2∗−δ,J12=2KL∗−m∗βf∗m∗2Km∗+s1∗+s2∗−βf∗m∗2L∗2∗m∗+s1∗+s2∗2,J13=−βf∗m∗2L∗2m∗+s1∗+s2∗2−βf∗m∗22Km∗+s1∗+s2∗,J14=−βf∗m∗2L∗2m∗+s1∗+s2∗2−βf∗m∗22Km∗+s1∗+s2∗,J21=βL∗m∗2+2s2∗22m∗+s1∗+s2∗−βf∗m∗2+2s2∗22Km∗+s1∗+s2∗,J22=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2+βf∗2Km∗L∗−m∗2−2s2∗22Km∗+s1∗+s2∗−δ,J23=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2−b∗fm∗2+2s2∗22Km∗+s1∗+s2∗,J24=−βf∗L∗m∗2+2s2∗22m∗+s1∗+s2∗2+βf∗4Ks2∗L∗−m∗2−2s2∗2Km∗+s1∗+s2∗,J33=−δ,J44=−δ.
The corresponding characteristic equation is(11)λ4+R1λ3+R2λ2+R3λ+R4=0,where(12)R1=−J11+J22+J33+J44,R2=J11J22+J11J33+J11J44+J22J33+J22J44+J33J44−J12J21,R3=J33+J44J12J21−J11+J22J33J44,R4=J11J22J33J44−J12J21J33J44.
According to Routh–Hurwitz criterion,f∗,m∗,s1∗,s2∗ is locally asymptotically stable if(13)R1>0,R3>0,R4>0,R1R2R3>R32+R12R4.Theorem 2.
In the case ofμ1=μ2=0, the trivial equilibrium 0,0,0,0 of TYC-SIT model (1) is globally asymptotically stable if β<δ/K.Proof.
Consider the Lyapunov functionVf,m,s1,s2=f+m+s1+s2. Note that Vf,m,s1,s2≥0 due to the positivity of its solutions and Vf,m,s1,s2 is radially unbounded. It is left to show that V′f,m,s1,s2<0 for all f,m,s1,s2≠0,0,0,0. Taking the derivative of Vf,m,s1,s2 about t yields(14)V′f,m,s1,s2=dVdt=dfdt+dmdt+ds1dt+ds2dt=fmβLmm+s1+s2+fs2βLs2m+s1+s2−δf−δm−δs1−δs2≤fmβ+fs2β−δf+m+s1+s2sinceL≤1,mm+s1+s2≤1ands2m+s1+s2≤1≤fmβ+fs2β−δm+s2=fβm+s2−δm+s2≤Kβm+s2−δm+s2≤Kβ−δm+s2.
Therefore, we only need to showKβ−δ<0 to get global stability, and this requires β<δ/K, 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 supermalesμ1 and μ2 are not known a priori and enter them into the system as time-dependent controls μ1t and μ2t. 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:(15)Objμ1t,μ2t=∫0T12f2+12μ1t2+12μ2t2dt,where f,m,s1,ands2 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 setU, which is given by(16)U=μ1,μ2μ1,μ2are measurable,0≤μ1t,0≤μ2t,t∈0,150.The goal is to find the optimal controlsμ1∗t,μ2∗t such that(17)Objμ1∗t,μ2∗t=minμ1t,μ2t∫0T12f2+12μ1t2+12μ2t2dt.Theorem 3.
There exists the optimal controlsμ1∗t and μ2∗t for the minization problem (17).Proof.
It is obvious thatObjμ1∗t,μ2∗t is concave in the argument μ1 and μ2; all state variables f,m,s1,ands2 are bounded by the carring capacity K, and the control variables μ1 and μ2 are also assumed bounded due to the cost of releasing sterile males and YY supermales; therefore, the existence of μ1∗t,μ2∗t for minimization problem (16) is guaranteed based on [33–35].Theorem 4.
An optimal controlμ1∗t,μ2∗t∈U of system (1) that minimizes the objective function Obj is characterized by(18)μ1∗t,μ2∗t=maxt0,−λ3,maxt0,−λ4.Proof.
Here, Pontryagin’s minimum principle is used to derive the necessary conditions on this problem. The HamiltonianH in this problem is(19)H=12f2+12μ1t2+12μ2t2+λ1f′+λ2m′+λ3s1′+λ4s2′.
The Hamiltonian is used to find the adjoint functionsλi,i=1,2,3,4:(20)λ1′=−∂H∂f=λ1δ+βm2f−KL2Km+s1+s2−λ2βLm2+2s222m+s1+s2+βfm2+2s222Km+s1+s2−f,λ2′=−∂H∂m=λ1βfm2L2m+s1+s22+βfmm−2KL2Km+s1+s2+λ2δ−βfmLm+s1+s2+βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2,λ3′=−∂H∂s1=λ1βfm2L2m+s1+s22+βfm22Km+s1+s2+λ2βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2+λ3δ,λ4′=−∂H∂s2=λ1βfm2L2m+s1+s22+βfm22Km+s1+s2+λ2−2βfs2Lm+s1+s2+βfLm2+2s222m+s1+s22+βfm2+2s222Km+s1+s2+λ4δ.
To find the optimalμ1∗t,μ2∗t, minimize H pointwise, that is,(21)∂H∂μ1=μ1+λ3,∂H∂μ2=μ2+λ4.
Note that1/2 cancels with 2 which comes from the square of the controls μ1 and μ2. Furthermore, the problem is indeed minimization as(22)∂2H∂μ12=1>0,∂2H∂μ22=1>0.
Hence, the optimal solutions are(23)∂H∂μ1>0⇒μ1∗t=0,∂H∂μ1=0⇒0≤μ1∗t=−λ3,∂H∂μ2>0⇒μ2∗t=0,∂H∂μ2=0⇒0≤μ2∗t=−λ4.
A compact way of writing the optimal controlμ1∗t,μ2∗t is(24)μ1∗t=maxt0,−λ3,μ2∗t=maxt0,−λ4.
Now, 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 Table2. 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.Table 2
Initial conditions. All parameters are nonnegative.
Initial conditionsParametersf0=100β=0.01m0=100δ=0.04s10=0,10,20,50,100μ1=0,5,10,15,20s20=0,10,20,50,100μ2=0,5,10,15,20The combined TYC-SIT system is modeled, and the initial conditions and parameters in Table2 are utilized to observe the relative population decline of females f in response to the addition of the radiation-based sterile males and YY supermales.Under the condition ofμ1=0,μ2=0, that is, sterile males or YY supermales are added to the population only once at time t=0 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.Figure 2
The population change of females over time underμ1=0 and μ2=0 with different number of initial introduction of sterile males and YY supermales. They all reach a stable state.Figure 3
The population change of females over time under continuous constant introduction of sterile males and YY supermales.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.,μ2=0) or solely continue adding YY supermales (i.e., μ1=0) 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 μ1 or μ2 is increased to 10, the initial decline rate with μ2=10 is faster than μ1=10, 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 μ1=0 or μ2=0 is similar if the influx of modified males is large enough.Figure 4
Investigation of purely introducing sterile males or purely introducing YY supermales after the initial introduction of the both. Blue star line is underμ2=0, and the orange star line is under μ1=0. Here, s10=s20=10. (a) Compared μ1=5,μ2=0 with μ1=0,μ2=5. (b) Compared μ1=10,μ2=0 with μ1=0,μ2=10. (c) Compared μ1=15,μ2=0 with μ1=0,μ2=15. (d) Compared μ1=20,μ2=0 with μ1=0,μ2=20.
(a)(b)(c)(d)The left panel of Figure5 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.Figure 5
The left panel shows the female density change varies with time under different influxes of sterile malesμ2=0. The right panel shows the female density change varies with time under different release rate of YY supermales μ1=0.Figure 6
The population change of females over time under continuous constant introduction of both sterile males and YY supermales.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 Figures7–9. The population can be driven to extinction with an optimal time controls μ1t or μ2t by using either SIT μ2=0, or TYC μ1=0, or the combined TYC-SIT approach.Figure 7
The optimal control analysis of the pure SIT strategy. (a) Female population change with increasing time. (b) Optimal controlμ2 vary with time.
(a)(b)Figure 8
The optimal control analysis of the pure TYC strategy. (a) Female population change with increasing time. (b) Optimal controlsμ1 vary with time.
(a)(b)Figure 9
The optimal control analysis of the combined TYC-SIT model. (a) Female population change with increasing time. (b) Optimal controlsμ1 and μ2 vary with time.
(a)(b)Figure9 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 t=4, 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 mosquitoesf, wild male mosquitoes m, radiation-based sterile males s1, and YY supermales s2. Six parameters, the birth coefficient β, the death coefficient δ, the logistic term L, the carrying capacity K, the influx of sterile males μ1, and the influx of sterile males μ2, 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 μ1=0 or μ2=0, 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.



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*Source: 2373350-2022-05-27.xml*