# An LMI Approach-Based Mathematical Model to ControlAedes aegypti Mosquitoes Population via Biological Control

Authors: J. Dianavinnarasi; R. Raja; J. Alzabut; M. Niezabitowski; G. Selvam; O. Bagdasar
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a novel age-structured delayed mathematical model to controlAedes aegypti mosquitoes via Wolbachia-infected mosquitoes is introduced. To eliminate the deadly mosquito-borne diseases such as dengue, chikungunya, yellow fever, and Zika virus, the Wolbachia infection is introduced into the wild mosquito population at every stage. This method is one of the promising biological control strategies. To predict the optimal amount of Wolbachia release, the time varying delay is considered. Firstly, the positiveness of the solution and existence of both Wolbachia present and Wolbachia free equilibrium were discussed. Through linearization, construction of suitable Lyapunov–Krasovskii functional, and linear matrix inequality theory (LMI), the exponential stability is also analyzed. Finally, the simulation results are presented for the real-world data collected from the existing literature to show the effectiveness of the proposed model.


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## Body

## 1. Introduction

Mosquito-borne diseases represent the vertical transmission of bacteria and viruses from mosquitoes to human while female mosquito taking a blood meal. Mosquito-borne diseases such as dengue, chikungunya, yellow fever, Zika virus, and Japanese encephalities cause over one million deaths per annum [1, 2]. Gubler in [3, 4] explained that the dengue and dengue hemorrhagic fever are the most common issues for public health. The primary vector for most of the mosquito-borne diseases is Aedes aegypti, and recently Aedes albopictus also add as a secondary vector [5]. In the past sixty years, the spread of mosquito-borne diseases has increased dramatically [6]. More than that, per year, dengue causes nearly 20 thousand deaths all over the world [7]. Also, nearly 112 countries are attacked by mosquito-borne diseases [8].In recent years, there are several articles are available to control vectors by genetic modifications [9]. Moreover, some biological control methods to replace the wild mosquitoes by releasing genetically modified mosquitoes are also tried by some researchers. Those biological control methods are sterilization of male mosquitoes [10, 11] and genetic modification to reduce the reproduction and increase the life-shortening bacteria Wolbachia [12]. Furthermore, via finding the reproduction number of a mathematical model which depicts the virus transmission via human sexual contact was analyzed in [13]. In [14], the author tried some other types of control agents such as bed nets, mosquito repellents, indoor residual spray, condoms during sex, medically treating infected human, and quarantine. However, in [15], the author tried to control mosquitoes by making modifications in feeding behaviours.In this environment, our main aim is to control the vector population that transmits the virus to the uninfected human while taking a blood meal. There is a life-shortening bacterium called Wolbachia which will be very useful to reach our aim; in [16], Mcmeniman et al. analyzed the stable introduction of Wolbachia in Aedes mosquitoes. Wolbachia is a Gram-negative bacterium, and it is first reported in the tissues of the mosquito Culex pipients (Hertig and Wolbach, 1924) [17]. In recent results, they found that yellow fever virus can also be blocked by Wolbachia [18].If a mosquito carries this bacterium, then the virus inside the mosquito does not get transmitted into the uninfected human. It blocks the virus inside the mosquito at salivary gland. This can be understood with Figure1.Figure 1
Dynamics of virus infection after Wolbachia introduction.In Figure1, the process of releasing Wolbachia bacteria into mosquito population is as follows:(1)
In laboratory, the Wolbachia pipients are injected into eggs, larvae, and pupae ofAedes aegypti via microinjection.(2)
Cytoplasmic incapability (CI): the adult Wolbachia-infected mosquitoes which are reared at the laboratory are released to the wild mosquito population ofAedes aegypti. Through this process, there exist three types of possibilities which are(i)
If the Wolbachia-infected female mosquitoes mate with the Wolbachia-infected male, then the progeny should have the Wolbachia by birth which is compatible.(ii)
If the Wolbachia-infected female crosses with Wolbachia-uninfected male, then the progeny face the same problems as in (i).(iii)
If Wolbachia-uninfected female crosses with Wolbachia-infected male, then there is no viable progeny.These two processes can be virtually understood by Figure2.Figure 2
Block diagram representing the mechanism of Wolbachia infection in mosquitoes.In eggs, larvae, and pupae population, we can microinject the Wolbachia and release this in patches at dengue-suspected areas. This process is practically done by placing “Zancu kits” around the people living areas. And the adult mosquitoes which are reared at lab can also be released into wild mosquito population. This process is called “introgression.”Various mathematical models have been studied to understand the interplay among Wolbachia and non-Wolbachia mosquitoes. In [19], the author considered the Wolbachia bacteria as a mechanism to control arbovirus, and his experimental studies show that the spread of Zika virus among mosquitoes and human was notably reduced. The same process for dengue virus spread was studied by Segoli et al. in [20]. In [21], the author has created a mathematical model considering only adult female mosquitoes and converted the model into endoepidemic model consisting of adult female mosquitoes and human population. In [22], the authors proposed a mathematical model depicting the life stages of mosquitoes with Wolbachia and proved that Wolbachia has excellent quality to control dengue virus spread. In that work, Koiler et al. discussed the virus as well as Wolbachia in both mosquitoes and human. Also, the objective is to predict the appropriate release of this bacterium, and the basic reproduction number was analyzed. Supriatna et al. in [23] developed a mathematical model to express the dynamics of dengue virus in both human and Aedes aegypti mosquitoes. In that, the human vaccination and Wolbachia introduction were used as optimal control methods. In [24], authors discussed the birth and death rate impulsive model to control mosquito-borne diseases using Wolbachia via the stroboscopic map method. The integer-order mathematical model which describes the interplay among the wild and Wolbachia-infected mosquitoes was analyzed in [25]. In that work, the author divided the mosquito population into two groups: one is aquatic and another one is adult. In [26], the author proposed a mathematical model to describe the persistence of Wolbachia via two-sex stage-structured model.Hence, with a full understanding of the interplay among the Wolbachia and non-Wolbachia mosquitoes in our work, we have created a mathematical model consisting of 10 stages to ensure the success of the proposed strategy. With reference of the practical results in [18], we can release the Wolbachia in every stages in the forms of “Zancu kits” and “Introgression.” So to obtain an optimal control, it is necessary to consider each and every variable. Because we know that in mathematical modeling, each and every parameter plays an important role. Up to our knowledge, this is the first article considering the control inputs in 10 stages incorporated with time-varying delays.By motivated by the above discussions, the main contribution of this paper is as follows:(i)
The main aim is to establish a novel mathematical model to describe the interplay among the both non-Wolbachia (wild mosquitoes) and Wolbachia-infected mosquitoes with time-varying delay.(ii)
We found that our method will increase Wolbachia-infected mosquitoes in terms of CI rescue team and non-Wolbachia mosquitoes go to annihilation.(iii)
We have proved that the releasing of adult female Wolbachia-infected mosquitoes is more beneficial than the releasing of adult male Wolbachia-infected mosquitoes.(iv)
There exists no literature on exponential stability results of delayed Wolbachia and non-Wolbachia age-structured model. This gap is filled by our work. Finally, by using real-world data, we checked the dynamics of the proposed model by using MATLAB LMI tool box.The rest of the paper is arranged as follows: in Section2, the novel mathematical model which describes the interplay among the Wolbachia free and Wolbachia present mosquito population is proposed. In Section 3, the analysis of the model such as positiveness of the solution and existence of equilibrium points are discussed. In Section 4, the exponential stability results of the linearized delayed system with time-varying delay is presented. In Section 5, numerical simulation results are presented. The work is concluded in Section 6.


## 2. Preliminaries and Model Formulation

In this section, some basic definitions and lemmas which are used to derive our results are presented. And the interaction between wild mosquitoes and Wolbachia-infected mosquitoes is modeled.Definition 1 (see [27]).
A model is said to be exponentially stable at its equilibrium point, if there existsγ>0 such that(1)yt≤Φe−2γt,for everyt>0.Lemma 1.
(Schur Complement, see [28]). Let us denote three n×n matrices as Ψ1,Ψ2,Ψ3, where Ψ1=Ψ1⊤ and Ψ2=Ψ2⊤>0. Then Ψ1+Ψ3⊤Ψ2−1Ψ3<0 if and only if Ψ1Ψ3⊤Ψ3−Ψ2<0 or −Ψ2Ψ3Ψ3⊤Ψ1<0.Lemma 2.
(see [29]). For any scalar ϵ>0, E,N∈ℝn, and matrix P1, then(2)E⊤P1N≤12ϵE⊤P1P1⊤E+ϵ2N⊤N.


### 2.1. Modeling the Life Stages of Wild Mosquito Population

In a common environment,Aedes aegypti mosquito population has five important life stages such as eggs Wet, larvae Wlt, pupae Wpt, matured female mosquitoes Wft, and adult male mosquitoes Wat (Figure 3). These life stages with respect to time t can be modeled as follows:(3)dWetdt=ΛweWfWaT−λweWe−γweWe,dWltdt=γweWe−λwlWl−γwlWl,dWptdt=γwlWl−λwpWp−γwpWp,dWftdt=ργwpWp−λwfWf,dWatdt=1−ργwpWp−λwaWa.Figure 3
Schematic representation of interaction between non-Wolbachia and Wolbachia-infected mosquitoes.


### 2.2. Modeling the Interaction between Wild and Wolbachia-Infected Mosquitoes

In this subsection, we modeled the release of Wolbachia-infected mosquitoes into the wild mosquitoes in the mosquito-borne disease suspected areas. The Wolbachia infection is released in both aquatic (Zanku Kits) and ariel stages. Therefore, by using the language of mathematics, we can model the interaction between wild and Wolbachia mosquitoes as follows:(4)dWetdt=ΛweWfWaT−λweWe−γweWe,dWltdt=γweWe−λwlWl−γwlWl+1−αγieIe,dWptdt=γwlWl−λwpWp−γwpWp+1−βγilIl,dWftdt=ργwpWp−λwfWf+1−εγipρiwIp,dWatdt=1−ργwpWp−λwaWa+1−εγip1−ρiwIp,dIetdt=ΛieIfWa+IaT−λieIe−αγieIe,dIltdt=αγieIe−λilIl−βγilIl,dIptdt=βγilIl−λipIp−εγipIp,dIftdt=ρiεγipIp−λifIf,dIatdt=1−ρiεγipIp−λiaIa.To understand the system of equations in (4), refer Figure 3.In the above mathematical model, the production of eggs can be found by the termΛweWfWa/T. That is, eggs without Wolbachia infection Wet are produced by the mating between wild female Wft and wild male Wat mosquitoes, where T is the total population which can be calculated by the following expression:(5)T=Wet+Wlt+Wpt+Wft+Wat+Iet+Ilt+Ipt+Ift+Iat.Along with this, the termsλwe (natural mortality rate of non-Wolbachia eggs) and γwe (maturation rate of non-Wolbachia eggs) denote the limitations in the growth of wild mosquito eggs. At the same time, after release of Wolbachia-infected mosquitoes (in both aquatic and ariel stages) in a common environment, the production of Wolbachia-infected mosquito eggs Iet depends on mating between Wolbachia-infected female Ift and non-Wolbachia male Wat and from mating between Wolbachia-infected female Ift and Wolbachia-infected male Iat. This implies that the birth rate of Wolbachia-infected mosquito egg population Iet with the reproduction rate Λie is(6)ΛieIfWa+IfIaT=ΛieIfWa+IaT.Similarly, the increase in growth is limited by the natural mortality rateλie and the maturation rate γie (that is, the rate in which the corresponding compartment moved into the next stage).Furthermore,1−αγieIe is added to the wild mosquito larvae population. Because the terms α and 1−α denote the probability of getting larvae with and without Wolbachia, respectively. Similarly, β and 1−β denote the probability of getting pupae with and without Wolbachia, respectively, ϵ and 1−ϵ denote the probability rate of having Wolbachia infection in adult mosquitoes by introgression. That is, ϵ be the probability of getting Wolbachia-infected adults (with ρiw = probability of getting male and 1−ρiw = probability of getting female). Because of these reasons, the terms 1−αγieIe, 1−βγilIl, 1−ϵγipρiwIp, and 1−ϵγip1−ρiwIp are added to the corresponding stages, and similarly, the terms αγieIe, βγilIl and ϵγipIip are removed from the corresponding stages. And the other parameters used in this model are described in Table 1.Table 1
Parameters involved in the population dynamics ofAedes aegypti mosquitoe.
Λwe, ΛieReproduction rate of non-Wolbachia mosquitoes and Wolbachia-infected mosquitoes, respectivelyλweThe mortality rate of non-Wolbachia eggsλwlThe mortality rate of non-Wolbachia larvaeλwpThe mortality rate of non-Wolbachia pupaeλwfThe mortality rate of non-Wolbachia adult female mosquitoesλwaThe mortality rate of non-Wolbachia adult male mosquitoesλieThe mortality rate of Wolbachia-infected eggsλilThe mortality rate of Wolbachia-infected larvaeλipThe mortality rate of Wolbachia-infected pupaeλifThe mortality rate of Wolbachia-infected adult female mosquitoesλiaThe mortality rate of Wolbachia-infected adult male mosquitoesγweThe rate at which the corresponding part of the wild mosquito eggs forms in which the larvae of wild mosquito emergeγwlThe rate at which the corresponding part of the wild mosquito larvae forms in which the pupae of wild mosquito emergeγwpThe rate at which the corresponding part of the wild mosquito pupae forms in which the immature female or male of wild mosquito emergeγieThe rate at which the corresponding part of the Wolbachia-infected mosquito eggs matured into Wolbachia-infected or uninfected larvaeγilThe rate at which the corresponding part of the Wolbachia-infected mosquito larvae matured into Wolbachia-infected or uninfected pupaeγipThe rate at which the corresponding part of the Wolbachia-infected mosquito pupae matured into Wolbachia-infected or uninfected adultsρThe probability of having male or female mosquitoesTTotal population


## 2.1. Modeling the Life Stages of Wild Mosquito Population

In a common environment,Aedes aegypti mosquito population has five important life stages such as eggs Wet, larvae Wlt, pupae Wpt, matured female mosquitoes Wft, and adult male mosquitoes Wat (Figure 3). These life stages with respect to time t can be modeled as follows:(3)dWetdt=ΛweWfWaT−λweWe−γweWe,dWltdt=γweWe−λwlWl−γwlWl,dWptdt=γwlWl−λwpWp−γwpWp,dWftdt=ργwpWp−λwfWf,dWatdt=1−ργwpWp−λwaWa.Figure 3
Schematic representation of interaction between non-Wolbachia and Wolbachia-infected mosquitoes.


## 2.2. Modeling the Interaction between Wild and Wolbachia-Infected Mosquitoes

In this subsection, we modeled the release of Wolbachia-infected mosquitoes into the wild mosquitoes in the mosquito-borne disease suspected areas. The Wolbachia infection is released in both aquatic (Zanku Kits) and ariel stages. Therefore, by using the language of mathematics, we can model the interaction between wild and Wolbachia mosquitoes as follows:(4)dWetdt=ΛweWfWaT−λweWe−γweWe,dWltdt=γweWe−λwlWl−γwlWl+1−αγieIe,dWptdt=γwlWl−λwpWp−γwpWp+1−βγilIl,dWftdt=ργwpWp−λwfWf+1−εγipρiwIp,dWatdt=1−ργwpWp−λwaWa+1−εγip1−ρiwIp,dIetdt=ΛieIfWa+IaT−λieIe−αγieIe,dIltdt=αγieIe−λilIl−βγilIl,dIptdt=βγilIl−λipIp−εγipIp,dIftdt=ρiεγipIp−λifIf,dIatdt=1−ρiεγipIp−λiaIa.To understand the system of equations in (4), refer Figure 3.In the above mathematical model, the production of eggs can be found by the termΛweWfWa/T. That is, eggs without Wolbachia infection Wet are produced by the mating between wild female Wft and wild male Wat mosquitoes, where T is the total population which can be calculated by the following expression:(5)T=Wet+Wlt+Wpt+Wft+Wat+Iet+Ilt+Ipt+Ift+Iat.Along with this, the termsλwe (natural mortality rate of non-Wolbachia eggs) and γwe (maturation rate of non-Wolbachia eggs) denote the limitations in the growth of wild mosquito eggs. At the same time, after release of Wolbachia-infected mosquitoes (in both aquatic and ariel stages) in a common environment, the production of Wolbachia-infected mosquito eggs Iet depends on mating between Wolbachia-infected female Ift and non-Wolbachia male Wat and from mating between Wolbachia-infected female Ift and Wolbachia-infected male Iat. This implies that the birth rate of Wolbachia-infected mosquito egg population Iet with the reproduction rate Λie is(6)ΛieIfWa+IfIaT=ΛieIfWa+IaT.Similarly, the increase in growth is limited by the natural mortality rateλie and the maturation rate γie (that is, the rate in which the corresponding compartment moved into the next stage).Furthermore,1−αγieIe is added to the wild mosquito larvae population. Because the terms α and 1−α denote the probability of getting larvae with and without Wolbachia, respectively. Similarly, β and 1−β denote the probability of getting pupae with and without Wolbachia, respectively, ϵ and 1−ϵ denote the probability rate of having Wolbachia infection in adult mosquitoes by introgression. That is, ϵ be the probability of getting Wolbachia-infected adults (with ρiw = probability of getting male and 1−ρiw = probability of getting female). Because of these reasons, the terms 1−αγieIe, 1−βγilIl, 1−ϵγipρiwIp, and 1−ϵγip1−ρiwIp are added to the corresponding stages, and similarly, the terms αγieIe, βγilIl and ϵγipIip are removed from the corresponding stages. And the other parameters used in this model are described in Table 1.Table 1
Parameters involved in the population dynamics ofAedes aegypti mosquitoe.
Λwe, ΛieReproduction rate of non-Wolbachia mosquitoes and Wolbachia-infected mosquitoes, respectivelyλweThe mortality rate of non-Wolbachia eggsλwlThe mortality rate of non-Wolbachia larvaeλwpThe mortality rate of non-Wolbachia pupaeλwfThe mortality rate of non-Wolbachia adult female mosquitoesλwaThe mortality rate of non-Wolbachia adult male mosquitoesλieThe mortality rate of Wolbachia-infected eggsλilThe mortality rate of Wolbachia-infected larvaeλipThe mortality rate of Wolbachia-infected pupaeλifThe mortality rate of Wolbachia-infected adult female mosquitoesλiaThe mortality rate of Wolbachia-infected adult male mosquitoesγweThe rate at which the corresponding part of the wild mosquito eggs forms in which the larvae of wild mosquito emergeγwlThe rate at which the corresponding part of the wild mosquito larvae forms in which the pupae of wild mosquito emergeγwpThe rate at which the corresponding part of the wild mosquito pupae forms in which the immature female or male of wild mosquito emergeγieThe rate at which the corresponding part of the Wolbachia-infected mosquito eggs matured into Wolbachia-infected or uninfected larvaeγilThe rate at which the corresponding part of the Wolbachia-infected mosquito larvae matured into Wolbachia-infected or uninfected pupaeγipThe rate at which the corresponding part of the Wolbachia-infected mosquito pupae matured into Wolbachia-infected or uninfected adultsρThe probability of having male or female mosquitoesTTotal population


## 3. Analysis of the Model

In this section, we analyze the positivity, existence of equilibrium points, and stability of the system of equations in (4).


### 3.1. Positivity of Solutions

Lemma 3.
For allt>0, the solutions We,Wl,Wp,Wf,Wa,Ie,Il,Ip,If,Ia are all nonnegative if the initial values We>0, Wl>0, Wp>0, Wf>0, Wa>0, Ie>0, Il>0, Ip>0, If>0, Ia>0.Proof.
Let us define(7)μ˜=supμ>0:We0>0,Wl0>0,Wp0>0,Wf0>0,Wa0>0,Ie0>0,Il0>0,Ip0>0,If0>0,Ia0>0∈0,μ.
From (7), μ˜>0. Let us consider the first equation of model (4), that is,(8)dWetdt=ΛweWfWaT−λweWe−γweWe,dWetdt+λwe+γweWe=ΛweWfWaT.
The integrating factor is(9)I.F=e∫0μ˜λwe+γwedμ.
Multiply (9) with (8) on both sides(10)dWetdte∫0μ˜λwe+γwedμ+λwe+γweWee∫0μ˜λwe+γwedμ=ΛweWfWaTe∫0μ˜λwe+γwedμ.
That is,(11)ddμWete∫0μ˜λwe+γwedμ=ΛweWfWaTe∫0μ˜λwe+γwedμ.
Integrating on both sides with respect toμ∈0,μ˜,(12)Wete∫0μ˜λwe+γwedμ0μ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds,Weμ˜e∫0μ˜λwe+γwedμ˜−We0e0=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds,Weμ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds+We0e−∫0μ˜λwe+γwedμ˜,Weμ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsdse−∫0μ˜λwe+γwedμ˜+We0e−∫0μ˜λwe+γwedμ˜.
This implies thatWeμ˜>0.
Similarly, we can prove that for the positive initial values, the solution is positive.


### 3.2. Existence of Equilibrium Points

In this section, the possible cases of existing equilibrium points are discussed. Furthermore, the null mosquitoes equilibrium point0,0,0,0,0,0,0,0,0,0 is omitted because this case does not exist in nature.


#### 3.2.1. Wolbachia Free Equilibrium

In this subsection, we can find Wolbachia free equilibrium by equating the system of equations in (2) to zero and putting Ie1∗=0, Il1∗=0, Ip1∗=0, If1∗=0, and Ia1∗=0.That is,(13)ΛweWf1∗Wa1∗T−λweWe1∗−γweWe1∗=0,γweWe1∗−λwlWl1∗−γwlWl1∗+1−αγieIe1∗=0,γwlWl1∗−λwpWp1∗−γwpWp1∗+1−βγilIl1∗=0,ργwpWp1∗−λwfWf1∗+1−εγipρiwIp1∗=0,1−ργwpWp1∗−λwaWa1∗+1−εγip1−ρiwIp1∗=0.By solving equations (13), we get the equilibrium point S1 as(14)S1=We1∗,γweWe1∗λwl+γwl,γwlγweWe1∗λwl+γwlλwp+γwp,ργwpγwlγweWe1∗λwfλwl+γwlλwp+γwp1−ργwpγwlγweWe1∗λwaλwl+γwlλwp+γwp,0,0,0,0,0,0,where(15)We1∗=Tλwfλwaλwe+γweλwl+γwl2λwp+γwp2ρ1−ρΛweγwp2γwe2γwl2.


#### 3.2.2. Wolbachia Present Equilibrium

In this subsection, we can find Wolbachia present equilibrium by equating the system of equations in (4) to zero. That is,(16)ΛweWfn∗Wan∗T−λweWen∗−γweWen∗=0,γweWen∗−λwlWln∗−γwlWln∗+1−αγieIen∗=0,γwlWln∗−λwpWpn∗−γwpWpn∗+1−βγilIln∗=0,ργwpWpn∗−λwfWfn∗+1−εγipρiwIpn∗=0,1−ργwpWpn∗−λwaWan∗+1−εγip1−ρiwIpn∗=0,ΛieIfn∗Wan∗+Ian∗T−λieIen∗−αγieIen∗=0,αγieIen∗−λilIln∗−βγilIln∗=0,βγilIln∗−λipIpn∗−εγipIpn∗=0,ρiεγipIpn∗−λifIfn∗=0,1−ρiεγipIpn∗−λiaIan∗=0.By solving the system of equations in (16), we get the following equilibrium point in terms of Ia∗:(17)Sn=Wen∗,Wln∗,Wpn∗,Wfn∗,Wan∗,Ien∗,Iln∗,Ipn∗,Ifn∗,Ian∗,n=2,3,Wen∗=λwl+γwlγweλwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−Ian∗γweB4λwl+γwlλwp+γwpγwl+λwl+γwl1−βγilγwlλiaB1βγil1−ρi+1−αγieλiaB1B2αγie1−ρi,Wln∗=λwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗−1−βγilγwlλiaB1βγil1−ρi,Wpn∗=TB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗,Wfn∗=ργwpλwfTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗+1−ϵρiwλiaϵλwf1−ρiIan∗,Wan∗=TB1B2B3λifρiΛie−Ian∗,Ien∗=B1B2λiaIan∗αγie1−ρi,Ipn∗=λia1−ρiϵγipIan∗,Ifn∗=ρiλiaIan∗1−ρiλif,with Ia3∗>Ia4∗, both roots can be found from the quadratic equation:(18)a1Ia∗2+a2Ia∗+a3=0,where(19)a1=ΛweρB4γwpTλwf;a2=λwe+γweTλwfλweλifρB1B2B3ρiΛieλwfλwa1−ρ+B4γwpλwl+γwlλwp+γwpB4γwl+λwl+γwl1−βλiaB1γwlβ1−ρi+1−αλiaB1B2α1−ρi;a3=ΛweρTB12B22B32λwaΛie21−ρρi2λwf.where(20)B1=1+λipϵγip;B2=1+λilβγil;B3=1+λieαγie;B4=1+1−ϵ1−ρiwλia1−ρ1−ρiϵγwp.


#### 3.2.3. Existence of Wolbachia Free and Wolbachia Present Equilibrium

The equilibrium points exist if it satisfies the following conditions:(i)
S1 exists provided that We1∗≠0 and 0<ρ<1.(ii)
S2 and S3 exist provided that Ia2∗,Ia3∗≠0 and 1−ρϵγipIp∗<λia.


## 3.1. Positivity of Solutions

Lemma 3.
For allt>0, the solutions We,Wl,Wp,Wf,Wa,Ie,Il,Ip,If,Ia are all nonnegative if the initial values We>0, Wl>0, Wp>0, Wf>0, Wa>0, Ie>0, Il>0, Ip>0, If>0, Ia>0.Proof.
Let us define(7)μ˜=supμ>0:We0>0,Wl0>0,Wp0>0,Wf0>0,Wa0>0,Ie0>0,Il0>0,Ip0>0,If0>0,Ia0>0∈0,μ.
From (7), μ˜>0. Let us consider the first equation of model (4), that is,(8)dWetdt=ΛweWfWaT−λweWe−γweWe,dWetdt+λwe+γweWe=ΛweWfWaT.
The integrating factor is(9)I.F=e∫0μ˜λwe+γwedμ.
Multiply (9) with (8) on both sides(10)dWetdte∫0μ˜λwe+γwedμ+λwe+γweWee∫0μ˜λwe+γwedμ=ΛweWfWaTe∫0μ˜λwe+γwedμ.
That is,(11)ddμWete∫0μ˜λwe+γwedμ=ΛweWfWaTe∫0μ˜λwe+γwedμ.
Integrating on both sides with respect toμ∈0,μ˜,(12)Wete∫0μ˜λwe+γwedμ0μ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds,Weμ˜e∫0μ˜λwe+γwedμ˜−We0e0=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds,Weμ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsds+We0e−∫0μ˜λwe+γwedμ˜,Weμ˜=∫0μ˜ΛweWfsWasTe∫0μ˜λwe+γwedsdse−∫0μ˜λwe+γwedμ˜+We0e−∫0μ˜λwe+γwedμ˜.
This implies thatWeμ˜>0.
Similarly, we can prove that for the positive initial values, the solution is positive.


## 3.2. Existence of Equilibrium Points

In this section, the possible cases of existing equilibrium points are discussed. Furthermore, the null mosquitoes equilibrium point0,0,0,0,0,0,0,0,0,0 is omitted because this case does not exist in nature.


### 3.2.1. Wolbachia Free Equilibrium

In this subsection, we can find Wolbachia free equilibrium by equating the system of equations in (2) to zero and putting Ie1∗=0, Il1∗=0, Ip1∗=0, If1∗=0, and Ia1∗=0.That is,(13)ΛweWf1∗Wa1∗T−λweWe1∗−γweWe1∗=0,γweWe1∗−λwlWl1∗−γwlWl1∗+1−αγieIe1∗=0,γwlWl1∗−λwpWp1∗−γwpWp1∗+1−βγilIl1∗=0,ργwpWp1∗−λwfWf1∗+1−εγipρiwIp1∗=0,1−ργwpWp1∗−λwaWa1∗+1−εγip1−ρiwIp1∗=0.By solving equations (13), we get the equilibrium point S1 as(14)S1=We1∗,γweWe1∗λwl+γwl,γwlγweWe1∗λwl+γwlλwp+γwp,ργwpγwlγweWe1∗λwfλwl+γwlλwp+γwp1−ργwpγwlγweWe1∗λwaλwl+γwlλwp+γwp,0,0,0,0,0,0,where(15)We1∗=Tλwfλwaλwe+γweλwl+γwl2λwp+γwp2ρ1−ρΛweγwp2γwe2γwl2.


### 3.2.2. Wolbachia Present Equilibrium

In this subsection, we can find Wolbachia present equilibrium by equating the system of equations in (4) to zero. That is,(16)ΛweWfn∗Wan∗T−λweWen∗−γweWen∗=0,γweWen∗−λwlWln∗−γwlWln∗+1−αγieIen∗=0,γwlWln∗−λwpWpn∗−γwpWpn∗+1−βγilIln∗=0,ργwpWpn∗−λwfWfn∗+1−εγipρiwIpn∗=0,1−ργwpWpn∗−λwaWan∗+1−εγip1−ρiwIpn∗=0,ΛieIfn∗Wan∗+Ian∗T−λieIen∗−αγieIen∗=0,αγieIen∗−λilIln∗−βγilIln∗=0,βγilIln∗−λipIpn∗−εγipIpn∗=0,ρiεγipIpn∗−λifIfn∗=0,1−ρiεγipIpn∗−λiaIan∗=0.By solving the system of equations in (16), we get the following equilibrium point in terms of Ia∗:(17)Sn=Wen∗,Wln∗,Wpn∗,Wfn∗,Wan∗,Ien∗,Iln∗,Ipn∗,Ifn∗,Ian∗,n=2,3,Wen∗=λwl+γwlγweλwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−Ian∗γweB4λwl+γwlλwp+γwpγwl+λwl+γwl1−βγilγwlλiaB1βγil1−ρi+1−αγieλiaB1B2αγie1−ρi,Wln∗=λwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗−1−βγilγwlλiaB1βγil1−ρi,Wpn∗=TB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗,Wfn∗=ργwpλwfTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗+1−ϵρiwλiaϵλwf1−ρiIan∗,Wan∗=TB1B2B3λifρiΛie−Ian∗,Ien∗=B1B2λiaIan∗αγie1−ρi,Ipn∗=λia1−ρiϵγipIan∗,Ifn∗=ρiλiaIan∗1−ρiλif,with Ia3∗>Ia4∗, both roots can be found from the quadratic equation:(18)a1Ia∗2+a2Ia∗+a3=0,where(19)a1=ΛweρB4γwpTλwf;a2=λwe+γweTλwfλweλifρB1B2B3ρiΛieλwfλwa1−ρ+B4γwpλwl+γwlλwp+γwpB4γwl+λwl+γwl1−βλiaB1γwlβ1−ρi+1−αλiaB1B2α1−ρi;a3=ΛweρTB12B22B32λwaΛie21−ρρi2λwf.where(20)B1=1+λipϵγip;B2=1+λilβγil;B3=1+λieαγie;B4=1+1−ϵ1−ρiwλia1−ρ1−ρiϵγwp.


### 3.2.3. Existence of Wolbachia Free and Wolbachia Present Equilibrium

The equilibrium points exist if it satisfies the following conditions:(i)
S1 exists provided that We1∗≠0 and 0<ρ<1.(ii)
S2 and S3 exist provided that Ia2∗,Ia3∗≠0 and 1−ρϵγipIp∗<λia.


## 3.2.1. Wolbachia Free Equilibrium

In this subsection, we can find Wolbachia free equilibrium by equating the system of equations in (2) to zero and putting Ie1∗=0, Il1∗=0, Ip1∗=0, If1∗=0, and Ia1∗=0.That is,(13)ΛweWf1∗Wa1∗T−λweWe1∗−γweWe1∗=0,γweWe1∗−λwlWl1∗−γwlWl1∗+1−αγieIe1∗=0,γwlWl1∗−λwpWp1∗−γwpWp1∗+1−βγilIl1∗=0,ργwpWp1∗−λwfWf1∗+1−εγipρiwIp1∗=0,1−ργwpWp1∗−λwaWa1∗+1−εγip1−ρiwIp1∗=0.By solving equations (13), we get the equilibrium point S1 as(14)S1=We1∗,γweWe1∗λwl+γwl,γwlγweWe1∗λwl+γwlλwp+γwp,ργwpγwlγweWe1∗λwfλwl+γwlλwp+γwp1−ργwpγwlγweWe1∗λwaλwl+γwlλwp+γwp,0,0,0,0,0,0,where(15)We1∗=Tλwfλwaλwe+γweλwl+γwl2λwp+γwp2ρ1−ρΛweγwp2γwe2γwl2.


## 3.2.2. Wolbachia Present Equilibrium

In this subsection, we can find Wolbachia present equilibrium by equating the system of equations in (4) to zero. That is,(16)ΛweWfn∗Wan∗T−λweWen∗−γweWen∗=0,γweWen∗−λwlWln∗−γwlWln∗+1−αγieIen∗=0,γwlWln∗−λwpWpn∗−γwpWpn∗+1−βγilIln∗=0,ργwpWpn∗−λwfWfn∗+1−εγipρiwIpn∗=0,1−ργwpWpn∗−λwaWan∗+1−εγip1−ρiwIpn∗=0,ΛieIfn∗Wan∗+Ian∗T−λieIen∗−αγieIen∗=0,αγieIen∗−λilIln∗−βγilIln∗=0,βγilIln∗−λipIpn∗−εγipIpn∗=0,ρiεγipIpn∗−λifIfn∗=0,1−ρiεγipIpn∗−λiaIan∗=0.By solving the system of equations in (16), we get the following equilibrium point in terms of Ia∗:(17)Sn=Wen∗,Wln∗,Wpn∗,Wfn∗,Wan∗,Ien∗,Iln∗,Ipn∗,Ifn∗,Ian∗,n=2,3,Wen∗=λwl+γwlγweλwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−Ian∗γweB4λwl+γwlλwp+γwpγwl+λwl+γwl1−βγilγwlλiaB1βγil1−ρi+1−αγieλiaB1B2αγie1−ρi,Wln∗=λwp+γwpγwlTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗−1−βγilγwlλiaB1βγil1−ρi,Wpn∗=TB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗,Wfn∗=ργwpλwfTB1B2B3λifλwaΛie1−ρρiγwp−B4Ian∗+1−ϵρiwλiaϵλwf1−ρiIan∗,Wan∗=TB1B2B3λifρiΛie−Ian∗,Ien∗=B1B2λiaIan∗αγie1−ρi,Ipn∗=λia1−ρiϵγipIan∗,Ifn∗=ρiλiaIan∗1−ρiλif,with Ia3∗>Ia4∗, both roots can be found from the quadratic equation:(18)a1Ia∗2+a2Ia∗+a3=0,where(19)a1=ΛweρB4γwpTλwf;a2=λwe+γweTλwfλweλifρB1B2B3ρiΛieλwfλwa1−ρ+B4γwpλwl+γwlλwp+γwpB4γwl+λwl+γwl1−βλiaB1γwlβ1−ρi+1−αλiaB1B2α1−ρi;a3=ΛweρTB12B22B32λwaΛie21−ρρi2λwf.where(20)B1=1+λipϵγip;B2=1+λilβγil;B3=1+λieαγie;B4=1+1−ϵ1−ρiwλia1−ρ1−ρiϵγwp.


## 3.2.3. Existence of Wolbachia Free and Wolbachia Present Equilibrium

The equilibrium points exist if it satisfies the following conditions:(i)
S1 exists provided that We1∗≠0 and 0<ρ<1.(ii)
S2 and S3 exist provided that Ia2∗,Ia3∗≠0 and 1−ρϵγipIp∗<λia.


## 4. Stability Analysis

In this section, the stability results of the system of equations in (2) by linearization, Lyapunov–Karasovskii functional, and LMI approach. To optimize our differential model, we consider the delay terms into account as discussed in [30] i.e., where Wet−τ1t is the produced wild egg density at time t−τ1t from which the wild larvae density at time t is produced, Wlt−τ2t is the produced wild larvae density at time t−τ2t from which the wild pupae density at time t is produced, Wpt−τ3t is the produced wild pupae density at time t−τ3t from which the wild female or male density at time t is produced, Iet−τ6t is the produced Wolbachia-infected egg density at time t−τ1t from which the Wolbachia-infected larvae density at time t is produced, Ilt−τ7t is the produced Wolbachia-infected larvae density at time t−τ7t from which the Wolbachia-infected pupae density at time t is produced, and Ipt−τ8t is the produced Wolbachia-infected pupae density at time t−τ8t from which the Wolbachia-infected female or male density at time t is produced. These considerations make our system of differential equations to system of delay differential equations:(21)dWetdt=ΛweWfWaT−λweWe−γweWe,dWltdt=γweWet−τ1t−λwlWl−γwlWl+1−αγieIe,dWptdt=γwlWlt−τ2t−λwpWp−γwpWp+1−βγilIl,dWftdt=ργwpWp1−τ3t−λwfWf+1−εγipρiwIp,dWatdt=1−ργwpWp1−τ3t−λwaWa+1−εγip1−ρiwIp,dIetdt=ΛieIfWa+IaT−λieIe−αγieIe,dIltdt=αγieIe1−τ6t−λilIl−βγilIl,dIptdt=βγilIl1−τ7t−λipIp−εγipIp,dIftdt=ρiεγipIp1−τ8t−λifIf,dIatdt=1−ρiεγipIp1−τ8t−λiaIa.By linearizing, the following system describes the interaction between wild mosquitoes and Wolbachia mosquitoes as(22)M˙t=AMt+BMt−τt+gMt+Cut,t≥0,Mt=πt,t∈−τ⌣,0,where(23)Mt=WetWltWptWftWatIetIltIptIftIat⊤;Mt−τ=Wet−τ1tWlt−τ2tWpt−τ3tWft−τ4tWat−τ5tIet−τ6tIlt−τ7tIpt−τ8tIft−τ9tIat−τ10t⊤,ut=00000u1u2u3u4u5⊤;0≤τ⌣≤τ^.A=−λwe−γwe0000000000−λwl−γwl0001−αγie000000−γwp−λwp0001−βγil000000−λwf0001−ϵγipρiw000000−λwa001−ϵγip1−ρiw0000000−λie−αγie0000000000−λil−βγil0000000000−λip−ϵγip0000000000−λif0000000000−λia,B=0000000000γwe0000000000γwl0000000000ργwp0000000001−ργwp0000000000000000000000αγie0000000000βγil0000000000ρiϵγip0000000001−ρiϵγip00,C=0000000000000000000000000000000000000000000000000000000100000000001000000000010000000000100000000001,gMt=Λwem4m5T0000Λiem9m5+m10T0000.Assume thatτt=maxτ1t,τ2t,τ3t,τ4t,τ5t,τ6t,τ7t,τ8t,τ9t,τ10t, and it is bounded, differentiable, and nonnegative. Now, consider an equilibrium point of system (22) as(24)M∗=We∗tWl∗tWp∗tWf∗tWa∗tIe∗tIl∗tIp∗tIf∗tIa∗t⊤,u∗=00000u1∗u2∗u3∗u4∗u5∗⊤.Let yt=Mt−M∗, and this implies that yt+M∗=Mt. And Ut=ut−u∗. Then the modified system is(25)y˙t=Ayt+Byt−τt+gyt+CUt,t≥0,yt=ψt,t∈−τ⌣,0,with ψt=π−M∗ be the initial condition.In general, stability is an important requirement to study dynamical systems [31–33]. In following theorem, we derived the sufficient for exponential stability results for system (25).Theorem 1.
The system (25) is said to be exponentially stable if there exist positive definite matrices P1 and P2 and positive scalars ϵ1 and ϵ2 such that(26)Ψ=2σP1+A⊤P1+K⊤C⊤P1+AP1+P1CK+BP2BP1P100∗−ϵ1000∗∗−ϵ200∗∗∗ϵ1−1−ηP2B0∗∗∗∗ϵ2>0,with τ˙t=η. Moreover,(27)yt≤e−2σtλMP1+λMP2ψ1−e2στt/2σψλMP1.Proof.
Let us consider the following Lyapunov–Krasovskii function:(28)Vyt=e2σty⊤tP1yt+∫t−τtty⊤sBe2σsP2ysds.
Now, the time derivative of the Lyapunov–Krasovskii functionalVyt along with the trajectories of system of equation (25) (29)V˙yt=2σe2σty⊤tP1yt+e2σty˙⊤tP1yt+e2σty⊤tP1y˙t+y⊤tBe2σtP2yt−1−τ˙ty⊤t−τtBe2σtP2yt−τt=e2σt2σy⊤tP1yt+y˙⊤tP1yt+y⊤tP1y˙t+y⊤tBP2yt−1−ηy⊤t−τtP2Byt−τt=e2σty⊤t2σP1yt+Ayt+Byt−τt+fyt+CUtTP1yt+y⊤tP1Ayt+Byt−τt+fyt+CUt+y⊤tBP2yt−1−ηy⊤t−τtP2Byt−τt.
PutUt=Kyt, the linear feedback control,(30)=e2σty⊤t2σP1yt+y⊤tA⊤P1yt+y⊤t−τtB⊤P1yt+f⊤ytP1yt+y⊤tK⊤C⊤P1yt+y⊤tAP1yt+y⊤tBP1yt−τt+y⊤tP1fyt+y⊤tP1CKyt+y⊤tBP2yt−1−ηy⊤t−τtP2Byt−τt=e2σty⊤t2σP1+A⊤P1+K⊤C⊤P1+AP1+P1CK+BP2yt+y⊤t−τtB⊤P1yt+y⊤tBP1yt−τt+f⊤ytP1yt+y⊤tP1fyt−y⊤t−τt1−ηP2Byt−τt.
By using Lemma2, we can get the following inequalities:(31)y⊤t−τtB⊤P1yt+y⊤tBP1yt−τt≤ϵ1y⊤t−τtyt−τt+ϵ1−1y⊤tBP1B⊤p1⊤yt,f⊤ytP1yt+y⊤tP1fyt≤ϵ2f⊤ytfyt+ϵ2−1y⊤tP1P1⊤yt≤e2σty⊤t2σP1+A⊤P1+K⊤C⊤P1+AP1+P1CK+BP2yt+ϵ1y⊤t−τtyt−τt+ϵ1−1y⊤tBP1B⊤P1⊤yt+ϵ2f⊤ytfyt+ϵ2−1y⊤tP1P1⊤yt=e2σty⊤t…2+ϵ1−1BP1P1⊤B⊤+ϵ2−1P1P1⊤yt+f⊤ytϵ2fyt+y⊤t−τtϵ1−1−ηP2Byt−τt.
V˙yt≤e2σtζ⊤tΨζt, where(32)Ψ=Ψ11000ϵ1−1−ηP2B000ϵ2,where Ψ11=2σP1+A⊤P1+K⊤C⊤P1+AP1+P1CK+BP2+ϵ1−1BP1P1⊤B⊤+ϵ2−1P1P1⊤.(33)Ψ=2σP1+A⊤P1+K⊤C⊤P1+AP1+P1CK+BP2BP1P100∗−ϵ1000∗∗−ϵ200∗∗∗ϵ1−1−ηP2B0∗∗∗∗ϵ2>0.Hence, by Lemma 1, Ψ>0⟹V˙yt<0. Moreover, Vyt≤Vy0. Here,(34)Vy0=e2σ0y⊤0P1y0+∫0−τt0y⊤sBe2σsP2ysds≤λMP1ψ+λMP2ψ2∫−τt0e2σsds=λMP1ψ+λMP2ψ21−e2στt2σ=λMP1+λMP2ψ1−e2στt2σψ.where λM⋅ is the maximum eigen value of ⋅.Here,(35)Vyt≥e2σtλMp1yt2.Now,(36)Vyt≤Vy0≤λMP1+λMP2ψ1−e2στt2σψ.This implies that(37)yt≤e−2σtλMP1+λMP2ψ1−e2στt/2σψλMP1,and by Definition 1, our system (25) is exponentially stable.Remark 1.
If the system (25) is without time-varying delay terms, then the results of Theorem 1 have changed as follows: the possible Lyapunov function is Vt=e2σty⊤tP1yt with the sufficient conditions(38)Ψ=2σP1+A⊤P1+P1A+P1CKP1P2∗−ϵ10∗∗−ϵ1<0.yt≤e−σtψ.Then the system (25) is without time-varying delay is exponentially stable.


## 5. Numerical Simulations

The numerical simulation results of system of equations in (21), based on the real-world data, are mentioned in Table 2.Table 2
The parameters used in numerical simulation.
ΛweReproduction rate of non-Wolbachia mosquitoes1.25/day [34]λwe, λwl, λwpNon-Wolbachia aquatic death rate1/7.78/day [35]γwe,γwl,γwpNon-Wolbachia maturation rate1/6.67/day [36]λwf,λwaNon-Wolbachia adult death rate1/14/day [35]λie,λil,λipWolbachia adult death rate1/7.78/day [35]λif,λiaWolbachia adult death rate1/7/day [37]ΛieWolbachia reproduction rate0.95∗Λwe/day [34]γie,γil,γipWolbachia-infected mosquitoes maturation rate1/6.67/day [37]Consider the following system:(39)y˙t=Ayt+Byt−τt+gyt+CUt,t≥0,yt=ψt,t∈−τ⌣,0,with numerical values(40)A=−0.27840000000000−0.27840000.0300000000−0.27840000.0300000000−0.07140000.0600000000−0.0714000.01500000000−0.24840000000000−0.24840000000000−0.20350000000000−0.14290000000000−0.1429,B=00000000000.149900000000000.149900000000000.11990000000000.030000000000000000000000000.119900000000000.119900000000000.06000000000000.015000;C=0000000000000000000000000000000000000000000000000000000100000000001000000000010000000000100000000001;with the control gain matrix(41)K=00000000000000000000000000000000−0.18820000000000000000000000000000000000000000000000000000001.28600.1611000000000.09791.1765;with ϵ1=0.7875 and ϵ2=0.9151.Remark 2.
The numerical simulations derived in our paper shows that the allowable upper bound of the time-varying delayτt=0.7899. However, the existing literature used the classical method such as Pontryagin maximum principle and Jacobian matrix method, by finding the sign of eigen values, fractional-order mathematical models [38, 39], and through finding the reproduction number to analyze the stability of the model. In our work, we have used LMI approach. However, compared with the other algebraic methods LMI approach is less conservative.
Figures4 and 5 depict the population dynamics of wild mosquitoes and Wolbachia-infected mosquitoes at the initial release of Wolbachia, respectively. From Figure 6, we can observe that the Wolbachia-infected population also tends to decrease in numbers. Our aim is to maintain the Wolbachia population in certain quantity to spread Wolbachia among wild mosquitoes naturally. Because of this reason, we have released another batch of Wolbachia-infected mosquitoes as an optimal control. After this release, we can observe from Figures 7 and 8 that the level of Wolbachia-infected mosquitoes is increased to certain quantity and wild mosquito population is decreased to zero.Figure 4
Population dynamics of wild mosquito.Figure 5
Population dynamics of Wolbachia-infected mosquitoes.Figure 6
Population dynamics of both wild and Wolbachia-infected mosquitoes.Figure 7
Population dynamics of Wolbachia-infected mosquitoes after the linear feedback control.Figure 8
Population dynamics of both wild and Wolbachia-infected mosquitoes after the linear feedback control.Remark 3.
For the optimal control to ensure the successful release of Wolbachia, the data from Table2 are substituted in the system of equations in (21). From this we obtained the following results: Figure 4 denotes the dynamics of non-Wolbachia mosquitoes, and Figure 5 denotes the dynamics of Wolbachia-infected mosquitoes. We can observe that the Wolbachia release in two main stages is not enough to control wild mosquitoes. In this, we noticed that the Wolbachia-infected mosquito population also goes to annihilation. Refer Figure 6, when comparing wild mosquitoes with Wolbachia-infected mosquitoes, it is increased and Wolbachia is not carried out to next generation. By implementing our proposed strategy, if we released Wolbachia in every stage, the Wolbachia-infected population is increased to certain level and maintained at that level (Figure 7) for the remaining period; at the same time, the Wolbachia population is annihilated without any increase (Figure 8).


## 6. Conclusion

A novel integer-ordered age-structured delayed mathematical model to describe the interplay among the wild and Wolbachia-infected mosquitoes was proposed. The positiveness of the solution and existence of both Wolbachia present and Wolbachia free equilibriums were analyzed. By using some mathematical techniques such as linearization, Lyapunov–Krasovskii functional, and LMI theory, the exponential stability results of the proposed model were established. By using MATLAB, the numerical simulation results were presented to show the effectiveness of the created mathematical model. Our results show that the release of adult female Wolbachia-infected mosquitoes is more beneficial than the release of adult male Wolbachia-infected mosquitoes.



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