Analysis and control of Aedes Aegypti mosquitoes using sterile-insect techniques with Wolbachia

1.   Introduction

Dengue hemorrhagic fever is emerging as one of the most serious vector-borne diseases worldwide. The Aedes Aegypti mosquito is a major vector for the transmission of dengue and other arboviral infections from an infected individual to a susceptible individual. In recent decades, people have been paying more attention to the prevalence, mortality, and massive economic impact of diseases like malaria, chikungunya, dengue, etc, due to their morbidity burdens; see ^[1,2,3,4]. For a long time, there is no available vaccines for the mosquito-borne diseases, even though the researchers is trying to find an effective vaccine. Multerer et al. ^[5] discussed the dynamical behaviors of Aedes Aegypti mosquito in terms of partial differential equations and also described an Allee effect to capture extinction events and optimal control analysis to identify the release strategy that eliminates the mosquitoes. Zheng et al. ^[6] investigated the existence and stability results for the mosquito and human populations model in which contains dengue serotype circulates, the delay terms capture the respective intrinsic and extrinsic incubation periods, as well as the maturation delay between mating and emergence of adult mosquitoes ^[7,8].

There are mainly two biological control techniques for mosquito populations: Incompatible Insect Techniques (IIT) and Sterile Insect Techniques (SIT). The IIT contains an intracellular bacterium, Wolbachia, commonly found in insects including mosquitoes. The technique of releasing mosquitoes carrying Wolbachia to replace the wild mosquito population is a type of population replacement strategy. If the females mosquitoes infected Wolbachia, then its produced offsprings will also be infected. If the females are uninfected, mate with male with infected Wolbachia, then the offsprings will be dead ^[9]. Ormaetxe et al. ^[10] reported the availability of different Wolbachia strains, which are successfully implanted to Aedes Aegypti mosquitoes. The important thing of such techniques is stopping the spread of mosquitoes and reducing the mosquitoes life span. The authors in ^[11], discussed the augmentation control strategies of Aedes Aegypti mosquitoes with infected Wolbachia. Li et al. ^[12] investigated the spread of Wolbachia in sex-structured mosquitoes model with birth pulse and studied the sufficient condition for stability of total replacement periodic solution. Zheng et al. ^[13] studied the qualitative properties of Aedes Aegypti mosquitoes with infected Wolbachia and maturation delay terms. Li et al. ^[14] introduced the stage-structured mosquitoes model with Wolbachia infection and key factors like, including male killing effect, cytoplasmic incompatibility (CI), fecundity cost due to fitness effect, different mortality rates for infected individuals and maternal transmission are discussed. The authors in ^[15] discussed the dynamical behaviours the Wolbachia infected mosquito model with delays and effective control techniques.

In the SIT system, natural mosquito reproduction is disrupted and modified by physical, chemical, and radical methods into male mosquitoes that are sterile. These sterile mosquitoes are then released into the environment to mate with wild mosquitoes that are present in the environment. Repeating the process of sterile mosquitoes releases may control or wipe out the wild mosquitoes population. The authors ^[16,17] reported the mathematical modelling of SIT to Aedes albopictus, which transmits the Chikungunya diseases and discussed pulsed periodic releases, which is useful to prevent, eliminate, reduce the diseases. Almeida et al. ^[18] discussed the control techniques, SIT and IIT to the mosquitoes model and optimizing the dissemination protocol for each of these strategies, in order to get as close as possible to these objectives. Nowadays, some researchers focused on the dynamical results of SIT into mosquitoes models; see ^{[19,20,21,22]}.

In the most recent literature, continuous dynamical models and continuous mosquito releases were discussed. Instantaneous releases are a key feature of sterile mosquito releases, and they should be repeated numerous times to bring the mosquito population under control. Generally, continuous dynamical models have disadvantages in defining such control analysis with the impulsiveness nature of releases. Impulsive sterile releases can be applied to make up for a lack of such capacity. Huang et al. ^[23] reported the dynamical behavior of the interaction of sterile and wild mosquitoes and also discussed the impact of periodic impulsive sterile mosquitoes releases. Zhang et al. ^[24] described the stroboscopic maps which used to define the numbers of mosquitoes with uninfected and infected Wolbachia immediately after each birth pulse at the discrete times. Li et al. ^[25] studied the dynamical behaviors of mosquitoes and Wolbachia model with impulsive general birth and death rate functions and the sufficient conditions of mosquito extinction are discussed. Bliman et al. ^[26] reported the open and closed-loop control strategies for releasing impulsive sterile male insect techniques in the wild mosquito population, and they studied the open-loop control technique. In cases where the size of the wild mosquito population cannot be determined in real-time, cyclical impulsive releases of sterile males with constant release sizes are used. A closed-loop control strategy is proposed if periodic assessments of wild population size (synchronized with releases or sparser) are available in real-time ^[27]. The authors in ^[28,29] described the modelling of sterile and incompatible insect techniques for the mosquitoes model and suppression of the population. Mosquitoes epidemic models described through the impulsive differential equations have received much attention from the researchers (see ^[30,31,32]).

Herein, we provide new models of Aedes Aegypti mosquitos with infected Wolbachia and impulsive discharges of sterile mosquitos, inspired by earlier studies. We investigate periodic impulsive sterile emissions and formulate parameters for effective eradication based on the magnitude and frequency of the releases. We use a combination of SIT and IIT approaches to produce high sterile male mosquitoes in these models. Periodic releases are calculated so as to maintain the mosquito-free equilibrium. In this environment, open-loop techniques can be designed that ensure mosquito eradication in a finite amount of time without estimating the size of the wild mosquito population. By using the open-loop control technique, sterile mosquitoes are released every $\tau$ days in order to eliminate wild mosquitoes. We can also create closed-loop control qualities to determine the size of the wild population.

In this paper, we develop a Wolbachia Aedes Aegypti Mosquito model and examine the findings of local stability in Section 2. The open and closed loop control features of the underlying model are studied in Section 3. In Section 4, the numerical results are discussed in order to confirm the theoretical results. Section 5 ends with a conclusion.

2.   Modeling of Aedes Aegypti mosquitoes with Wolbachia

Assume that $M_u(t), M_w(t)$ and $F_u(t), F_w(t)$ denote the population density of male and female Aedes Aegypti mosquitoes with uninfected, infected Wolbachia respectively. Consider a Aedes Aegypti Mosquitoes with Wolbachia model of the form

This model assumes that all females mate equally. The parameter $\beta$ includes direct or indirect effects of competition at different stages (larvae, pupae, adults). Let $r \in (0, 1)$ define the primary sex ratio in offspring, and $\rho_u, \rho_w$ represent the mean number of eggs produced by a single female (uninfected and infected Wolbachia) who can deposit on average per day. $\gamma_w$ measures the competition between male mosquitoes with uninfected and infected Wolbachia for the female mates. $\mu_{M_u}, \mu_{F_u}, \mu_{M_w}, \mu_{F_w}$ are mean death rates of adult mosquitoes with uninfected, infected Wolbachia respectively. The basic offspring numbers are described as $N_{M_u} = \frac{r\rho_u}{\mu_{M_u}}, N_{F_u} = \frac{(1-r)\rho_u}{\mu_{F_u}}, N_{M_w} = \frac{r\rho_w}{\mu_{M_w}},$ and $N_{F_w} = \frac{(1-r)\rho_w}{\mu_{F_w}}$. Generally, the male mortality is larger than female, so let us take $\mu_{M_u}\geq \mu_{F_u}, \mu_{M_w}\geq \mu_{F_w}.$

Now, we discuss the local asymptotic stability of (2.1). The model (2.1) has four steady states:

(i) Mosquito free equilibrium point $\mathcal{E}_{0} = (0, 0, 0, 0)$.

(ii) If $N_{F_u} > 1$, Wolbachia free equilibrium point $\mathcal{E}_{1} = (M_{u}^{*}, F_{u}^{*}, 0, 0),$

where $M_{u}^{*} = \frac{N_{M_u}}{N_{M_u}+N_{F_u}}\frac{1}{\beta} ln N_{F_u}, F_{u}^{*} = \frac{N_{F_u}}{N_{M_u}+N_{F_u}}\frac{1}{\beta} ln N_{F_u}$ and $M_u^*+F_u^* = \frac{1}{\beta} ln N_{F_u}.$

(iii) If $N_{F_w} > 1$, all Wolbachia infected equilibrium point $\mathcal{E}_{2} = (0, 0, M_{w}^{*}, F_{w}^{*})$ where $M_{w}^{*} = \frac{N_{M_w}}{N_{M_w}+N_{F_w}}\frac{1}{\beta} ln N_{F_w}, F_{w}^{*} = \frac{N_{F_w}}{N_{M_w}+N_{F_w}}\frac{1}{\beta} ln N_{F_w}$ and $M_w^*+F_w^* = \frac{1}{\beta} ln N_{F_w}.$

$(iv)$ If $N_{F_u}, N_{F_w} > 1$, the interior equilibrium point $\mathcal{E}_{3} = (M_{u}^{*}, F_{u}^{*}, M_{w}^{*}, F_{w}^{*}),$ where

The Jacobian matrix of the model (2.1) is described by

where

The Jacobian matrix at $\mathcal{E}_{0}$ is

Then, the trivial equilibrium point $\mathcal{E}_{0}$ is locally stable if $N_{F_w} < 1.$

The characteristic equation at Wolbachia free equilibrium point $\mathcal{E}_{1}$ is defined as

where

$p_1 > 0, p_2 > 0$ are necessary conditions for all roots in the characteristic equation $\lambda^2+p_1 \lambda +p_2 = 0$ to have negative real parts.

Lemma 1. Suppose that $N_{F_u} > 1.$ Then Wolbachia free equilibrium point $\mathcal{E}_{1}$ is locally asymptotically stable if $p_1 > 0, p_2 > 0.$

The characteristic equation at all Wolbachia infected equilibrium point $\mathcal{E}_{2}$ is defined as

where

$r_1 > 0, r_2 > 0$ are necessary conditions for all roots in the characteristic equation $\lambda^2+r_1 \lambda +r_2 = 0$ to have negative real parts.

Lemma 2. Suppose that $N_{F_w} > 1,$ then all Wolbachia infected equilibrium point $\mathcal{E}_{2}$ is locally asymptotically stable if $r_1 > 0, r_2 > 0.$

The characteristic equation at interior equilibrium point $\mathcal{E}_{3}$ is defined as

where

According to the Routh-Hurwitz criterion, all roots $\lambda_{1, 2, 3, 4}$ of the characteristic equation $\lambda^4+s_1\lambda^3+ s_2 \lambda^2+s_3 \lambda +s_4 = 0$ must be negative real parts. The conditions are

We arrive at the following Lemma.

Lemma 3. The interior steady state $\mathcal{E}_{3}$ is locally asymptotically stable if $s_1 > 0, s_3 > 0, s_4 > 0, s_1s_2s_3 > s_3^2+s_1^2s_4$.

2.1.   Continuous releases of sterile male mosquitoes

Here, we incorporate continuous releases sterile male mosquitoes into model (2.1) and assume $\gamma_w = 1$, the revised model takes the form

The sterile male population density at time $t$ is $M_s(t)$. At the beginning of each release period, $\Lambda$ is the number of sterile male mosquitoes released. The mean death rate of sterile mosquitoes is $\mu_{M_s}$. $\gamma_s$ be a relative reproductive efficiency and the value is smaller than one. The recruitment terms in (2.2) only include the successful mating of uninfected females $F_u$ and Wolbachia infected females $F_w$, i.e., those leading to viable offspring, that are detected with probabilities $\frac{M_u}{M_u+M_w +\gamma_s M_s}$and $\frac{M_u+ M_w}{M_u+M_w+\gamma_s M_s}$, respectively. From (2.2), the equilibrium point $M_{s}^{*} = \frac{\Lambda}{\mu_{M_s}}.$ Moreover, $\mu_{M_s}\geq \mu_{M_u}$ and $\mu_{M_s}\geq \mu_{M_w}.$

Theorem 1. Assume $N_{F_w} > 1$, then the $\Lambda^{crit}$ value described as

Proof: From the model (2.2)

and

If $N_{F_w} > 1$, there exists $\Lambda^{crit} > 0$ such that (2.2) have one non negative steady state at $\Lambda = \Lambda^{crit}$ and no non negative steady state at $\Lambda > \Lambda^{crit}$. The asymptotic stability results of the model (2.2) are similar to those in the previous subsection, so it is omitted.

2.2.   Impulsive releases of sterile male mosquitoes

Now, we incorporate periodic impulsive sterile male mosquitoes into model (2.2), the model becomes

Make $\Lambda_n$ a constant and drop consequently the subindex $n$. $M_{s}(n \tau^\pm)$ represents the right and left limits of $M_{s}(t)$ at $t = n\tau.$ In other terms, model (2.3) evolves according first four equations of (2.3) on the union of open intervals ($n \tau$, $(n+1) \tau$). While $M_{s}$ undergoes jumps at each $n\tau$, accounting for the released sterile male mosquitoes. For such release schedule, when $t \rightarrow \infty,$ the function $M_s$ converges towards the following periodic solution

We define the periodic system

The model (2.3) has the same mosquito free equilibrium point $\mathcal{E}_{0}$. Now, we are going to study the conditions under which mosquitoes free steady state is asymptomatically stable. For the such study, we find the mean value of $\frac{1}{M_{s}^{per}}$,

3.   Stability analysis and main results

Theorem 2. Assume that

Therefore, the solution of (2.4) converges globally exponentially to the steady-state $\mathcal{E}_{0}$.

Proof: From (2.4),

For any $F_u \geq 0$ and $t \geq 0$ and use $\alpha = \{x e^{-\beta x}; \quad x \geq 0\} = \frac{1}{e \beta}$

Integrate from $n\tau$ to $t$ with $n\tau < t$, we get

Thus, the sequence $\{F_u(n\tau)\}_{n \in \mathbb{N}}$ approaches to $0$,

From the first equation of (2.4), similarly, we can prove $\Big < \frac{1}{ M_s^{per}}\Big > < e \beta \gamma_s \frac{1}{N_{M_u}}$. Let us consider

Integrate from $n\tau$ to $t$ with $n\tau < t$,

Thus, the sequence $\{F_w(n\tau)\}_{n \in \mathbb{N}}$ decreases towards $0$, provided that

Similarly, from (2.4), we have $\Big < \frac{1}{ M_s^{per}}\Big > < e \beta \gamma_s \frac{1}{N_{M_w}}$.

Providing the necessary conditions $\Big < \frac{1}{ M_s^{per}}\Big >$ leads to sufficient conditions for asymptotic stability at $\mathcal{E}_{0};$

Lemma 4. Let $k_1, k_2$ be a real number such that $0 < k_1 < \frac{1}{N_{F_u}}, 0 < k_2 < \frac{1}{N_{F_w}}.$ Then, every solution of (2.2) such that $\frac{M_u}{M_u+M_w+\gamma M_{s}} \leq k_1$ and $\frac{M_u+M_w}{M_u+M_w+\gamma M_{s}} \leq k_2, t \geq 0,$ converges exponentially to $\mathcal{E}_{0}$.

Proof: By using the assumptions $\frac{M_u}{M_u+M_w+\gamma M_{s}} \leq k_1$ and $\frac{M_u+M_w}{M_u+M_w+\gamma M_{s}} \leq k_2,$ the model (2.2) becomes

The autonomous linear system

is monotone (Metzler matrix involved) (see ^[33]), and it can be used as a comparison system for the evolution of (2.2). Thus, it is deduced that

Here, $(M_u^{'}, F_u^{'}, M_w^{'}, F_w^{'})$ be the solution of the linear system (3.2) obtained by same initial condition as the solution $(M_u, F_u, M_w, F_w)$ of (2.2). Furthermore, the linear system (3.2) is asymptotically stable if $0 < k_1 < \frac{1}{N_{F_u}}, 0 < k_2 < \frac{1}{N_{F_w}}.$, i.e., $(M_u^{'}, F_u^{'}, M_w^{'}, F_w^{'})$ asymptotically converges to $\mathcal{E}_{0}$. Based on this, $(M_u, F_u, M_w, F_w)$ also asymptotically converges to $\mathcal{E}_{0}$.

Remark 1. The upper bound of $k_1$ and $k_2$ are fixed on the ratio $\frac{M_u}{M_u+M_w+\gamma M_{s}}$ and $\frac{M_u+M_w}{M_u+M_w+\gamma M_{s}}$ respectively, in order to make the apparent basic offspring number $k_1 N_{F_u}$ and $k_2 N_{F_w}$ is smaller than $1.$

Here, we want to verify the condition $\frac{M_u+M_w}{M_u+M_w+\gamma M_{s}} \leq k_2$, based on the sufficient impulse sterile releases $\Lambda_n$. Before, the value of $M_{s}$ on $(n\tau, (n+1)\tau]$ is described as

We impose the stronger condition instead of $\frac{M_u+M_w}{M_u+M_w+\gamma M_{s}} \leq k_2$, on $(n\tau, (n+1)\tau]$

where $M_u^{'}(t), M_w^{'}(t)$ refers to super solution of $M_u(t), M_w(t)$ (Lemma 4).

Lemma 5. The solution of (3.2) on $(n\tau, (n+1)\tau]$ with initial conditions \\ $(M_u^{'}(n \tau), F_u^{'}(n \tau), M_w^{'}(n \tau), F_w^{'}(n \tau)) = (M_u(n \tau), F_u(n \tau), M_w(n \tau), F_w(n \tau))$ is defined by

where $p_1 = e^{- \mu_{M_u}(t-n \tau)}, p_2 = e^{- (\mu_{F_{u}}-(1-r)\rho_u k_1)(t-n \tau)}, p_3 = e^{- \mu_{M_w}(t-n \tau)}, p_4 = e^{- (\mu_{F_{w}}-(1-r)\rho_w k_2)(t-n \tau)},$ $d_1 = \frac{r \rho_u k_1}{\mu_{M_{u}}-\mu_{F_{u}}+(1-r)\rho_u k_1} (e^{- (\mu_{F_{u}}-(1-r)\rho_u k_1)(t-n \tau)}-e^{- \mu_{M_{u}}(t-n \tau)})$, and $d_2 = \frac{r \rho_w k_2}{\mu_{M_{w}}-\mu_{F_{w}}+(1-r)\rho_w k_2} (e^{- (\mu_{F_{w}}-(1-r)\rho_w k_2)(t-n \tau)}-e^{- \mu_{M_{w}}(t-n \tau)}).$

Now, we define the feedback control analysis, on any $(n\tau, (n+1)\tau],$ substitute the values of (3.3) and (3.5) into (3.4), we get

We arrive the following theorem:

Theorem 3. For a given $k_1\in \Big(0, \frac{1}{N_{F_u}} \Big), k_2 \in \Big(0, \frac{1}{N_{F_w}} \Big)$, assuming, for $n \in \mathbb{N}$,

Then, every solution of (2.3) exponentially converges to $\mathcal{E}_{0}$ with a rate of convergence restricted from below by a value unrelated to the initial value. Moreover,

then the series $\sum_{n = 0}^{+\infty}\Lambda_n$ also converges.

Proof: Suppose $(M_u(n\tau), F_u(n\tau), M_w(n\tau), F_w(n\tau)) = (0, 0, 0, 0)$, there is no impulsion effect $\Lambda_n$ on the evolution of $(M_u, F_u, M_w, F_w).$ Let us consider $(M_u(n\tau), F_u(n\tau), M_w(n\tau), F_w(n\tau))\neq (0, 0, 0, 0)$.

Case(i) Assume the strict inequality of (3.6), one can easily get

where the solution $(M_u^{'}, F_u^{'}, M_w^{'}, F_w^{'})$ of (3.2) starting from $(M_u(n\tau), F_u(n\tau), M_w(n\tau), F_w(n\tau))$ at $t = n\tau.$ Initially, we establishes like that

Let $t_0 \in [ n\tau, (n+1)\tau)$ such that $M_u(t_0) \leq M_u^{'}(t_0), F_u(t_0) \leq F_u^{'}(t_0), M_w(t_0) \leq M_w^{'}(t_0), F_w(t_0) \leq F_w^{'}(t_0)$ with at least one equality. Let us show the existence of $t_1$ such that $t_0 < t_1 < (n+1)\tau$,

Based on (3.8) and the definition of $t_0$,

when $t_0 = n\tau,$ and $M_{s}(t_0) = M_{s}(n\tau^+).$ The functions $M_u(t), F_u(t), M_w(t), F_w(t)$ and $M_{s}(t)$ are continuous on $(n\tau, (n+1)\tau),$ there exists $t_1$ such that $t_0 < t_1 < (n+1)\tau$

It can be shown as in Lemma 4 that $(M_u^{'}(t), F_u^{'}(t), M_w^{'}(t), F_w^{'}(t))\geq (M_u(t), F_u(t), M_w(t), F_w(t))$ for any $t \in (t_0, t_1)$, Also $(M_u^{'}(t), F_u^{'}(t), M_w^{'}(t), F_w^{'}(t)) > (M_u(t), F_u(t), M_w(t), F_w(t))$ because the functions defining the right hand sides of (2.3) take on strictly smaller values than those defining the r.h.s of (3.2). Therefore, for any $t_0 \in {(n\tau^{+})}\cup (n\tau, (n+1)\tau),$ there exists $t_1 > t_0$ such that (3.10) holds. From (3.10) and the fact that $(M_u(n\tau), F_u(n\tau), M_w(n\tau), F_w(n\tau)) = (M_u^{'}(n\tau), F_u^{'}(n\tau), M_w^{'}(n\tau), F_w^{'}(n\tau)),$ one deduces that (3.10) is true with $t_1 = (n+1)\tau$, and (3.9) is true. Finally, putting together (3.8) and (3.9) yields

Case(ii) Assume (3.6) holds, and instead of (3.11), considering the $\Lambda_n$ values convergent from above to the quantity in the r.h.s of this inequality, and trust the flow's consistency with respect to $\Lambda_n$.

From (3.12), for any $t \in (n\tau, (n+1)\tau ]$, $\frac{M_u}{M_u+M_w+M_{s}} \leq k_1$ and $\frac{M_u+M_w}{M_u+M_w+M_{s}} \leq k_2$.

From the values of $\Gamma$, there exists $\epsilon_1, \epsilon_2 > 0$ such that $\mu_{F_u}-(1-r)\rho_u k_1 > \epsilon_1, \mu_{F_w}-(1-r)\rho_w k_2 > \epsilon_2$ and then $\dot{F}_u \leq -\epsilon_1 F_u, \dot{F}_w \leq -\epsilon_2 F_w.$ Therefore, $F_u(t), F_w(t)$ exponentially converges to $0$. It is then deduced from first and third equation of (2.3) that $M_u(t), M_w(t)$ also exponentially converges to $0$. $(M_u(t), F_u(t), M_w(t), F_w(t))$ converges to $\mathcal{E}_{0}$.

Finally, $\Lambda_n$ satisfies (3.6) and (3.7), based on exponentially stability concepts, there exist $C, \epsilon > 0$ such that $M_u(t) < C e^{-\epsilon t}, F_u(t) < C e^{-\epsilon t}, M_w(t) < C e^{-\epsilon t}, F_w(t) < C e^{-\epsilon t}, t\geq 0.$ We can easily obtain that

The above series is convergent.

4.   Numerical simulations

We consider the parameter values of the model: $r = 0.5$, $\rho_u = 6.4$, $\rho_w = 3.5$, $\gamma_s = 0.9$, $\mu_{M_{u}} = 0.07$, $\mu_{F_{u}} = 0.06$, $\mu_{M_{w}} = 0.04$, $\mu_{F_{w}} = 0.03$, $\mu_{M_s} = 0.11$, $\sigma = 0.05$, $K = 155$, $\beta = \frac{\sigma}{K} = 3.22\times 10^{-4}.$ The basic offspring numbers: $N_{M_{u}}\approx 45.71$, $N_{F_{u}}\approx 53.33$, $N_{M_{w}}\approx 43.75$, $N_{F_{w}}\approx 58.33$. The basic offspring figures show the average number of children produced over a person's lifespan. Figures 1 and 2 display the stable behaviour of the steady states $\mathcal{E}_1, \mathcal{E}_2$ and $\mathcal{E}_3$, respectively. Let's starting with the regular rash discharges of sterile male mosquitoes. The releasing method derived in Theorem 2 is demonstrated. For open-loop periodic impulsive releases carried out every 7 and 14 days. Consider the smallest value in (3.1) to estimate the number of sterile male mosquitoes to release, i.e., $\Lambda_{per}^{crit}\times 7 = 1525$ $\times$ $7 = 10,675$ and $\Lambda_{per}^{crit}\times 14 = 1595\times 14 = 22,330$, sterile male mosquitoes per hectare and per two weeks, respectively. Figure 3 depicts the corresponding simulations.

The closed-loop method can be utilized to reduce the total number of sterile insects released. Theorem 3 shows how the method can be used to lower the total number of sterile insects released. We consider the wild population every $\tau$ days both here and in the examination of the feedback technique. We also look at the values of $k_1$ and $k_2$ to show the tradeoff between treatment duration and control effort. A small $k_1$ and $k_2$ results in a larger control effort and a faster convergence to ${\mathcal E}_0$. For $k_1N_{F_u} = 0.2$, $k_2N_{F_w} = 0.25$, $\tau = 7$ and $k_1N_{F_u} = 0.2$, $k_2N_{F_w} = 0.25$, $\tau = 14$, the diagrams displayed in Figure 4 show that that the wild population is close to extinction with the help of SIT treatment. The control effort is smaller and convergence should be delayed for bigger values of $k_1$ and $k_2$. For $k_1N_{F_u} = 0.7$, $k_2N_{F_w} = 0.8$, $\tau = 7$ and $k_1N_{F_u} = 0.7$, $k_2N_{F_w} = 0.8$, $\tau = 14$, the diagrams are displayed in Figure 5. The size $\Lambda_n$ of the $n^{th}$ release is taken equal to the right-hand side of (3.6). As it can be clearly seen that $k_1$, $k_2$ and $\tau$ have a significant impact on the mosquito population convergence to $\mathcal{E}_0$.

5.   Concluding remarks

Our study utilized a Wolbachia-infected Aedes Aegypti mosquitoes model, followed by continuous and impulsive releases of sterile male mosquitoes. Mosquitoes can be controlled by releasing sterile mosquitoes or by replacing the wild population with one that carries Wolbachia bacteria, which prevents the transmission of viruses from mosquitoes to humans. Our analysis suggests that despite the lower fitness of the Wolbachia-carrying population in comparison to the wild population, the CI-reproductive phenotype gives the system an advantage when shifting from a wild to a Wolbachia-carrying population. A number of interesting sufficient conditions have been derived for the model's local asymptotic stability. The conditions required for open-loop or closed-loop control systems were also assessed using sterile male mosquitoes released impulsively. Based on wild population estimates, release sizes are determined in closed-loop control. As shown by (3.6), the released volume is essentially proportional to the measured population. Using sterile insect techniques may lead to periodic outbreaks of mosquitoes due to a periodic oscillation in the system. The theoretical results confirm the numerical solution to the proposed model.

The use of control variables (open, closed-loop, and mixed control techniques) will be considered in future research to determine the best SIT technique, mixed control, and mosquito population elimination strategies.

Acknowledgments

This research was funded by the UAEU Research, fund # 12S005-2021.

Conflict of interest

The authors declare no conflicts of interest.