# Approximate Solution of the Fractional Order Sterile Insect Technology Model via the Laplace–Adomian Decomposition Method for the Spread of Zika Virus Disease

Authors: William Atokolo; Remigius Okeke Aja; Stephen Ekwueme Aniaku; Ifeanyi Sunday Onah; Godwin C. E. Mbah
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

Sterile insect technology (SIT) is an environmental-friendly method which depends on the release of sterile male mosquitoes that compete with the wild male mosquitoes and mate with wild female mosquitoes, which leads to the production of no offspring and as such reduces the population of Zika virus vector population over time, thereby eliminating the spread of Zika virus in a population. The fractional order sterile insect technology (SIT) model to reduce the spread of Zika virus disease is considered in this present work. We employed the use Laplace–Adomian decomposition method (LADM) to determine an analytical (approximate) solution of the model. The Laplace–Adomian decomposition method (LADM) produced a solution in form of an infinite series that further converges to the exact value. We compared solutions of the fractional model with the classical case using our plots and discovered that the fractional order has more degree of freedom and as such the system can be varied to get many preferred responses of the different classes of the model as the fraction(β) could be varied to the desired rate, say 0.7, 0.4, etc. We have been able to show that LADM can be used to solve an SIT model which has never been done before in literature.


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## Body

## 1. Introduction

Zika is a viral infection that is usually spread in human population by the bite of infected mosquitoes. It was discovered in 1947 in Uganda [1]. The most common way to contact Zika virus is from the bites of an infected mosquito. Two species of mosquitoes spread the virus to people; the yellow fever mosquitoes (Aedes aegypti) and the Asian tiger mosquitoes (Aedes albopictus). Both are native to Texas [1, 2]. The sterile insect technology is an environmental-friendly insect pest control method involving the mass rearing and sterilization using radiation, of a target pest, followed by the systematic area-wide release of the sterile males by air over defined areas, where they mate with wild females, resulting in no offspring and a declining pest population [3, 4]. Mathematical modeling of disease transmission dates back to 1766 as was first presented by D. Bernoulli. This has metamorphosed to a deep study of disease transmission and control especially with infectious diseases like HIV/AIDS, Lassa fever, cholera, malaria, and others [5–10]. In recent years, fractional calculus has attracted great attention from researchers and different aspects of the said subject are under consideration for research; this is due to the fact that the fractional derivative is an important tool to explain the dynamical behaviour of various physical systems [11]. The strength of these differential operator as presented in [11] is their nonlocal characteristics which do not exist in the integer order differential operators and that fractional order models are more realistic and practical than the classical integer order model.The technique of Laplace–Adomian decomposition method (LADM) involves the combination of the Adomian decomposition method (ADM) and the Laplace transforms. Adomian’s is an effective technique for obtaining solutions of model or a system of ordinary differential equations. Laplace transform is an efficient method used in engineering and applied sciences. The coupling of these two methods leads to the Laplace–Adomian decomposition method (LADM). The Laplace–Adomian decomposition method has been applied to many problems in physics, biology, applied mathematics, and engineering [11]. The basic idea of this method is to assume an infinite solution of the kind: q=∑n=0∞qn, then apply Laplace transformation to the differential equation. The nonlinear terms of the model are then decomposed in terms of the Adomian polynomials, and then an iterative procedure is formulated for the determination of the qn in a recursive form. This method can be used for a system of linear and nonlinear ordinary and partial differential equations of the classical and fractional order. The method does not require any perturbation and also has no need for a predefined step size. The method is an effective method for numerical and explicit solutions of a system of differential equations representing physical problems [11].Fazal et al. [11] presented a numerical solution of the fractional order smoking model via the Laplace–Adomian decomposition method; the model solution was obtained in form of an infinite series which converges rapidly to its exact value. Ogun [12] presented the Laplace–Adomian decomposition method to solve a model for HIV infection where the approximate solution of the model was determined. Adejoh and Mbah [13] also presented the application of fractional differential equations to obtain an approximate and numerical solution of a cancer disease model incorporating control measures. The Laplace–Adomian decomposition method was also used by Fazal et al. [14] to determine a numerical solution of a fractional order epidemic model of a childhood disease. The Laplace–Adomian decomposition method unlike other numerical methods requires no discretization and linearization and as such the results obtained from it are more effective and realistic. In fact, models such as the ones in [11–14] are veritable tools toward studying the application of LADM in solving linear and nonlinear differential equations.The fractional order model gives a better description of the entire space of a system; unlike the integer model that describes only the local properties of a system, it also gives a better description of a real system with memory effects [15–18]. The Caputo derivative and the Riemann–Liouville derivative are regarded as singular kernels fractional derivative relative to biological problems, we also have others which are nonsingular such as Mittag-Leffler and the Atangana–Baleanu operators [17, 19]. The fractional order and classical derivatives have been used in studying transmission dynamics in SIR models and the like, but it has not been adopted in an SIT model. Also, an SIT model solution has not been determined using the LADM. So, in this work, we consider a fractional order model (using the Caputo derivative), which is an extension of the classical order sterile insect technology model for the control of Zika virus disease presented by Atokolo et al. [1], whose approximate solution would be determined using the Laplace–Adomian decomposition method (LADM).


## 2. Model Formulation and Procedures

The mosquito life cycle is generally divided into two stages, the aquatic and nonaquatic class. The aquatic class is denoted by a single compartmentA. The nonaquatic mosquito class is divided into seven compartments consisting of the male mosquitoes MM, female mosquitoes not yet laying eggs FM, female nonsterile mosquitoes, FNM, female sterile mosquitoes FSM, sterile male mosquitoes MS, female infected nonsterile mosquitoes, FINM, and female infected sterile mosquitoes FISM. The human population is divided into susceptible human SH, exposed human EH, infected human IH, infected human but on treatment IHT, and recovered human RH. The aquatic stage of the mosquitoes which consists of eggs, larva, and pupae population increases from the oviposition by reproductive mosquitoes. It reduces due to natural death of the mosquitoes at the rate of μA and by density dependence death rate of μρ. The female mosquitoes FM are recruited at the rate of Aϕγ, where γ is the maturity rate of aquatic mosquitoes to adult mosquitoes and ϕ is the proportion of emerging females; it is reduced by the mating rate at the level of β for female mosquitoes to be with wild male mosquitoes or sterile male mosquitoes with mating probabilities ρω and ρs, respectively. The population is reduced by death induced due to the attempt to seek for blood meals at the rate of δM and finally reduced by natural death at the rate of μF. The male mosquitoes MM are recruited by the proportion of the emerging male mosquitoes 1−ϕ that mature to adult mosquitoes at the rate of γ which also reduces by natural death at the rate of μM. The female nonsterile mosquito FNM population is increased by the female mosquitoes’ probability to mate with the wild male mosquitoes which is given by the rate MM/MM+MS, with a mating rate of β1. This population is reduced by ω1, the rate at which the female nonsterile mosquitoes FNM are infected and moved to the female infected nonsterile mosquitoes class FINM. It is also reduced by the death induced due to the attempt to seek for blood meals at the rate of δM; this class is finally reduced by natural death rate at the rate of δM. The female sterile mosquito FSM population is increased by the wild female mosquitoes’ probability to mate with the sterile mosquitoes which is given by the rate MS/MM+MS, with a mating rate of β2. The class reduces by ω2, the rate at which the FSM becomes infected and moves to the FISM class. The class reduces by death induced due to the attempt to seek for blood meals at the rate of δM and reduces finally by natural death at the rate of μF. The population of female infected nonsterile mosquitoes FINM is recruited at the rate at which the female nonsterile mosquitoes FNM are infected at the rate of ω1 The population is reduced by death induced due to the attempt to seek for blood meals at the rate of δM and finally reduced by natural death at the rate of μF. The population of female infected sterile mosquitoes FISM is recruited at the rate at which the female sterile mosquitoes FSM are infected at the rate of ω2. The population is reduced by death induced due to the attempt to seek for blood meals at the rate of δM and finally by natural death at the rate of μF. The sterile male mosquitoes MS are released into the population at the rate Λ2. However, due to some environmental and geographical factors that may affect the mixing of sterile and wild mosquitoes, such as location of mosquitoes breeding site, it is convenient to assume that only a fraction p of the released mosquitoes will join wild mosquitoes population. Second, because of the differences in physiology of wild and sterile mosquitoes, a parameter q is used to capture the mean mating competitiveness of sterile mosquitoes, so that the actual number of sterile male mosquitoes competing with wild mosquitoes is pqMS, and as such, the available injected sterile male mosquitoes MS into the wild population of mosquitoes that can competitively mate with wild female mosquitoes is pqΛ2. This population is reduced by natural death at the rate of μS. The susceptible human population is recruited at the level of Λ3, of which a fraction l of those infected at birth joined the infectious human population. The population reduces by the rate at which infectious mosquitoes (female infected nonsterile mosquitoes FINM or female infected sterile mosquitoes FISM) infects susceptible human at the levels of α1 and α2, respectively. Also, it reduces by the rate at which the infectious human (infected class IH, recovered class RH or infected but on treatment class IHT(IHT) infects susceptible human through sex at the level of or α3 or α4, α5, respectively. This is in line with the clinical studies that high viral load was found in the semen and saliva of recovered patients weeks after recovery, (WHO, 2016), which means, Zika can be transmitted sexually. The population finally reduces by natural death at the rate of μH. The population of the exposed human EH is generated by infection of susceptible individuals at the rate of α. This population reduces by natural death at the of rate μH and by the rate at which the exposed are finally infectious at the rate of σ. The infected human IH class is generated by the incoming of infected babies from infected mothers at the rate of Λ3, due to vertical transmission. In addition, the population increases at the rate by which the exposed become infected at the level σ. The class reduces at the rate θ by which the infected are taken for treatment and by natural recovery rate of τ1. This class reduces finally by both natural and disease-induced death rates at the levels of μH and δ1, respectively. The infected but on treatment class IHT is recruited by the incoming of the infected who are taken for treatment at the rate of θ; this class reduces at the rate by which the infected but on treatment class recovers due to supportive treatment at the rate of τ2. It reduces finally by natural death and disease induced death at the rates of μH and δ2, respectively, where δ2 is assumed to be less than δ1. The recovered human is recruited at the rate by which the infected human recovers naturally at the rate of τ1 or due to supportive treatment at the rate of τ2. The population reduces by natural death at the rate of μH.


### 2.1. Assumptions of the Model

(1)
Zika virus can be transmitted through the bite of Aedes mosquitoes or through sexual activities with an infected human.(2)
There is both vertical and horizontal transmission.(3)
The mating competitiveness of the sterile and nonsterile mosquitoes are not equal.(4)
Mosquitoes do not recover from infection.(5)
Aquatic mosquitoes have a density dependent death rate which is a nonlinear decreasing function.(6)
Aquatic and nonaquatic mosquitoes do not have the same death rate.(7)
Female mosquitoes do not have the same death rate with male mosquitoes.(8)
There is a disease-induced death rate by the female mosquitoes seeking blood meals.


### 2.2. Mathematical Model

The mathematical equations that incorporate the above formulations, assumptions, and from Figure1, we have the following:(1)dAdt=Λ1FNM+FINM−1−ϕγA−ϕγA−μAA−μρA2,dFMdt=ϕγA−β1MMMM+MS+β2MSMM+MSFM−δMFM−μFFM,dMMdt=1−ϕγA−μMMM,dFSMdt=β1MMMM+MSFM−ω1FNM−δMFNM−μFFNM,dFINMdt=β2MMMM+MSFM−ω2FSM−δMFSM−μFFSM,dFISMdt=ω1FNM−δMFINM−μFFINM,dMSdt=ω2FSM−δMFISM−μFFISM,dSHdt=pqΛ2−μSMS,dEHdt=1−ℓΛ3−αSH−μHSH,dEHdt=αSH−σEH−μHEH,dIHdt=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,dIHTdt=θIH−τ2IHT−∂2IHT−μHIHT,dRHdt=τ1IH−τ2IHT−μHRH,where(2)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+α2IHTNH,ω=ω1,=ω2.Figure 1
Flow diagram.From the earlier assumption, the mating competitiveness of both sterile and nonsterile mosquitoes are not equal, that is,β1≠β2.Also, from (1), we let MS/MM+MS=ρS and MM/MM+MS=ρω.Therefore, from the second equation of (1), we have as follows:(3)β1MMMM+MS+β2MSMM+MS=β1ρω+β2ρs,where ρs is the female mating probability with the sterile mosquitoes and ρω is the female mating probability with the wild male mosquitoes. Moreso, the rate of infection of FNM and FSM after biting an infectious human and then moving to the FINM and FISM classes, respectively, is equal, that is, to say ω=ω1=ω2. We also decoupled the sterile male mosquitoes population MS equation from the entire system since it is independent of other compartments and the size of its population is controlled by human intervention. Hence, we can rewrite the mathematical equations that represent the sterile insect technology (SIT) model for the control of Zika virus disease as presented by Atokolo et al. in [1] as follows:(4)dAdt=Λ1FNM+FINM−1−ϕγA−ϕγA−μAA−μρA2,dFMdt=ϕγA−β1MMMM+MS+β2MSMM+MSFM−δMFM−μFFM,dMMdt=1−ϕγA−μMMM,dFSMdt=β1MMMM+MSFM−ω1FNM−δMFNM−μFFNM,dFINMdt=β2MMMM+MSFM−ω2FSM−δMFSM−μFFSM,dFISMdt=ω1FNM−δMFINM−μFFINM,dMSdt=ω2FSM−δMFISM−μFFISM,dSHdt=pqΛ2−μSMS,dEHdt=1−ℓΛ3−αSH−μHSH,dEHdt=αSH−σEH−μHEH,dIHdt=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,dIHTdt=θIH−τ2IHT−∂2IHT−μHIHT,dRHdt=τ1IH−τ2IHT−μHRH,where(5)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+λ2IHTNH.The model variables and parameters descriptions are presented in Appendix.


### 2.3. Fractional Order Zika Model

The Caputo derivative is considered as a differential operator in our model. In a Caputo fractional initial value problem, the initial condition can be expressed with an initial integer order, whose physical interpretation is very easy for real life interpretation, and hence is suitable for the Zika virus model. We now present in this section, some well known definitions and results that we will use throughout this paper.Definition 1.
The Caputo fractional order derivative of a functionf on the interval 0,T is defined as follows:(6)D0βcft=1Γn−β∫0tt−sn−β−1fnsds,where n=β+1 and β represent the integer parts of β. In particularly, for 0<β<1, the Caputo derivative becomes as follows:(7)D0βcft=1Γn−β∫0tfst−sβds.Definition 2.
Laplace transform of Caputo derivatives is defined as follows:(8)ℒDβcqt=SβhS−∑k=0nSβ−i−1yk0,n−1<β<n,n∈N,for arbitrary ci∈R, i=0,1,2,…,n−1, where n=β+1 and β represent the noninteger parts of β.Lemma 1.
The following result holds for fractional differential equations:(9)IβDβcht=ht+∑i=0n−1hi0i!ti,for arbitrary β>0, i=0,1,2,…,n−1, where n=β+1 and β represent the integer parts of β.
Introducing the fractional order into the model (4), we now present a new model described by the following set of fractional differential equations of order β, for 0<β<1(10)DβA=Λ1FNM+FINM−γA−μAA−μρA2,DβFM=ϕγA−β1ρω+β2ρsFM−δMFM−μFFM,DβMM=1−ϕγA−μMMM,DβFNM=β1ρωFM−ωFNM−δMFNM−μFFNM,DβFSM=β2ρsFM−ωFSM−δMFSM−μFFSM,DβFINM=ωFNM−δMFINM−μFFINM,DβFISM=ωFSM−δMFISM−μFFISM,DβSH=1−ℓΛ3−αSH−μHSH,DβEH=αSH−σEH−μHEH,DβIH=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,DβIHT=θIH−τ2IHT−∂2IHT−μHIHT,DβRH=τ1IH−τ2IHT−μHRH,where(11)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+λ2IHTNH.


## 2.1. Assumptions of the Model

(1)
Zika virus can be transmitted through the bite of Aedes mosquitoes or through sexual activities with an infected human.(2)
There is both vertical and horizontal transmission.(3)
The mating competitiveness of the sterile and nonsterile mosquitoes are not equal.(4)
Mosquitoes do not recover from infection.(5)
Aquatic mosquitoes have a density dependent death rate which is a nonlinear decreasing function.(6)
Aquatic and nonaquatic mosquitoes do not have the same death rate.(7)
Female mosquitoes do not have the same death rate with male mosquitoes.(8)
There is a disease-induced death rate by the female mosquitoes seeking blood meals.


## 2.2. Mathematical Model

The mathematical equations that incorporate the above formulations, assumptions, and from Figure1, we have the following:(1)dAdt=Λ1FNM+FINM−1−ϕγA−ϕγA−μAA−μρA2,dFMdt=ϕγA−β1MMMM+MS+β2MSMM+MSFM−δMFM−μFFM,dMMdt=1−ϕγA−μMMM,dFSMdt=β1MMMM+MSFM−ω1FNM−δMFNM−μFFNM,dFINMdt=β2MMMM+MSFM−ω2FSM−δMFSM−μFFSM,dFISMdt=ω1FNM−δMFINM−μFFINM,dMSdt=ω2FSM−δMFISM−μFFISM,dSHdt=pqΛ2−μSMS,dEHdt=1−ℓΛ3−αSH−μHSH,dEHdt=αSH−σEH−μHEH,dIHdt=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,dIHTdt=θIH−τ2IHT−∂2IHT−μHIHT,dRHdt=τ1IH−τ2IHT−μHRH,where(2)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+α2IHTNH,ω=ω1,=ω2.Figure 1
Flow diagram.From the earlier assumption, the mating competitiveness of both sterile and nonsterile mosquitoes are not equal, that is,β1≠β2.Also, from (1), we let MS/MM+MS=ρS and MM/MM+MS=ρω.Therefore, from the second equation of (1), we have as follows:(3)β1MMMM+MS+β2MSMM+MS=β1ρω+β2ρs,where ρs is the female mating probability with the sterile mosquitoes and ρω is the female mating probability with the wild male mosquitoes. Moreso, the rate of infection of FNM and FSM after biting an infectious human and then moving to the FINM and FISM classes, respectively, is equal, that is, to say ω=ω1=ω2. We also decoupled the sterile male mosquitoes population MS equation from the entire system since it is independent of other compartments and the size of its population is controlled by human intervention. Hence, we can rewrite the mathematical equations that represent the sterile insect technology (SIT) model for the control of Zika virus disease as presented by Atokolo et al. in [1] as follows:(4)dAdt=Λ1FNM+FINM−1−ϕγA−ϕγA−μAA−μρA2,dFMdt=ϕγA−β1MMMM+MS+β2MSMM+MSFM−δMFM−μFFM,dMMdt=1−ϕγA−μMMM,dFSMdt=β1MMMM+MSFM−ω1FNM−δMFNM−μFFNM,dFINMdt=β2MMMM+MSFM−ω2FSM−δMFSM−μFFSM,dFISMdt=ω1FNM−δMFINM−μFFINM,dMSdt=ω2FSM−δMFISM−μFFISM,dSHdt=pqΛ2−μSMS,dEHdt=1−ℓΛ3−αSH−μHSH,dEHdt=αSH−σEH−μHEH,dIHdt=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,dIHTdt=θIH−τ2IHT−∂2IHT−μHIHT,dRHdt=τ1IH−τ2IHT−μHRH,where(5)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+λ2IHTNH.The model variables and parameters descriptions are presented in Appendix.


## 2.3. Fractional Order Zika Model

The Caputo derivative is considered as a differential operator in our model. In a Caputo fractional initial value problem, the initial condition can be expressed with an initial integer order, whose physical interpretation is very easy for real life interpretation, and hence is suitable for the Zika virus model. We now present in this section, some well known definitions and results that we will use throughout this paper.Definition 1.
The Caputo fractional order derivative of a functionf on the interval 0,T is defined as follows:(6)D0βcft=1Γn−β∫0tt−sn−β−1fnsds,where n=β+1 and β represent the integer parts of β. In particularly, for 0<β<1, the Caputo derivative becomes as follows:(7)D0βcft=1Γn−β∫0tfst−sβds.Definition 2.
Laplace transform of Caputo derivatives is defined as follows:(8)ℒDβcqt=SβhS−∑k=0nSβ−i−1yk0,n−1<β<n,n∈N,for arbitrary ci∈R, i=0,1,2,…,n−1, where n=β+1 and β represent the noninteger parts of β.Lemma 1.
The following result holds for fractional differential equations:(9)IβDβcht=ht+∑i=0n−1hi0i!ti,for arbitrary β>0, i=0,1,2,…,n−1, where n=β+1 and β represent the integer parts of β.
Introducing the fractional order into the model (4), we now present a new model described by the following set of fractional differential equations of order β, for 0<β<1(10)DβA=Λ1FNM+FINM−γA−μAA−μρA2,DβFM=ϕγA−β1ρω+β2ρsFM−δMFM−μFFM,DβMM=1−ϕγA−μMMM,DβFNM=β1ρωFM−ωFNM−δMFNM−μFFNM,DβFSM=β2ρsFM−ωFSM−δMFSM−μFFSM,DβFINM=ωFNM−δMFINM−μFFINM,DβFISM=ωFSM−δMFISM−μFFISM,DβSH=1−ℓΛ3−αSH−μHSH,DβEH=αSH−σEH−μHEH,DβIH=ℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,DβIHT=θIH−τ2IHT−∂2IHT−μHIHT,DβRH=τ1IH−τ2IHT−μHRH,where(11)α=α1FINM+α2FISM+α3IH+α4IHT+α5RHNH,ω=λ1IH+λ2IHTNH.


## 3. Laplace–Adomian Decomposition Method(LADM) Implementation

In this section, we discuss the general procedure of this method with the given initial conditions. Applying the Laplace transform to both sides of (10), we obtain the following:(12)SβℒA−Sβ−1A0=ℒΛ1FNM+FINM−γA−μAA−μρA2,SβℒFM−Sβ−1FM0=ℒϕγA−β1ρω+β2ρsFM−δMFM−μFFM,SβℒMM−Sβ−1MM0=ℒ1−ϕγA−μMMM,SβℒFNM−Sβ−1FNM0=ℒβ1ρωFM−λ1IH+λ2IHTNHFNM−δMFNM−μFFNM,SβℒFSM−Sβ−1FSM0=ℒβ2ρsFM−λ1IH+λ2IHTNHFSM−δMFSM−μFFSM,SβℒFINM−Sβ−1FINM0=ℒλ1IH+λ2IHTNHFNM−δMFINM−μFFINM,SβℒFISM−Sβ−1FISM0=ℒλ1IH+λ2IHTNHFSM−δMFISM−μFFISM,SβℒSH−Sβ−1SH0=ℒ1−ℓΛ3−α1FINM+α2FISM+α3IH+α4IHT+α5RHNHSH−μHSH,SβℒEH−Sβ−1EH0=ℒα1FINM+α2FISM+α3IH+α4IHT+α5RHNHSH−σEH−μHEH,SβℒIH−Sβ−1IH0=ℒℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,SβℒIHT−Sβ−1IHT0=ℒθIH−τ2IHT−∂2IHT−μHIHT,SβℒRH−Sβ−1RH0=ℒτ1IH−τ2IHT−μHRH,with initial conditions(13)A0=n1,FM0=n2,MM0=n3,FNM0=n4,FSM0=n5,FINM0=n6,FISM0=n7,SH0=n8,EH0=n9,IH0=n10,IHT0=n11,RH0=n12.Dividing (12) by Sβ yields(14)ℒAt=n1S+1SβℒΛ1FNM+FINM−γA−μAA−μρA2,ℒFMt=n2S+1SβℒϕγA−β1ρω+β2ρsFM−δMFM−μFFM,ℒMMt=n3S+1Sβℒ1−ϕγA−μMMM,ℒFNMt=n4S+1Sβℒβ1ρωFM−λ1IH+λ2IHTNHFNM−δMFNM−μFFNM,ℒFSMt=n5S+1Sβℒβ2ρsFM−λ1IH+λ2IHTNHFSM−δMFSM−μFFSM,ℒFINMt=n6S+1Sβℒλ1IH+λ2IHTNHFNM−δMFINM−μFFINM,ℒFISMt=n7S+1Sβℒλ1IH+λ2IHTNHFSM−δMFISM−μFFISM,ℒSHt=n8S+1Sβℒ1−ℓΛ3−α1FINM+α2FISM+α3IH+α4IHT+α5RHNHSH−μHSH,ℒEHt=n9S+1Sβℒα1FINM+α2FISM+α3IH+α4IHT+α5RHNHSH−σEH−μHEH,ℒIHt=n10S+1SβℒℓΛ3+σEH−τ1IH−θIH−∂1IH−μHIH,ℒIHTt=n11S+1SβℒθIH−τ2IHT−∂2IHT−μHIHT,ℒRHt=n12S+1Sβℒτ1IH−τ2IHT−μHRH.We now decompose the nonlinear terms of system (9), we assume that the solutions of At, FMt, MMt, FNMt, FSMt, FINMt, FISMt, SHt, EHt, IHt, IHTt, RHt are in the form of infinite series given by:(15)At=∑n=0∞An,FMt=∑n=0∞FMn,MMt=∑n=0∞MMn,FNMt=∑n=0∞FNMn,FSMt=∑n=0∞FSMn,FINMt=∑n=0∞FINMn,FISMt=∑n=0∞FISMn,SHt=∑n=0∞SHn,EHt=∑n=0∞EHn,IHt=∑n=0∞IHn,IHTt=∑n=0∞IHTn,RHt=∑n=0∞RHn.Moreover, the nonlinear terms involved in the models are as follows:(16)AtAt,FINMtSHt,FISMtSHt,IHtSHt,IHTtSHt,RHtSHt,IHtFNMt,IHTtFNMt,IHtFSMt,IHTtFSMt.The nonlinear terms in (16) are decomposed by the Adomian polynomial as follows:(17)AtAt=∑n=0∞Bn,FINMtSHt=∑n=0∞Cn,FISMtSHt=∑n=0∞Dn,IHtSHt=∑n=0∞En,IHTtSHt=∑n=0∞Fn,RHtSHt=∑n=0∞Gn,IHtFNMt=∑n=0∞Hn,ItFNMt=∑n=0∞In,IHtFSMt=∑n=0∞Jn,IHTtFSMt=∑n=0∞Kn.where Bn, Cn, Dn, En, Fn, Gn, Hn, In, Jn, Kn are the Adomian polynomial defined as follows:(18)Bn=1Γn+1dndλn∑k=0nλkAk∑k=0nλkAk|λ=0,Cn=1Γn+1dndλn∑k=0nλkFINMk∑k=0nλkSHk|λ=0,Dn=1Γn+1dndλn∑k=0nλkFISMk∑k=0nλSHkk|λ=0,En=1Γn+1dndλn∑k=0nλkIHk∑k=0nλkFHk|λ=0.Fn=1Γn+1dndλn∑k=0nλkIHTk∑k=0nλkFHk|λ=0.Gn=1Γn+1dndλn∑k=0nλkIHk∑k=0nλkFHk|λ=0.Hn=1Γn+1dndλn∑k=0nλkIHk∑k=0nλkFNMk|λ=0.In=1Γn+1dndλn∑k=0nλkIHTk∑k=0nλkFNMk|λ=0,Jn=1Γn+1dndλn∑k=0nλkIHk∑k=0nλkFSMk|λ=0.In=1Γn+1dndλn∑k=0nλkIHTk∑k=0nλkFSMk|λ=0.We substitute (14) for n=0 into (15) and (17) to have the following:(19)ℒA0=n1s,ℒFM0=n2s,ℒMM0=n3s,ℒFNM0=n4s,ℒFSM0=n5s,ℒFINM0=n6s,ℒFISM0=n7s,ℒSH0=n8s,ℒEH0=n9s,ℒIH0=n10s,ℒIHT0=n11s,ℒRH0=n12s.Similarly, forn=1 to n=n+1, we have as follows:(20)ℒA1=1SβℒΛ1FNM0+FINM0−γA0−μAA0−μρB0,ℒFM1=1SβℒϕγA0−β1ρω+β2ρsFM0−δMFM0−μFFM0,ℒMM1=1Sβℒ1−ϕγA0−μMMM0,ℒFNM1=1Sβℒβ1ρωFM0−1NHλ1H0+λ2I0−δMFNM0−μFFNM0,ℒFSM1=1Sβℒβ2ρsFM0−1NHλ1J0+λ2K0−δMFSM0−μFFSM0,ℒFINM1=1Sβℒ1NHλ1H0+λ2I0−δMFINM0−μFFINM0,ℒFISM1=1Sβℒ1NHλ1J0+λ2K0−δMFISM0−μFFISM0,ℒSH1=1Sβℒ1−ℓΛ3−1NHα1C0+α2D0+α3E0+α4F0+α5G0−μHSH0,ℒEH1=1Sβℒ1NHα1C0+α2D0+α3E0+α4F0+α5G0−σEH0−μHEH0,ℒIH1=1SβℒℓΛ3+σEH0−τ1IH0−θIH0−∂1IH0−μHIH0,ℒIHT1=1SβℒθIH0−τ2IHT0−∂2IHT0−μHIHT0,ℒRH1=1Sβℒτ1IH0−τ2IHT0−μHRH0,⋯=⋯ℒAn+1=1SβℒΛ1FNMn+FINMn−γAn−μAAn−μρBn,ℒFMn+1=1SβℒϕγAn−β1ρω+β2ρsFMn−δMFMn−μFFMn,ℒMMn+1=1Sβℒ1−ϕγAn−μMMMn,ℒFNMn+1=1Sβℒβ1ρωFMn−1NHλ1Hn+λ2In−δMFNMn−μFFNMn,ℒFSMn+1=1Sβℒβ2ρsFMn−1NHλ1Jn+λ2Kn−δMFSMn−μFFSMn,ℒFINMn+1=1Sβℒ1NHλ1Hn+λ2In−δMFINMn−μFFINMn,ℒFISMn+1=1Sβℒ1NHλ1Jn+λ2Kn−δMFISMn−μFFISMn,ℒSHn+1=1Sβℒ1−ℓΛ3−1NHα1Cn+α2Dn+α3En+α4Fn+α5Gn−μHSHn,ℒEHn+1=1Sβℒ1NHα1Cn+α2Dn+α3En+α4Fn+α5Gn−σEHn−μHEHn,ℒIHn+1=1SβℒℓΛ3+σEHn−τ1IHn−θIHn−∂1IHn−μHIHn,ℒIHTn+1=1SβℒθIHn−τ2IHTn−∂2IHTn−μHIHTn,ℒRHn+1=1Sβℒτ1IHn−τ2IHTn−μHRHn.Taking the inverse Laplace transform of (20), we have the following:(21)A1=Λ1FNM0+FINM0−γA0−μAA0−μρB0tβΓβ+1,FM1=ϕγA0−β1ρω+β2ρsFM0−δMFM0−μFFM0tβΓβ+1,MM1=1−ϕγA0−μMMM0tβΓβ+1,FNM1=β1ρωFM0−1NHλ1H0+λ2I0−δMFNM0−μFFNM0tβΓβ+1,FSM1=β2ρsFM0−1NHλ1J0+λ2K0−δMFSM0−μFFSM0tβΓβ+1,FINM1=1NHλ1H0+λ2I0−δMFINM0−μFFINM0tβΓβ+1,FISM1=1NHλ1J0+λ2K0−δMFISM0−μFFISM0tβΓβ+1,SH1=1−ℓΛ3−1NHα1C0+α2D0+α3E0+α4F0+α5G0−μHSH0tβΓβ+1,EH1=1NHα1C0+α2D0+α3E0+α4F0+α5G0−σEH0−μHEH0tβΓβ+1,IH1=ℓΛ3+σEH0−τ1IH0−θIH0−∂1IH0−μHIH0tβΓβ+1,IHT1=θIH0−τ2IHT0−∂2IHT0−μHIHT0tβΓβ+1,RH1=τ1IH0−τ2IHT0−μHRH0tβΓβ+1,⋯=⋯An+1=Λ1FNMn+FINMn−γAn−μAAn−μρBn,FMn+1=ϕγAn−β1ρω+β2ρsFMn−δMFMn−μFFMntβΓβ+1,MMn+1=1−ϕγAn−μMMMntβΓβ+1,FNMn+1=β1ρωFMn−1NHλ1Hn+λ2In−δMFNMn−μFFNMntβΓβ+1,FSMn+1=β2ρsFMn−1NHλ1Jn+λ2Kn−δMFSMn−μFFSMntβΓβ+1,FINMn+1=1NHλ1Hn+λ2In−δMFINMn−μFFINMntβΓβ+1,FISMn+1=1NHλ1Jn+λ2Kn−δMFISMn−μFFISMntβΓβ+1,SHn+1=1−ℓΛ3−1NHα1Cn+α2Dn+α3En+α4Fn+α5Gn−μHSHntβΓβ+1,EHn+1=1NHα1Cn+α2Dn+α3En+α4Fn+α5Gn−σEHn−μHEHntβΓβ+1,IHn+1=ℓΛ3+σEHn−τ1IHn−θIHn−∂1IHn−μHIHntβΓβ+1,IHTn+1=θIHn−τ2IHTn−∂2IHTn−μHIHTntβΓβ+1,RHn+1=τ1IHn−τ2IHTn−μHRHntβΓβ+1.Recalling the earlier stated initial conditions and substituting these conditions into (15), we have equation (22) as follows:A0=n1, FM0=n2, MM0=n3, FNM0=n4, FSM0=n5, FINM0=n6, FISM0=n7, SH0=n8, EH0=n9, IH0=n10, IHT0=n11, RH0=n12.(22)At=n1+Λ1n4+n6−γn1−μAn1−μρn1n1tβΓβ+1+Λ1β1ρωn2−λ1n10n4−λ2n11n4,−δMn4−μFn4+λ1n10n4+λ2n11n4−δMn6−μFn6tβΓβ+1−γ+μAΛ1n4+n6−γn1−μAn1,−μρn1n1tβΓβ+1−μρ2n1Λ1n4+n6−γn1−μAn1−μρn1n1tβΓβ+1tβΓβ+1,FMt=n2+ϕγn1−β1ρω+β2ρsn2−δMn2−μFn2tβΓβ+1ϕγΛ1n4+n6−γn1−μAn1−μρn1n1tβΓβ+1−β1ρω+β2ρsϕγn1−β1ρω−β2ρsn2−δMn2−μFn2tβΓβ+1,−δM+μFβ1ρω+β2ρsϕγn1−β1ρω+β2ρsn2−δMn2−μFn2tβΓβ+1tβΓβ+1,MMt=n3+1−ϕγn1−μMn3tβΓβ+1+1−ϕγΛ1n4+n6−γn1−μAn1−μρn1n1tβΓβ+1−μM1−ϕγn1−μMn3tβΓβ+1tβΓβ+1,FNMt=n4+β1ρωn2−1NHλ1n10n4+λ2n11n4−δMn4−μFn4tβΓβ+1+β1ρωϕγn1−β1ρωn2−δMn2−μFn2tβΓβ+1−λ1n10+n4β1ρωn2−λ1n10n4−λ2n11n4−δMn4−μFn4tβΓβ+1,−λ2n11+n4β1ρωn2−λ1n10n4−λ2n11n4−δMn4−μFn4tβΓβ+1−δM+μFβ1ρωn2,−λ1n10n4−λ2n11n4−δMn4−μFn4−δMn4−μFn4tβΓβ+1tβΓβ+1,FSMt=n5+β2ρsn2−1NHλ1n10n5+λ2n11n5−δMn5−μFn5tβΓβ+1+β2ρsβ1ρωn2−λ1n10n4−λ2n11n4−δMn4tβΓβ+1−λ1n10+n5β1ρωn2−λ1n10n4−λ2n11n4−δMn4,−μFn4tβΓβ+1−λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn5−μFn5tβΓβ+1,−δM+μFβ2ρsn2−λ1n10n5+λ2n11n5−δMn6−μFn6tβΓβ+1tβΓβ+1,FINMt=n6+1NHλ1n10n4+λ2n11n4−δMn6−μFn6tβΓβ+1+1NHλ1n10+n4β1ρωn2−λ2n11n5−δMn5−μFn5tβΓβ+1+λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn4,−δMn4−μFn4tβΓβ+1−δM+μFλ1n10n5+λ2n11n5−δMn6−μFn6tβΓβ+1tβΓβ+1,FISMt=n7+1NHλ1n10n5+λ2n11n5−δMn7−μFn7tβΓβ+1+1NHλ1n10+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn5−μFn5tβΓβ+1+λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn4,−μFn5tβΓβ+1−δM+μFλ1n10n5+λ2n11n5−δMn6−μFn6tβΓβ+1tβΓβ+1,SHt=n8+1−ℓΛ3−1NHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−μHn8tβΓβ+1+1−ℓΛ3−1NHα1n6+n8λ1n10n4+λ2n11n4−δMn6−μFn6−tβΓβ+1α2n7+n8λ1n10n5+λ2n11n5−δMn7−μFn7−tβΓβ+1α31−ℓΛ3−α1n10+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8−tβΓβ+1α4n11+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8tβΓβ+1−α5n8+n10τ1n10−τ2n11−μHn12−μH1−ℓΛ3−α1n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8tβΓβ+1,EHt=n9+1NHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9tβΓβ+1+1NHα1n6+n8λ1n10n4+λ2n11n4−δMn6−μFn6+tβΓβ+1α2n7+n8λ1n10n5+λ2n11n5−δMn7−μFn7+tβΓβ+1α31−ℓΛ3−α1n10+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8tβΓβ+1+α4n11+n8n6n8−α2n7n8−α3n10n8−α4n11n8+α5n12n8−μHn8+α5n8+n10τ1n10−τ2n11−μHn12−σ+μHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9tβΓβ+1,IHt=n10+ℓΛ3+σn9−τ1n10−θn10−∂1n10−μHn10tβΓβ+1+ℓΛ3+σα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9−τ1+θ+∂1+μHℓΛ3+σn9−n10τ1+θ+∂1+μHtβΓβ+1,IHTt=n11+n10−τ2n11−∂2n11−μHn11tβΓβ+1+θΛ3+σn9−τ1n10−θn10−∂2n10−μHn10−τ2+∂2+μHϕn10−τ2n11−∂2n11−μHn11tβΓβ+1,RHt=n12+τ1n10−τ2n11−μHn12tβΓβ+1+τ1ℓΛ3+σn9−τ1n10−θn10−∂2n10−μHn10−τ2ϕn10−τ2n11−∂2n11−μHn11−μHτ1n10−τ2n11−μHn12tβΓβ+1.Equation (22) can be further expressed as follows:(23)At=n1+Λ1n4+n6−γn1−μAn1−μρn1n1tβΓβ+1+Λ1β1ρωn2−λ1n10n4−λ2n11n4,−δMn4−μFn4+λ1n10n4+λ2n11n4−δMn6−μFn6−γ+μAΛ1n4+n6−γn1−μAn1,−μρn1n1−μρ2n1Λ1n4+n6−γn1−μAn1−μρn1n1t2βΓ2β+1,FMt=n2+ϕγn1−β1ρω+β2ρsn2−δMn2−μFn2tβΓβ+1ϕγΛ1n4+n6−γn1−μAn1−μρn1n1−β1ρω+β2ρsϕγn1−β1ρω−β2ρsn2−δMn2−μFn2,−δM+μFβ1ρω+β2ρsϕγn1−β1ρω+β2ρsn2−δMn2−μFn2t2βΓ2β+1,MMt=n3+1−ϕγn1−μMn3tβΓβ+1+1−ϕγΛ1n4+n6−γn1−μAn1−μρn1n1−μM1−ϕγn1−μMn3t2βΓ2β+1,FNMt=n4+β1ρωn2−1NHλ1n10n4+λ2n11n4−δMn4−μFn4tβΓβ+1+β1ρωϕγn1−β1ρωn2−δMn2−μFn2−λ1n10+n4β1ρωn2−λ1n10n4−λ2n11n4−δMn4−μFn4tβΓβ+1,−λ2n11+n4β1ρωn2−λ1n10n4−λ2n11n4−δMn4−μFn4−δM+μFβ1ρωn2,−λ1n10n4−λ2n11n4−δMn4−μFn4−δMn4−μFn4t2βΓ2β+1,FSMt=n5+β2ρsn2−1NHλ1n10n5+λ2n11n5−δMn5−μFn5tβΓβ+1+β2ρsβ1ρωn2−λ1n10n4−λ2n11n4−δMn4−λ1n10+n5β1ρωn2−λ1n10n4−λ2n11n4−δMn4,−μFn4−λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn5−μFn5tβΓβ+1,−δM+μFβ2ρsn2−λ1n10n5+λ2n11n5−δMn6−μFn6t2βΓ2β+1,FINMt=n6+1NHλ1n10n4+λ2n11n4−δMn6−μFn6tβΓβ+1+1NHλ1n10+n4β1ρωn2−λ2n11n5−δMn5−μFn5+λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn4,−δMn4−μFn4−δM+μFλ1n10n5+λ2n11n5−δMn6−μFn6t2βΓ2β+1,FISMt=n7+1NHλ1n10n5+λ2n11n5−δMn7−μFn7tβΓβ+1+1NHλ1n10+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn5−μFn5+λ2n11+n5β2ρsn2−λ1n10n5−λ2n11n5−δMn4,−μFn5−δM+μFλ1n10n5+λ2n11n5−δMn6−μFn6t2βΓ2β+1,SHt=n8+1−ℓΛ3−1NHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−μHn8tβΓβ+1+1−ℓΛ3−1NHα1n6+n8λ1n10n4+λ2n11n4−δMn6−μFn6−α2n7+n8λ1n10n5+λ2n11n5−δMn7−μFn7−α31−ℓΛ3−α1n10+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8−α4n11+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8−α5n8+n10τ1n10−τ2n11−μHn12−μH1−ℓΛ3−α1n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8t2βΓ2β+1,EHt=n9+1NHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9tβΓβ+1+1NHα1n6+n8λ1n10n4+λ2n11n4−δMn6−μFn6+α2n7+n8λ1n10n5+λ2n11n5−δMn7−μFn7+α31−ℓΛ3−α1n10+n8n6n8−α2n7n8−α3n10n8−α4n11n8−α5n12n8−μHn8+α4n11+n8n6n8−α2n7n8−α3n10n8−α4n11n8+α5n12n8−μHn8+α5n8+n10τ1n10−τ2n11−μHn12−σ+μHα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9t2βΓ2β+1,IHt=n10+ℓΛ3+σn9−τ1n10−θn10−∂1n10−μHn10tβΓβ+1+ℓΛ3+σα1n6n8+α2n7n8+α3n10n8+α4n11n8+α5n12n8−σn9−μHn9−τ1+θ+∂1+μHℓΛ3+σn9−n10τ1+θ+∂1+μHt2βΓ2β+1,IHTt=n11+n10−τ2n11−∂2n11−μHn11tβΓβ+1+θℓΛ3+σn9−τ1n10−θn10−∂2n10−μHn10−τ2+∂2+μHϕn10−τ2n11−∂2n11−μHn11t2βΓ2β+1,RHt=n12+τ1n10−τ2n11−μHn12tβΓβ+1+τ1ℓΛ3+σn9−τ1n10−θn10−∂2n10−μHn10−τ2ϕn10−τ2n11−∂2n11−μHn11−μHτ1n10−τ2n11−μHn12t2βΓ2β+1.


### 3.1. Numerical Solution of the Laplace–Adomian Decomposition Method (LADM)

In this section, we shall consider the numerical solution of the model. Using the initial conditions presented in Appendix, the Laplace–Adomian decomposition method (LADM) gives us an approximate solution in terms of an infinite series presented as follows:(24)At=2500+44100.3tβΓβ+1−37045.1t2βΓ2β+1…FMt=500+332tβΓβ+1+1864.39t2βΓ2β+1…MMt=160+18.43tβΓβ+1+1053.16t2βΓ2β+1…FNMt=250+74.331tβΓβ+1−175.22t2βΓ2β+1…FSMt=120+83.82tβΓβ+1−100t2βΓ2β+1…FINMt=125−36.835tβΓβ+1+11.21t2βΓ2β+1…FISMt=40−11.672tβΓβ+1+3.910t2βΓ2β+1…SHt=1000+35.310tβΓβ+1+36.46t2βΓ2β+1…EHt=30+0.3052tβΓβ+1+0.19t2βΓ2β+1…IHt=20+0.020tβΓβ+1+2.032t2βΓ2β+1…IHTt=15−2.3752tβΓβ+1−3.824t2βΓ2β+1…RHt=0.412tβΓβ+1+0.3848t2βΓ2β+1…The solution of the model forβ=1 is given as follows:(25)At=2500+44100.3t−18522t2⋯FMt=500+332t−932.19t2⋯MMt=160+18.42t+526.58t2⋯FNMt=250+74.331t−87.61t2⋯FSMt=120+83.82t−50t2⋯FINMt=125−36.835t+5.6t2⋯FISMt=40−11.672t+1.96t2⋯SHt=1000+35.310t+18.23t2⋯EHt=30+0.3052t+0.095t2⋯IHt=20+0.020t+1.016t2⋯IHTt=15−2.3752t+1.91t2⋯RHt=0.412t+0.1924t2⋯


### 3.2. Numerical Simulation

Here, we present the numerical simulation of our fractional order sterile insect technology model so as to compare with the classical order. All the variables of our model are presented from Figures2–7 with variation in the value of the fractional order β. Figures 2–7, shows that the fractional order sterile insect technology model has more degrees of freedom as such the order β can be varied to determine various degrees of responses of the different classes that makes up the model. This is seen when we compared the fractional order (ie β=0.4,0.7 with the classical case where β=1.0. The effect of the sterile mosquitoes on the female mosquitoes are shown earlier when β<1 than when β=1, which agrees with our earlier result that the fractional order allows for determination of responses at different levels when compared with the classical case. The fractional order indeed helps us to get different responses in real time of the different classes of our model as shown in Figures 2–7. We also carefully chose our initial values and time to avoid having negative population of mosquitoes or human which might not be biologically meaningful.Figure 2
.Population of aquatic mosquitoes and male mosquitoes with variations in values ofβ. (a) Population of aquatic mosquitoes with variation of β. (b) Population of male mosquitoes with different values of β.
(a)(b)Figure 3
.Population of female mosquitoes not yet laying eggs and female nonsterile mosquitoes with variation inβ. (a) Population of female mosquitoes not yet laying eggs with variation of β. (b) Population of female nonsterile mosquitoes with different values of β.
(a)(b)Figure 4
population of female sterile mosquitoes and female infected nonsterile mosquitoes with variation inβ. (a) Population of female sterile mosquitoes with variation of β. (b) Population of female infected nonsterile mosquitoes with different values of β.
(a)(b)Figure 5
Population of female infected sterile mosquitoes and sterile male mosquitoes with variation inβ. (a) Population of female infected sterile mosquitoes with variation of β. (b) The population of sterile male mosquitoes with different values of β.
(a)(b)Figure 6
Population of susceptible humans and exposed humans with variation inβ. (a) Population of susceptible humans with variation of β. (b) Population of exposed humans with different values of β.
(a)(b)Figure 7
Population of infected humans and infected humans on treatment with variation inβ. (a) Population of infected humans with variation of β. (b)Population of infected humans on treatment with different values of β.
(a)(b)


### 3.3. Convergence Analysis for the Laplace–Adomian Decomposition Method (LADM)

The solution of (17) is expressed in forms of infinite series as presented in equation (18) which converges uniformly to its exact solution. To verify the convergence of the series (17), we employ the method used in [20]. For sufficient conditions of convergence of the LADM, we present the following theorem:Theorem 1 (see [11]).
LetX be a Banach space and T:X⟶X be a constructive nonlinear operator such that for all x,x′∈X,Tx−Tx′≤kx−x′,0<k<1. Then, T has a unique point x such that Tx=x, where x=A,FM,MM,FNM,FSM,FINM,FISM,SH,EH,IH,IHT,RH.
The series given in (?) can be written by applying the Adomian decomposition method as follows:(26)xn=Txn−1,xn−1,=∑i=1n−1xi,n=1,2,3,…and we assume that x0∈Brx, where Brx=x′∈X:x′−x<r; then, we have as follows:(i)
xn∈Brx.(ii)
limn⟶∞xn=x.Proof.
For condition (i), invoking mathematical induction forn=1, we have as follows:(27)x0−x=Tx0−Tx≤x0−x.
If this is true form−1, then(28)x0−x≤km−1x0−x.
This gives the following:(29)xm−x=Txm−1−Tx≤kxm−1−x≤knx0−x.
Therefore,(30)xm−x≤knx0−x≤knr<r.
This directly implies thatxn∈Brx.
Also, for (ii), we have that sincexm−x≤knx0−x and limn⟶∞kn=0, we can write limn⟶∞xn=x.


## 3.1. Numerical Solution of the Laplace–Adomian Decomposition Method (LADM)

In this section, we shall consider the numerical solution of the model. Using the initial conditions presented in Appendix, the Laplace–Adomian decomposition method (LADM) gives us an approximate solution in terms of an infinite series presented as follows:(24)At=2500+44100.3tβΓβ+1−37045.1t2βΓ2β+1…FMt=500+332tβΓβ+1+1864.39t2βΓ2β+1…MMt=160+18.43tβΓβ+1+1053.16t2βΓ2β+1…FNMt=250+74.331tβΓβ+1−175.22t2βΓ2β+1…FSMt=120+83.82tβΓβ+1−100t2βΓ2β+1…FINMt=125−36.835tβΓβ+1+11.21t2βΓ2β+1…FISMt=40−11.672tβΓβ+1+3.910t2βΓ2β+1…SHt=1000+35.310tβΓβ+1+36.46t2βΓ2β+1…EHt=30+0.3052tβΓβ+1+0.19t2βΓ2β+1…IHt=20+0.020tβΓβ+1+2.032t2βΓ2β+1…IHTt=15−2.3752tβΓβ+1−3.824t2βΓ2β+1…RHt=0.412tβΓβ+1+0.3848t2βΓ2β+1…The solution of the model forβ=1 is given as follows:(25)At=2500+44100.3t−18522t2⋯FMt=500+332t−932.19t2⋯MMt=160+18.42t+526.58t2⋯FNMt=250+74.331t−87.61t2⋯FSMt=120+83.82t−50t2⋯FINMt=125−36.835t+5.6t2⋯FISMt=40−11.672t+1.96t2⋯SHt=1000+35.310t+18.23t2⋯EHt=30+0.3052t+0.095t2⋯IHt=20+0.020t+1.016t2⋯IHTt=15−2.3752t+1.91t2⋯RHt=0.412t+0.1924t2⋯


## 3.2. Numerical Simulation

Here, we present the numerical simulation of our fractional order sterile insect technology model so as to compare with the classical order. All the variables of our model are presented from Figures2–7 with variation in the value of the fractional order β. Figures 2–7, shows that the fractional order sterile insect technology model has more degrees of freedom as such the order β can be varied to determine various degrees of responses of the different classes that makes up the model. This is seen when we compared the fractional order (ie β=0.4,0.7 with the classical case where β=1.0. The effect of the sterile mosquitoes on the female mosquitoes are shown earlier when β<1 than when β=1, which agrees with our earlier result that the fractional order allows for determination of responses at different levels when compared with the classical case. The fractional order indeed helps us to get different responses in real time of the different classes of our model as shown in Figures 2–7. We also carefully chose our initial values and time to avoid having negative population of mosquitoes or human which might not be biologically meaningful.Figure 2
.Population of aquatic mosquitoes and male mosquitoes with variations in values ofβ. (a) Population of aquatic mosquitoes with variation of β. (b) Population of male mosquitoes with different values of β.
(a)(b)Figure 3
.Population of female mosquitoes not yet laying eggs and female nonsterile mosquitoes with variation inβ. (a) Population of female mosquitoes not yet laying eggs with variation of β. (b) Population of female nonsterile mosquitoes with different values of β.
(a)(b)Figure 4
population of female sterile mosquitoes and female infected nonsterile mosquitoes with variation inβ. (a) Population of female sterile mosquitoes with variation of β. (b) Population of female infected nonsterile mosquitoes with different values of β.
(a)(b)Figure 5
Population of female infected sterile mosquitoes and sterile male mosquitoes with variation inβ. (a) Population of female infected sterile mosquitoes with variation of β. (b) The population of sterile male mosquitoes with different values of β.
(a)(b)Figure 6
Population of susceptible humans and exposed humans with variation inβ. (a) Population of susceptible humans with variation of β. (b) Population of exposed humans with different values of β.
(a)(b)Figure 7
Population of infected humans and infected humans on treatment with variation inβ. (a) Population of infected humans with variation of β. (b)Population of infected humans on treatment with different values of β.
(a)(b)


## 3.3. Convergence Analysis for the Laplace–Adomian Decomposition Method (LADM)

The solution of (17) is expressed in forms of infinite series as presented in equation (18) which converges uniformly to its exact solution. To verify the convergence of the series (17), we employ the method used in [20]. For sufficient conditions of convergence of the LADM, we present the following theorem:Theorem 1 (see [11]).
LetX be a Banach space and T:X⟶X be a constructive nonlinear operator such that for all x,x′∈X,Tx−Tx′≤kx−x′,0<k<1. Then, T has a unique point x such that Tx=x, where x=A,FM,MM,FNM,FSM,FINM,FISM,SH,EH,IH,IHT,RH.
The series given in (?) can be written by applying the Adomian decomposition method as follows:(26)xn=Txn−1,xn−1,=∑i=1n−1xi,n=1,2,3,…and we assume that x0∈Brx, where Brx=x′∈X:x′−x<r; then, we have as follows:(i)
xn∈Brx.(ii)
limn⟶∞xn=x.Proof.
For condition (i), invoking mathematical induction forn=1, we have as follows:(27)x0−x=Tx0−Tx≤x0−x.
If this is true form−1, then(28)x0−x≤km−1x0−x.
This gives the following:(29)xm−x=Txm−1−Tx≤kxm−1−x≤knx0−x.
Therefore,(30)xm−x≤knx0−x≤knr<r.
This directly implies thatxn∈Brx.
Also, for (ii), we have that sincexm−x≤knx0−x and limn⟶∞kn=0, we can write limn⟶∞xn=x.


## 4. Conclusion

In this work, we extended the work of Atokolo et al. [1] by formulating a fractional order sterile insect technology model. To solve this model, we developed a numerical scheme that gives an analytical (approximate) solution of our model using the Laplace–Adomian decomposition method. Our solution showed quantitative agreement with other numerical solutions. We also showed that our approximate solution converges to an exact solution. Finally, we showed from our numerical simulation that the fractional order gives more degrees of freedom as the order can be varied to show responses by the different classes in real time. It is also seen that reduction of the fractional order β gives a corresponding reduction in the population value of each of the classes considered, which implies that the fractional order has a direct implication in our model which is aimed at reducing the mosquito population. We have been able to show that the LADM can be used to solve an SIT model which has never been done before in literature. We also hope to apply the homotopy perturbation method in solving our model and compare the result with the LADM so as to make strong recommendation on the best method to use and why.We therefore conclude that the fractional order model solved via the Laplace–Adomian decomposition method gives a higher degree of freedom as compared to the integer order model, and as such, linear and nonlinear ordinary and partial differential equations of classical and fractional orders can be solved using the Laplace–Adomian decomposition method [21–27].



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*Source: 2297630-2022-07-12.xml*