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.
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: , 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 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 compartment . The nonaquatic mosquito class is divided into seven compartments consisting of the male mosquitoes , female mosquitoes not yet laying eggs , female nonsterile mosquitoes, , female sterile mosquitoes , sterile male mosquitoes , female infected nonsterile mosquitoes, , and female infected sterile mosquitoes . The human population is divided into susceptible human , exposed human , infected human , infected human but on treatment , and recovered human . 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 and by density dependence death rate of . The female mosquitoes are recruited at the rate of , where is the maturity rate of aquatic mosquitoes to adult mosquitoes and is the proportion of emerging females; it is reduced by the mating rate at the level of for female mosquitoes to be with wild male mosquitoes or sterile male mosquitoes with mating probabilities and , respectively. The population is reduced by death induced due to the attempt to seek for blood meals at the rate of and finally reduced by natural death at the rate of . The male mosquitoes are recruited by the proportion of the emerging male mosquitoes that mature to adult mosquitoes at the rate of which also reduces by natural death at the rate of . The female nonsterile mosquito population is increased by the female mosquitoes’ probability to mate with the wild male mosquitoes which is given by the rate , with a mating rate of . This population is reduced by , the rate at which the female nonsterile mosquitoes are infected and moved to the female infected nonsterile mosquitoes class . It is also reduced by the death induced due to the attempt to seek for blood meals at the rate of ; this class is finally reduced by natural death rate at the rate of . The female sterile mosquito population is increased by the wild female mosquitoes’ probability to mate with the sterile mosquitoes which is given by the rate , with a mating rate of . The class reduces by , the rate at which the becomes infected and moves to the class. The class reduces by death induced due to the attempt to seek for blood meals at the rate of and reduces finally by natural death at the rate of . The population of female infected nonsterile mosquitoes is recruited at the rate at which the female nonsterile mosquitoes are infected at the rate of The population is reduced by death induced due to the attempt to seek for blood meals at the rate of and finally reduced by natural death at the rate of . The population of female infected sterile mosquitoes is recruited at the rate at which the female sterile mosquitoes are infected at the rate of . The population is reduced by death induced due to the attempt to seek for blood meals at the rate of and finally by natural death at the rate of . The sterile male mosquitoes are released into the population at the rate . 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 of the released mosquitoes will join wild mosquitoes population. Second, because of the differences in physiology of wild and sterile mosquitoes, a parameter 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 , and as such, the available injected sterile male mosquitoes into the wild population of mosquitoes that can competitively mate with wild female mosquitoes is . This population is reduced by natural death at the rate of . The susceptible human population is recruited at the level of , of which a fraction of those infected at birth joined the infectious human population. The population reduces by the rate at which infectious mosquitoes (female infected nonsterile mosquitoes or female infected sterile mosquitoes ) infects susceptible human at the levels of and , respectively. Also, it reduces by the rate at which the infectious human (infected class , recovered class or infected but on treatment class (IHT) infects susceptible human through sex at the level of or or , , 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 . The population of the exposed human is generated by infection of susceptible individuals at the rate of . This population reduces by natural death at the of rate and by the rate at which the exposed are finally infectious at the rate of . The infected human class is generated by the incoming of infected babies from infected mothers at the rate of , due to vertical transmission. In addition, the population increases at the rate by which the exposed become infected at the level . The class reduces at the rate by which the infected are taken for treatment and by natural recovery rate of . This class reduces finally by both natural and disease-induced death rates at the levels of and , respectively. The infected but on treatment class is recruited by the incoming of the infected who are taken for treatment at the rate of ; this class reduces at the rate by which the infected but on treatment class recovers due to supportive treatment at the rate of . It reduces finally by natural death and disease induced death at the rates of and , respectively, where is assumed to be less than . The recovered human is recruited at the rate by which the infected human recovers naturally at the rate of or due to supportive treatment at the rate of . The population reduces by natural death at the rate of .
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 Figure 1, we have the following:where
From the earlier assumption, the mating competitiveness of both sterile and nonsterile mosquitoes are not equal, that is, .
Also, from (1), we let and .
Therefore, from the second equation of (1), we have as follows:where is the female mating probability with the sterile mosquitoes and is the female mating probability with the wild male mosquitoes. Moreso, the rate of infection of and after biting an infectious human and then moving to the and classes, respectively, is equal, that is, to say . We also decoupled the sterile male mosquitoes population 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:where
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 function on the interval is defined as follows:where and represent the integer parts of . In particularly, for , the Caputo derivative becomes as follows:
Definition 2. Laplace transform of Caputo derivatives is defined as follows:for arbitrary , , where and represent the noninteger parts of .
Lemma 1. The following result holds for fractional differential equations:for arbitrary , , where and represent the integer parts of .
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 , for where
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:with initial conditions
Dividing (12) by yields
We now decompose the nonlinear terms of system (9), we assume that the solutions of , , , , , , , , , , , are in the form of infinite series given by:
Moreover, the nonlinear terms involved in the models are as follows:
The nonlinear terms in (16) are decomposed by the Adomian polynomial as follows:where , , , , , , , , , are the Adomian polynomial defined as follows:
We substitute (14) for into (15) and (17) to have the following:
Similarly, for to , we have as follows:
Taking the inverse Laplace transform of (20), we have the following:
Recalling the earlier stated initial conditions and substituting these conditions into (15), we have equation (22) as follows:
, , , , , , , , , , , .
Equation (22) can be further expressed as follows:
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:
The solution of the model for is given as follows:
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 Figures 2–7 with variation in the value of the fractional order . Figures 2–7, shows that the fractional order sterile insect technology model has more degrees of freedom as such the order 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 with the classical case where . The effect of the sterile mosquitoes on the female mosquitoes are shown earlier when than when , 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.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(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]). Let be a Banach space and be a constructive nonlinear operator such that for all . Then, has a unique point such that , where .
The series given in (?) can be written by applying the Adomian decomposition method as follows:and we assume that , where ; then, we have as follows:(i).(ii).
Proof. For condition (i), invoking mathematical induction for , we have as follows:If this is true for , thenThis gives the following:Therefore,This directly implies that .
Also, for (ii), we have that since and , we can write .
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 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].
Appendix
Here, we present Table 1 showing definition of variables used in our model, Table 2 showing definition of parameters as used in our model equations, and Table 3 that shows initial conditions for our model variables and parameter values, respectively.
Data Availability
The data used to support the findings of this study are available from the corresponding author on request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.