Modeling and Analysis of an Age-Structured Malaria Model in the Sense of Atangana–Baleanu Fractional Operators

. In this paper, integer-and fractional-order models are discussed to investigate the dynamics of malaria in a human host with a varied age distribution. A system of diferential equation model with fve human state variables and two mosquito state variables was examined. Preschool-age (0–5) and young-age individuals make up our model’s division of the human population. We investigated the existence of an area in which the model is both mathematically and epidemiologically well posed. According to the fndings of our mathematical research, the disease-free equilibrium exists whenever the fundamental reproduction number R 0 is smaller than one and is asymptotically stable. Te disease-free equilibrium point is unstable when R 0 > 1. We showed that the endemic equilibrium point is unique for R 0 > 1. Also, the most infuential control parameters of the spread of malaria were identifed. Numerical simulations of both classical and fractional order were conducted, and we used ODE (45) for classical part and numerical technique developed by Toufk and Atangana for fractional order. Te infected population will grow because of the high biting frequency of the mosquito and the high likelihood of transmission from the infected mosquito to the susceptible human. R � 1 . 622, which is more than one, indicating that the mosquito vector keeps on growing. Tis supports the stability of the endemic equilibrium point theorem, which states that the disease becomes endemic when R � 1. Te susceptible human population will decrease because of the presence of the infective mosquito, which has a high biting frequency for the frst couple of days. Since the infective mosquito bit the susceptible human, the susceptible human became infected and went to the infected human compartments. Ten, the susceptible population will decrease and the infested human population will increase. After a certain amount of time, it becomes zero due to the growth of protected classes. In this case, a disease-free equilibrium point exists and is stable. Tis condition exists because R 0 � 2 . 827 × 10 − 5 is less than 1. Tis supports the theorem that the stability of the disease-free equilibrium point is obtained when R 0 < 1. Depending on equation, we have shown that the possibility of some endemic equilibria exists when R 0 < 1, that is, it undergoes backward bifurcation, even when the disease-free equilibrium is locally stable, and the result means that the society may misunderstand the level of malaria prevalence in the community


Introduction
Vector-borne diseases (VBDs) result from an infection communicated by vectors such as mosquitoes, ticks, lice, and feas.Tese vectors carry pathogenic organisms such as bacteria, viruses, fungi, protists, and parasitic worms which can be transferred from one host to another.Some examples of VBDs are dengue fever, Lyme disease, malaria, West Nile virus, Rift Valley fever, and Japanese encephalitis [1].In many tropical and subtropical regions, malaria is a prevalent and potentially fatal infectious disease.It is brought on by the Plasmodium parasite, which is spread when female Anopheles mosquitoes bite people to obtain blood for their eggs [2].Te most prevalent species of Plasmodium are Plasmodium vivax in temperate zones and Plasmodium falciparum in tropical areas [3,4].About half of the world's population is in danger of malaria, according to the WHO (World Health Organization) malaria report [5].Globally, there were estimated 228 million cases of malaria and 405000 deaths from it in 2018.Most of these cases and deaths accounted for 93% and 94% of all malaria cases globally in 2018.Te projected number of cases and fatalities from malaria in 2019 was 229 million worldwide [6].Globally, there were reportedly 247 million cases of malaria in 2021, with 619000 deaths attributed to the disease.
A disproportionately large amount of the worldwide malaria burden is placed on the WHO African Region.95% of malaria cases and 96% of malaria deaths in 2021 occurred in the area.Almost 80% of all malaria deaths in the WHO African Area occurred in children under the age of fve.Two-thirds of recorded deaths are children [6].As they have not yet acquired immunity to illnesses, children under the age of fve are more susceptible to malaria than adults [7].As a result, the age distribution in a community afects the spread of malaria.In Ethiopia, 60% and 40% of malaria cases are caused by the species Plasmodium falciparum and Plasmodium vivax, respectively [8,9].Many scientifc attempts have been made, including the creation of mathematical models, to lessen the impact of malaria on the global community.Ross, in 1911 [10], applied deterministic compartmental epidemic models to illustrate the dynamics of malaria infection between vector and host populations.Macdonald and Ross's model [11] was modifed by adding biological data about mosquito latency brought on by the growth of the malaria parasite.Nonetheless, eforts to stop the spread of malaria have resulted in the creation of effective vector control measures, including larvicide, indoor residual spraying, and insecticide-treated nets (ITNs) [6,12].Non-integer-order calculus has more than 300-year history.Many theories are being added to the literature on fractional calculus every day.In the seventeenth century, German mathematician Gottfried Leibniz and well-known British scientist Isaac Newton developed the idea of fractional calculus as a result of calculus's ramifcations.In the form of generalized fractional order, fractional calculus deals with the defnitions of classical calculus [13].In order to comprehend, forecast, and manage the spread of diseases among populations, mathematical modeling of infectious diseases is essential.Trough the integration of mathematical tools and epidemiological knowledge, researchers are able to conduct scenario simulations, investigate diverse intervention options, and facilitate the making of public health decisions.Tese models are useful instruments for educating decisionmakers and supporting the creation of winning plans to fght infectious illnesses and safeguard the public's health [14].
A novel mathematical model was recently presented by Mohammed-Awel and Gumel [15], because of the widespread use of indoor residual spraying (IRS) and insecticidetreated nets (ITNs) for malaria control, for evaluating the impact of pesticide resistance in the mosquito population.In [1,16], fractional-order derivatives are described as nonlinear systems in a more realistic way in comparison with integer-order derivatives, and a comparison of temperature distribution via Atangana-Baleanu non-integer-order fractional derivatives is used to illustrate the application of the mathematical technique of Laplace transform.Many fractional derivatives have been developed by researchers and used in a variety of scientifc and engineering felds [17][18][19], and the most frequently used derivatives in the various branches of science, particularly in mathematical epidemiology, are Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB).Diferent kernel properties apply to each of these three fractional derivatives.In contrast to Caputo-Fabrizio, which uses an exponentially decaying type kernel (which is nonsingular but nonlocal), AB derivative in the Caputo sense uses a Mittag-Lefer type kernel.Odibat just developed a brand-new fractional derivative of the generalized Caputo type [20].Te features of this novel generalized Caputo derivative are comparable to those of Caputo derivatives [21].A nonlinear fractional-order model for analyzing the dynamical behavior of vector-borne diseases within the frame of Caputo-fractional derivative was analyzed, numerical simulations for diferent values of fractional-order derivative were performed, and a comparison with the results of the integer-order derivative was made.In this study, the nature of our malaria model is read at noninteger-order values using Atangana-Baleanu fractional derivatives with a high efciency rate.
Te advantage of using CF and AB fractional derivatives to solve the projected malaria disease model is that they provide strong approaches for the arbitrary order case, memory efects, and crossover behavior of the model.
Te beneft of using the Atangana-Baleanu operator is that it incorporates the crossover behavior of the malaria disease dynamics model as well as memory results.It also has a nonsingular and non-nearby kernel, which enables us to explain complex structures that uniquely, incredibly, and efciently observe both the law of electricity and exponential decay at the same time.Here, we consider the integer-order model proposed by "Klinck" in [15] and modify it to become fractional-order models in the Atangana-Baleanu-Caputo sense.After recalling some defnitions and results concerning integer-order and fractional-order derivatives, we prove the existence and give conditions under the fractional models that admit a unique solution.To illustrate our analytical results, we shall adopt the Toufk-Atangana method to perform numerical simulations for the fractional model.Researchers in [15] worked on the dynamics of malaria in an agestructured human host; in their model, human population was partitioned into two compartments: preschool age (0− 5) and the rest of the human population.Tey have divided the human population into two classes: H 1 and H 2 , having S, I, and R compartments in each class.Tus, the human population N H is divided into six compartments, and we modify such integer-order model proposed in [15] by including the parameter of natural recovery rate of both age groups in addition to the recovery rate due to treatment and the protected group of human population to measure the efect of intervention mechanisms such as insecticide-treated nets (ITNs) and indoor residual spraying (IRS) in the transmission dynamics of malaria and fractional-order models in the Atangana-Baleanu-Caputo sense to describe the memory efects and crossover behavior of the malaria model.For simplicity, we consider only one susceptible and recovery compartment, respectively, for both age groups of human population.Te aim of our study is to understand the dynamics of malaria through integer-order and fractionalorder analysis of age-structured malaria model.Tis study is organized as follows.In Section 2, we develop mathematical model formulation.Section 3 gives the model analysis.Section 4 presents the numerical simulation of the integer-order malaria model.Section 5 gives the fractional malaria model and analysis.Section 6 presents the numerical scheme and simulation of the fractional-order model.Section 7 gives the result and discussion.Section 8 draws the conclusion.

Formulation of Modified Mathematical Model
Trough birth, people are added to society at a constant rate (Λ h ).Of those added, those protected by specifc protective measures belong to the protected class P h , while the remaining ((1 − c)Λ h ) belong to the susceptible class (S h ).Te susceptible people who heard recommendations and implement protective measures will join the protected class P h at the rate of τ.In our model, individuals belonging to the susceptible class are at risk of infection at a rate of (λ c ) for the infectious class I c (pre-school age) and at a rate of λ y for the infectious class I y (young -age).Te infected individuals of both age levels recover spontaneously at the natural recovery rate of ω 1 and ω 2 and treatment recovery rate of δ 1 and δ 2 , respectively, to join the recovery class R h .Some studies [22,23] indicated that the recovered humans have some immunity to the disease and do not get clinically ill, but they still harbor low levels of the parasite in their bloodstream and can pass the infection to mosquitoes.After a certain amount of time, they lose their immunity at a rate β and the proportion βϕ returns to the susceptible class and the remaining (1 − ϕ)βR who take some protective measurements enter into the protected class.Since the malaria interventions might face serious obstacles in the form of heterogeneity in parasite, vector, and human population [24], the protected humans may become susceptible again and move to the susceptible class S h at the rate ϕ.Humans leave the total population through natural death rate μ h and malaria death rate (disease-induced death rate) μ d .When a susceptible mosquito S m bites an infectious human, it enters into class I c and I y with fraction of bite K 2 .Mosquitoes are assumed to sufer death due to natural causes and due to the use of insecticide spray at a rate μ m or mortality due to insecticides but cannot die directly from the malaria parasite infection [25]; female mosquitoes enter their population through the susceptible compartment at per capita rate Λ m .It is assumed that there is no immigration of infectious individuals in the human population.Te death related to the disease is diferent between children (preschool aged) and young-aged people, i.e., μ d 1 is greater than μ d 2 [26].We also assume that infectious preschool-age children mature and join the corresponding infectious young-age class at the rate of η.
In Figure 1, red lines show disease progression and solid and black lines show human or mosquito progression from one compartment to another compartment.Based on the above assumptions and fow diagram, the dynamics of the disease were described by the following nonautonomous deterministic system of nonlinear DEs. where represent the force of infection of preschool age, young age, and mosquito.All parameters in Table 1 are positive.

Model Analysis
3.1.Positivity of the Solution of the Model.For the system of diferential equations in (6), to ensure that the solutions of the system with positive initial conditions remain positive  and the initial value for the malaria model (2) be S h (0) > 0, P h (0) > 0, S m (0) > 0, I c (0) ≥ 0, Ten, the solution of S h (t), P h (t), I c (t), I y (t), R h (t), S m (t), and I m (t) of the nonlinear system of diferential equation above is positive for all t > 0.
Similarly, all state variables at t could not be zero and positive.From this, we conclude that all the solutions of (2) are in R 7  + for all t > 0 provided that initial conditions are positive.

Invariant Region.
Te invariant region is a region where solutions of model equation (2) exist biologically [27].Biological entities cannot be negative; therefore, all the solutions of model equation (2) are positive for all time t ≥ 0 [27].Te total population size N h and N m can be defned as in equation (1).In the absence of malaria disease, the DEs for N h are given as Te DEs for N m are also given as be any solution of the system with nonnegative initial condition.Using (7), Terefore, as t ⟶ 0, the human population N h approaches Λ h /μ h , and it follows that [24] lim Terefore, the feasible solution set for model (2) is given by Hence, the compact set M is positively invariant, and the solutions are bounded (i.e., all solutions with initial conditions in M remain in M for all time t).□ 3.3.Disease-Free Equilibrium Point.At the disease-free equilibrium, all the disease classes are zero.It is a scenario which depicts an infection-free state in the community or society.Further, at the disease equilibrium point of people and mosquitoes, I c � 0, I y � 0, I m � 0. Disease-free equilibrium of system is given by ε 0 � (S 0 h , P 0 h , 0, 0, 0, S 0 m , 0), where

Te Basic Reproduction Number.
Te average number of secondary cases a typical infected person produces in their entire life as infectious or an infectious period when introduced or allowed to exist in a group of susceptible individuals is known as the basic reproduction number or R 0 [28].R 0 is a threshold quantity computed using the nextgeneration method which is used to handle the future dynamical behavior of the pandemic and used to study the spread of an infectious disease in epidemiological modeling [28,29].It is defned as using the next-generation method.
Te dominant eigenvalue or reproduction number becomes for which is the average number of secondary infections caused by a single infective in a totally susceptible population.

Local Stability of Disease-Free Equilibrium Point
Theorem 3. Te disease-free equilibrium point of the system of ordinary diferential equation ( 2) is locally asymptotically Proof.To show the local stability of disease-free equilibrium point, we use (7 × 7) Jacobian matrix and the Routh-Hurwitz (RH) criterion.
We consider only the frst and the second column of 7 × 7 matrix; when we consider the ffth and the seventh column, we will get zero matrix because of zero column matrix.Trough the reduction process, we obtain two negative eigenvalues λ 1 � − μ m and λ 2 � − (β + μ h ) and the reduced submatrix becomes and the characteristic equation of the frst submatrix, is where and all coefcients A i of submatrix (21) of the characteristic equation and the frst column of the RH array are positive, so by the RH stability criterion, the two eigenvalues λ 3 and λ 4 of Jacobian have negative real part.Te second submatrix is given by and the characteristic equation of the second submatrix is where All the frst columns of the RH array are positive; then, the remaining eigenvalues of the Jacobian are negative real part for R 0 < 1. Tus, the disease-free equilibrium point ε 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1. □

Global Stability of Disease-Free Equilibrium Point
Theorem 4. If the reproduction number R 0 < 1, the diseasefree equilibrium point ε 0 of model ( 2) is globally asymptotically stable in the feasible region M.
Proof.To prove the global asymptotic stability of the disease-free equilibrium point ε 0 , we use the method of Lyapunov function.Let us defne an appropriate Lyapunov function V(t) by applying the approach in [27].
By substituting expressions for dI c /dt, dI y /dt, and dI m /dt from (2) in (26) and by collecting like terms of the equation, we obtain and by taking coefcients of I c and I y equal to zero for dv/dt ≤ 0, From ( 28), By substituting C 2 in (30), we obtain After substituting C 1 and C 2 in (29) and after some calculation, we obtain [R 2 0 − 1]I m ≤ 0; from this, for I m � 0, R 0 ≤ 1.Finally, we obtain dv/dt ≤ 0; then, dv/dt � 0. Tis shows that the disease-free equilibrium is globally asymptotically stable.□ 3.6.Endemic Equilibrium Point.Endemic equilibrium points are steady-state solutions where the disease persists in the population.Te solution for the endemic equilibrium is obtained in terms of the infected humans, which is also expressed in terms of R 0 .
Theorem 5.If R 0 > 1, then the system of DEs of the model has a unique endemic equilibrium point.
Proof.After some algebraic manipulation, we have 8 Journal of Mathematics and after some steps and simplifcation, the degree three polynomial is reduced to quadratic: and at I * � 0, there was DFE. where From quadratic equation ( 35), the endemic equilibrium exists for Te number of possible positive real roots for (35) depends on the signs of A, B, and C. Tis can be analyzed by using the Descartes rule of signs on the quadratic As indicated in [30], Descartes's rule of sign is used to determine the number of real zeros of a polynomial function; it indicates that the number of positive real zeros in a polynomial function f(I * m ) is equal to or less than the number of coefcient sign changes, on an even number basis.
In Table 2, the existence of multiple endemic equilibria when R 0 < 1 suggests the possibility of backward bifurcation.Te change of stability occurring at R 0 � 1 is often followed by the emergence of branch of steady states.Tis is referred to as bifurcation; this may happen for values of R 0 slightly greater than one which is called forward bifurcation, and if R 0 is slightly less than one, this is called backward bifurcation.In quadratic equation (15), If R 0 > 1 or C < 0, then (15) has a unique positive root: where ∆ > 0. If R 0 � 0 or C � 0, then (15) has a unique positive solution.I * m � − B/2A, provided that B < 0. Here, if B � 0, then I * m � 0 which shows DFE ε 0 , and if B > 0, then I * m < 0, and this does not show meaning in epidemiology.For R 0 < 1 or C > 0 and Δ > 0, we consider two cases.(15) has two endemic equilibria.
By considering such diferent cases of the solution of ( 15), a theorem is established as follows.
□ Theorem 6. Te age-structured malaria model has

3) No endemic equilibrium in all other ways.
In the theorem for R 0 < 1, stable DFE and stable EE come together; this indicates the probability of backward bifurcation.Analysis of backward bifurcation was carried out by employing center manifold theory.

Center Manifold Teory.
Computation of eigenvalues of the Jacobian matrix can be used to determine the stability of the disease at an endemic equilibrium point.Te bifurcation analysis is performed at the disease-free equilibrium by using center manifold theory as presented in Martcheva [28].To apply the center manifold theory, the following simplifcation and change of variables are made on the model which are rewritten by using state variables of malaria model and center manifold approach on the system. Let (41) Further by using the vector, Te system can be written in the form Journal of Mathematics and as follows writing the system in vector forms: Choose k 1 as bifurcation parameter, and solving for R 0 � 1, Te Jacobian matrix evaluated at disease-free equilibrium: Eigenvalues of Jacobian are (48) Using RH criteria, the remaining eigenvalues of Jacobian are negative real for R 0 < 1.Hence, the center manifold theory can be used to analyze the dynamics of the system for the case when R 0 − 1, and it can be shown that the Jacobian matrix has a right eigenvector.
Similarly, the components of the left eigenvector of J correspond to zero eigenvalue, and it can be done by transposing Jacobian matrix. where Now, we shall establish the conditions on parameter values that cause a backward bifurcation to occur in system (45) based on the use of center manifold theory in Martcheva [28].
Computation of a and b for the transformed system of (45) is associated with nonzero partial derivatives of f evaluated at the DFE (S 0 h , P 0 h , 0, 0, 0, S 0 m , 0).
It is not necessary to calculate the derivatives of 12 Journal of Mathematics when we come to a, Let Let and by considering F 1 and F 2 , a is positive if As we observed b is positive, according to center manifold theory, if a > 0, b > 0, then the given age-structured malaria model undergoes backward bifurcation at R 0 � 1 whenever b > 0 and As we observed, an age-structured malaria model exhibits backward bifurcation whenever a > 0, and the epidemiological signifcance of backward bifurcation is that, in addition to generating R 0 < 1, more action is necessary to reduce the dynamics of malaria transmission in communities.Figure 2 shows the backward bifurcation phenomenon as evidence for the malaria model analysis.Te stable equilibrium is represented by the solid line and the unstable equilibrium is represented by the dotted line.It confrms the results of the analysis, showing an endemic equilibrium.

Te Local Stability of the Endemic Equilibrium Point.
We conduct linear stability on the endemic equilibrium point using the Jacobian of the malaria model of the equations.Ten, the following stability theorem is stated.Proof.To show the local stability of the endemic equilibrium point, we use the method of the Jacobian matrix and RH stability criterion.Te Jacobian of the malaria model at any point is where m /N 0 h .By considering the frst column and corresponding row of 7 × 7 matrix, and if we consider |J − λI| � 0, the frst column has a diagonal entry.Terefore, one of the eigenvalues is given by λ 1 � D 2 � − (φ + μ h ).Te reduced matrix becomes  Journal of Mathematics and if we consider |J − λI| � 0, the third column has a diagonal entry.Terefore, the second eigenvalue is Te reduced matrix becomes .
By considering the second column of 7by7 Jacobian matrix, the reduced matrix after manipulation becomes (2) − (φτ) By taking the common of (1) and ( 2), By considering the ffth column of 7 × 7 of the Jacobian matrix, the reduced matrix after expanding diferent columns with corresponding rows becomes And by considering the seventh column, the corresponding row of the reduced matrix is By considering (3), (4), and ( 5), After manipulation and rearranging, we obtain the characteristic equation: Using the RH stability criteria, we prove that when R 0 > 1, all roots of the polynomial equations have negative real parts.Tus, the endemic equilibrium point ε * is locally asymptotically stable if R 0 > 1. of the system is globally asymptotically stable if R 0 > 1.
Proof.Let us defne an appropriate Lyapunov function V(x) by applying the approach [28] such that where X i represent the population of the compartment X * i and are endemic equilibrium points in R 7 + , and thus, (64)

Journal of Mathematics
By diferentiating (64) with respect to t and replacing the derivatives in the equation from their respective expressions in the equation of the system, we obtain Terefore, by the principle of LaSalle [31], the endemic equilibrium ε 0 is globally asymptotically stable in the invariant region □ 3.9.Sensitivity Analysis.Te normalized direct sensitivity index of the variable R 0 depends on a parameter (P r ) defned as Tese small shear sensitivities allow us to determine the relative importance of diferent parameters on malaria transmission and prevalence.Te most sensitive parameter in Table 3 has a sensitivity index greater than all other parameters.

Numerical Simulation of Integer-Order Malaria Model
Te numerical simulations examine the efect of combinations of parameters of the modifed model on the transmission of the disease by using MATLAB.Te simulation is carried out by taking diferent values of parameters.Te set of parameter values is given in Table 4 whose sources are mainly from literature as well as assumptions.We used diferential equation solver ODE (45).Te simulations and analysis made are based on these parameter values and initial conditions below.Te following initial conditions have been considered:

Numerical Simulation with Sensitive
Parameter.We consider two sensitive parameters, namely, number of bites on preschool-age human per female mosquito per time and the number of bites on young-age human per female mosquito per time.

The Fractional Malaria Model and Analysis
Tere are a certain number of limitations of the models developed via classical diferential equations, such as the absence of memory efects and being not able to capture the crossover behavior of a physical or a biological process."Te fractional operator, specifcally the ABC operator, comprises Journal of Mathematics the memory efects and the crossover behavior of the model."Memory efect means that the future state of the fractional operator of a given function depends on the current state and the historical behavior of the state [32].Terefore, to explore the malaria dynamics more realistically, "Some basics of fractional calculus" are reformulated with the replacement of classical derivative by the one having fractional order in ABC sense.Tus, the fractional epidemic model for age-structured malaria model with the nonlocal kernel is formulated through the following system.

Some Basic
Defnition 10 (see [33]).Let f: [a, b] ⟶ R be bounded and continuous function; then, the corresponding fractional integral concerning AB fractional-order derivatives is defned as ABC a Theorem 11.Let f: [a, b] ⟹ R be bounded and continuous function; then, the following result holds as in [32]: Furthermore, the Atangana-Baleanu derivative fulflls the Lipschitz condition [32] for two functions f 1 , f 2 ∈ L(a, b), b > a; then, the AB fractional derivative satisfes the following inequality: where 0 < α ≤ 1 is the order of fractional derivatives.Te fractional-order system of the diferential equation of malaria is proposed as follows: Te Atangana-Baleanu (AB) derivative is described by the system of DEs, and the mathematical model can be written as ABC 0

Existence and Uniqueness of Solutions.
To show the existence of solution of the given model, we use the Banach fxed point theorem, and to show the existence and uniqueness of the solution, we apply AB fractional integral to the proposed model [34].Let be the Banach space of real-valued continuous functions defned on an interval E(J) � [0, T] with the corresponding norm defned by and the associated sup norm [35].
For each kernel in the fractional model above, there exists a Lipschitz constant L i > 0, i � 1, 2, 3, 4, 5, 6, 7, such that Proof.Let the kernel of the frst compartment, F 1 , satisfy the Lipschitz condition and contraction if the inequality given below holds: Suppose that where Journal of Mathematics is bounded function, so and thus for F 1 , the Lipschitz condition is obtained, and if then F 1 is contraction.Similarly, are bounded functions; if 0 ≤ L i < 1, i � 2, 3, 4, 5, 6, 7, then F i , i � 2, 3, 4, 5, 6, 7, are contraction.Consider the following recursive form for any positive integer n: and we express the diference between the successive terms by using recursive formula in (80).
From ( 81), the diference between successive terms is expressed as follows: Journal of Mathematics 19 with initial conditions Equation ( 70) can be reduced using defnition of the norm. 20

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Applying triangular inequality, and by integrating (73), we obtained Similarly, □ Theorem 13.Te mathematical model involving Atangana-Baleanu fractional model given in (69) has solution if there exists y 0 such that Proof.Using techniques of recursive formula, we obtain Similarly, Now we are going to show functions which are S h , P h , I c , I y , R h , S m , I m that are solutions of (69). Assume and by repeating the process of recursive formula, we obtain for t � y 0 , and (92) becomes

Journal of Mathematics
and by taking the limit of (93), Tis completes the proof of existence of the solution of the given model using the Banach fxed point theorem; the same is true for the remaining expressions.

□ Theorem 14. Te Atangana-Baleanu fractional model has a unique solution if
By taking the norm of both sides and after integrating, we obtain We have ‖X(t) − X * (t)‖ which is common for both sides since and we get ‖X(t) − X * (t)‖ � 0; then, we have X(t) � X * (t), and thus (69) has unique solution.

Theorem 15. Te epidemiologically feasible region of AB fractional model is given by
To show positivity, we have to consider the following lemma.
Lemma 16 (see [36]) (generalized mean value theorem).Let f(x) is nonincreasing.Let us show that M is positively invariant; using the above lemma, we have Similarly, each of the remaining solutions of the model is nonnegative and remains in M. To show that the solution of the system is bounded, we have to obtain the fractional derivatives of total population by summing up all the relations in the system, so By applying Laplace transform on both sides of the above inequality,

22
Journal of Mathematics where and by applying the inverse Laplace transform [37], the solution is given by where E α,β refers to Mittag-Lefer function, and it has asymptotic behavior.
as, t ⟶ ∞, N h (t) ≤ Λ h /μ h as, t ⟶ 0 hence it is a biologically feasible region that means for t ≥ 0 we have 0 < N h (t) ≤ (N h (t))/μ h this indicates that the total human population is bounded.In the same way, mosquito population is also bounded because N m (t) ≤ Λ m /μ m as t ⟶ 0.

Local Stability of Disease-Free Equilibrium Point of
Fractional Model.As for the case of the model with integer derivative (45), the fractional model (69) admits always DFE, and disease-free equilibrium point of ( 69) is given in (14).As in the case of ODE model (45), we compute the reproduction number R 0 using the next-generation matrix approach [38], and the reproduction number of fractional model ( 69) is given by Theorem 17. Te disease-free equilibrium of ( 69) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
To show the local stability of disease-free equilibrium of (69), we use 7 × 7 Jacobian matrix and RH criterion.As indicated in (11), the disease-free equilibrium (ε 0 ) of ( 69) is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.
By using the same Lyapunov type function as in ( 14) of classical model (45), we prove that DFE of fractional model (69) is globally asymptotically stable in M whenever R 0 ≤ 1. Tus, the following result is valid.Theorem 18.For any α ∈ (0, 1], the disease-free equilibrium point ε 0 of model ( 69 To show the local stability of the endemic equilibrium point, we used the method of the Jacobian matrix and RH stability criterion. As indicated in (21), endemic equilibrium point ε * is locally asymptotically stable if R 0 > 1.

Theorem 20. Te endemic equilibrium point ε
As we observe from (26) of this paper, ε 0 is globally asymptotically stable in the invariant region M if G 1 < G 2 for R 0 > 1.

Numerical Scheme and Simulation of
Fractional-Order Model Te numerical scheme for (112) is defned as in [34].
By adopting the procedure in [40], the numerical scheme of each compartment in the fractional model (69) takes for step size h(t m , t m− 1 ).
In the numerical simulation of dynamics of malaria disease, two categories of the species are considered: the host population which contains the human population and the vector which consists of mosquito.Te two species are interconnected with each other.Te efect of embedded parameter is shown on the dynamics of both species.We used the numerical technique developed by Toufk and Atangana.

Result and Discussion
Te susceptible human population decreases rapidly as shown in Figure 3. Tis indicates that the susceptible human population will continue to join the infected class; as a result, the infected population will increase due to high biting rate of mosquito and high probability transmission rate from the infected mosquito to the susceptible human.R 0 � 1.622 which is greater than one; this indicates that the mosquito vector is continuously increasing.It supports the theorem for stability of endemic equilibrium point that the disease is endemic when R 0 > 1.
Figure 4 shows the distribution of population for different classes with time.Tus, the susceptible human population decreases due to the presence of infective mosquito with high biting rate of mosquito for the frst few days.Since the infective vector bites the susceptible human, the susceptible human becomes infected and goes to the infected human compartments; then, the susceptible population decreases and the infected human population increases.After some interval of time, they go to zero due to increment of protected class, that is, as protected class increases, susceptible vector and infected vector class decrease due to lack of meal for their egg; then, the disease-free equilibrium point exists and is stable.Te existence of this condition is due to the fact that R 0 � 2.827 × 10 − 5 which is less than one.Tis supports the theorem that the stability of disease-free equilibrium point exists when R 0 < 1, i.e., the society is free from the disease when R 0 < 1.
We also evaluated sensitivity indices of the parameter values shown in Table 2.In the case of malaria transmission, the most sensitive parameter is the rate of mosquito bites or the number of bites in people of preschool age (ε c ) and young age (ε y ); other parameters include the probability of transmission from an infectious mosquito to a susceptible person or a portion of the bite that successfully infects human (K 1 ) and the probability of disease transmission from infectious human to susceptible vector or a portion of the bite that successfully infects mosquito.As shown in Figure 4, the number of bites of mosquito increases and the susceptible human population decreases because when the contact between mosquito and the Journal of Mathematics two human age levels increases, the force of infection increases and individuals go to the infected class to increase the infected human class as shown in Figure 5.As the number of bites increases, the population of susceptible vector decreases because of increase in infected vector for some time interval initially and then decreases because infected vector goes to the susceptible human class to increase force of infection as shown in Figures 6-8.Susceptible human population decreases in three directions: frstly because of natural death rate, secondly because of increment in force of infection as shown in Figure 4, and thirdly because of increment in transfer rate tau (τ) of human from susceptible class to protected class as shown in  3 and 12, the number of infected humans decreases as the number of recovered humans increases because of increment in natural recovery rate and treatment rate, so this increment in recovery rate is one way to increase protected class individuals; increment in protected class is our target to control malaria disease; because of increment in protection class and recovery rate, infected preschool-age and young-age human population decreases as shown in Figures 13-15.
In Figures 16 and 17, we show the global asymptotic stability of the proposed model by varying the initial conditions of each compartment for time being, and to save time and space, we show only two compartments, namely, susceptible humans and infected preschool humans.
As indicated in Figures 18 and 19, as fractional order α approaches 1 or integer order, the susceptible human population decreases which is similar to that of classical model result, that is, as the biting rate of mosquito increases, the susceptible human population decreases.Te majority of              28 Journal of Mathematics endemic equilibrium when R 0 < 1 experiences backward bifurcation; this suggests that society may not fully comprehend the extent of malaria prevalence in the population.When the level of malaria endemicity expressed only by the size of the basic reproductive number is less than one, the disease can disappear but still persist (at very high endemic levels).

Conclusion
Integer-and fractional-order models were presented for the dynamics of malaria in human hosts with varying ages.A system of diferential equations model with fve human state variables and two mosquito state variables was examined.We demonstrated the existence of an area in which the model is both mathematically and epidemiologically well posed.Te equilibrium point devoid of disease was discovered, and its stability was examined.We identifed the basic reproduction number R 0 in terms of the model parameters that measure the intensity of the transmission of the disease.It has been demonstrated that during the course of the infectious period, R 0 predicts the anticipated number of additional infections (in mosquitoes and people) from one infectious individual (human or mosquito).It was also established that for the basic reproduction number R 0 < 1, the disease-free equilibrium point emerges.We showed that the endemic equilibrium point is unique for R 0 > 1, and also we showed numerical result for both fractional-and integer-order models.For the numerical simulation of integer order, we used ODE (45), and we used numerical technique developed by Toufk and Atangana for fractional order.Plasmodium parasites cause malaria.Since malaria is a global problem, the following measures should be taken to eradicate the disease.Te biological explanation of ( 21) is that when backward bifurcation occurs, malaria can still exist in the community even when R 0 < 1, and such situation can lead to misunderstandings about malaria eradication programs.Responsible bodies like policy makers may think they have succeeded in bringing R 0 under one and hope malaria will disappear.Unfortunately, if backward bifurcation occurs, there will be a large endemic equilibrium due to the hysteresis that occurs when R 0 < 1.
(i) Our results in (21) indicate that responsible bodies need to increase the culture of using diferent control mechanisms like ITNs and IRS including diferent medical treatments to avoid backward bifurcation or inverse bifurcation.(ii) Since the model indicates that the probability of disease transmission rate and mosquito biting rate play a major role in the disease's spread, eforts should be made to reduce mosquito populations and biting rates through biological or chemical means, or any other method that will lower the rate of malaria infection.(iii) Government agencies with accountability should start and continue efcient programs to guarantee that public health decision makers take into account intervention strategies aimed at reducing mosquito populations and biting rates when controlling malaria.Furthermore, we plan to expand the model in subsequent work to encompass (a) Efects of diferent constant control mechanisms on malaria prevalence, optimal control, and cost analysis.(b) Te efects of temperature and rainfall on the spread of malaria on the mortality and survival probabilities via optimal control.(c) Diferent fractional-order derivatives and integerorder derivatives, comparing their results and performing backward and forward bifurcation to identify the prevalence of malaria disease.
where C 1 , C 2 , C 3 are positive constants and I c , I y , and I m are positive state variables.

□ 3 . 8 . 8 .
Te Global Stability of the Endemic Equilibrium Point Theorem Te endemic equilibrium point ε * � S * h Figures 9 and 10 to increase protected class individuals as shown in Figure 11.As explained in assumption part, protection class is the class with individuals who use intervention mechanisms like ITNs and IRS; if individuals who use such control mechanisms increase, the force of infection decreases because of decrease in mosquito vector with lack of meal for their egg production.As shown in Figures

Figure 3 :Figure 4 :
Figure 3: Human mosquito plot shows that susceptible human populations decrease rapidly.Tis indicates that the susceptible human population will continue to join the infected class R 0 � 1.622.

Figure 5 :
Figure 5: As the number of bites of mosquito increases, the susceptible human population decreases.

Figure 6 :
Figure 6: Te number of infected young humans increases as the number of bites increases.

Figure 7 :
Figure 7: As the number of bites increases, the population of susceptible vector decreases because of the increment of infected vector.

Figure 8 :
Figure 8: Te result shows that the infected vector population is increased for initial time interval then after decrease as time increase.

Figure 9 :
Figure 9: Te population of infected vector increases for initial time interval and then decreases.

Figure 10 :
Figure 10: Susceptible population decreases as increases transfer rate (τ) of human from susceptible to protected class.

Figure 11 :
Figure 11: Te dynamics of susceptible population with fractional order α which show the decay behavior for the given time t.

Figure 12 :
Figure 12: Te result of recovered human population with fractional order Îś shows that the recovery is increased for some time interval.

Figure 13 :
Figure 13: Te dynamics of protected human with fractional order α which show protected individuals increased for initial time interval and decreasing behavior for some time; and then increase fnally.

Figure 14 :
Figure 14: Te result shows that the infected preschool-age human increased for the initial interval of time and then strictly slow or decrease as time increase.

Figure 15 :Figure 16 :
Figure15: As the result showed that the infected young human increased for the initial interval of time and then decreased as that of infected preschool-age human, but the decrement is not strict.

Figure 17 :
Figure 17: Global stability of susceptible human compartment.

Figure 18 :
Figure 18: As the value of τ increases, the protected class also increases.

Figure 19 :
Figure19: Te result of susceptible vector population with fractional order alpha shows that the susceptible vector population decreases because of infection in vector population.
Te human population, denoted by N h , is divided into fve epidemiological categories: the susceptible class S h , the protected class P h , the infectious class of preschool age I c , the infectious class of young age I y , and the recovered class R h .We also divide the mosquito population into two major stages: the mature stage and the aquatic stage (egg, larvae, and pupae), but we consider the mature stage which is divided into two compartments, namely, susceptible class of mosquitoes denoted by S m and infectious class of mosquitoes denoted by I m .Te mosquitoes' population does not have a recovered class because their infective period ends with their death.At any time t, the total size of the human N h (t) and mature mosquitoes N m (t) is, respectively, denoted by

Table 1 :
Description of parameters.
1 Natural recovery rate of preschool age δ 1 Recovery rate by treatment of preschool age ω 2 Natural recovery rate of young age δ 2 Recovery rate by treatment of young age φ Transfer rate of human from P h to S h τ Transfer rate of human from S h to P h Λ h Constant recruitment rate for humans β Rate of loss of immunity ϕ Proportion of humans who lose their immunity that become S h (1 − ϕ) Proportion of humans who lose their immunity that become P h c Proportion of new recruitments that are protected λ C Force of infection for preschool age λ y Force of infection of young age λ m Force of infection of mosquitoes η Maturation rate from I C to I y μ d 1 Disease-induced mortality rate of preschool age μ d 2 Disease-induced mortality rate of young age μ m Mortality rate of mosquitoes μ h Natural mortality rate of human ε c Number of bites on preschool-age people ε y Number of bites on young-age people

Table 2 :
Number of possible real roots of f(I* m ) for R 0 > 1 and R 0 < 1.

Theorem 7 .
Te endemic equilibrium point ε * � (S * m ) of the malaria model is locally asymptotically stable if and only if R 0 > 1.
Based on this, we can observe that the biggest compact invariant singleton set in M � S h (t), P h (t), I c (t), I y (t), R simplify equation (65) by gathering negative and positive terms, and yielding: dV/ dt � G 1 − G 2 , where G 1 denotes positive terms and G 2 denotes negative terms.Terefore, if G 1 < G 2 , dV/ dt ≤ 0 and dV/ dt � 0 if and only if S h � S * h , P h � P * h , I c � I * c , I y � I * y , R h � R * h , S m � S * m , I m � I * m , hence V is, therefore, the Lyapunov function on M. h  (t), S m (t),
Concepts from Fractional Calculus Defnition 9. Let f: [a, b] ⟶ R, a < b, be abounded and continuous function and let α ∈ [0, 1].Te Atangana-Baleanu fractional derivative for a given function of order α in Caputo sense is defned by ABC a