A mathematical approach for studying the fractal-fractional hybrid Mittag-Le ffl er model of malaria under some control factors

: Malaria disease, which is of parasitic origin, has always been one of the challenges for human societies in areas with poor sanitation. The lack of proper distribution of drugs and lack of awareness of people in such environments cause us to see many deaths every year, especially in children under the age of five. Due to the importance of this issue, in this paper, a new five-compartmental ( c 1 , c 2 )-fractal-fractional SIR - SI -model of malaria disease for humans and mosquitoes is presented. We use the generalized Mittag-Le ffl er fractal-fractional derivatives to design such a mathematical model. In di ff erent ways, we study all theoretical aspects of solutions such as the existence, uniqueness and stability. A Newton polynomial that works in fractal-fractional settings is shown, which allows us to get some numerical trajectories. From the trajectories, we saw that an increase in antimalarial treatment in consideration to memory e ff ects reduces the peak of sick individuals, and mosquito insecticide spraying minimizes the disease burden in all compartments.


Introduction
The well-known disease malaria is one of the illnesses threatening human health that appears under the influence of a parasitic infectious agent in female Anopheles mosquitoes. In fact, Plasmodium parasites are the main cause of this infection, which causes disease by implanting in the red blood cells of an infected person, and among the different types of these parasites, two of them are among the most common pathogenic parasites: Plasmodium vivax and Plasmodium falciparum [1]. The malaria parasite is transmitted to the human body and its bloodstream by biting infected female Anopheles mosquitoes. Also, the process of transmitting the parasite can occur through blood transfusions or even infection of the fetus through its pregnant mother who carries the infection [2].
Although this disease will not be dangerous if it is detected in time and treated properly, it has negative effects on human health and social life. The disease is most commonly reported in the tropical regions of Africa and Asia and imposes heavy financial burdens on families and governments. According to extensive studies on the spread and control of the disease by various medical institutes and associations, it still threatens public health and causes deaths in children under five in disadvantaged and less developed countries [3]. Irregular vaccination or improper distribution of antimalarial drugs can spread the disease. Even due to the nature of the parasites, sometimes, with widespread changes in the climate and the surrounding environment, the drug resistance of the parasites is impaired and causes the drugs to lose their effectiveness. Currently, the simplest advice for people in high-risk areas is to use bedside nets and window sills, which, to a large extent, prevent the number of bites during sleep or indoors. With widespread social and environmental changes and the effects of climate change in recent years, there has been a need to study the biology of host parasites carefully. The study of behaviors and dynamics of these parasites in the context of different methods of disease control and treatment has been one of the important points in recent research studies that have attracted the attention of researchers.
In this regard, various mathematical models came to the aid of researchers to simulate the exact dynamic behaviors of the transmission and spread of different types of viruses and infectious parasites. Also, by providing and designing different treatment and control methods, one can evaluate and predict the amount of prevalence during a specific time period. In 2001, Yang [4] considered a mathematical structure to model the transmission of malaria based on two factors, i.e., global warming and socioeconomic conditions; further, Yang completed the study by providing the sensitivity analysis. In 2008, Chiyaka et al. [5] turned to control strategies on a deterministic model of malaria during two latent periods and analyzed some qualitative criteria to calculate the vaccination rate. One year later, Rafikov et al. [6] implemented some strategies on malaria to compute the optimal control index with the help of some genetic modifications on vector mosquitoes. In 2011, Mandal et al. [7] published a review paper on the different models of malaria disease. After that, in 2012, Agusto et al. [8] extended control strategies on malaria by adding three indexes, including bed nets, treatment and the use of the spray. In 2013, Abdullahi et al. [9] designed their mathematical model of malaria by investigating the effectiveness of drugs. Recently, in 2017, Senthamarai et al. [10] gave a multi-compartmental model of malaria and predicted the behavior of solutions based on numerical algorithms obtained by the homotopy method.
However, for the sake of the existing limitations in the above methods and models, and due to the locality of the integer-order operators, in recent years, the tools and operators in fractional calculus have begun to provide considerable simulation and prediction power for mathematicians and physicians malaria until now. In relation to our main contribution to the present study, it is necessary to emphasize that we divide our target population into five different groups of humans and mosquitoes and then provide parameters and rates for which we can measure the effects of vaccination, antimalarial drugs and spraying on the control and reduction of this disease. Our model will be discussed from several perspectives. Because our fractal-fractional model is newly structured, we investigate existence theory via the Leray-Schauder alternative fixed-point theorem. Furthermore, the Banach contraction is utilized to obtain a unique solution. Further, other types of stable solutions are studied here for the suggested model. To simulate it, we use the new method of Newton polynomials in the fractal-fractional version. The findings and effects of fractal-fractional orders on the dynamics of solutions are analyzed, and the graphs are plotted using MATLAB.

Preliminaries
In the present section, we state some definitions of the generalized fractal-fractional operators. We refer the readers to [43] for more information.
Let a continuous map z : The (c 1 , c 2 )fractal-fractional derivative of the function z of the generalized Mittag-Leffler-type kernel in the Riemann-Liouville (RL) sense is given by which is the fractal derivative; also, and AB(0) = AB(1) = 1 [43].
Simply, we see that the fractal-fractional derivative FFML D (c 1 ,c 2 ) t 0 ,t is transformed into the standard c th 1 -RL derivative RL D c 1 t 0 ,t by assuming c 2 = 1. By considering such a function z with the above properties, the (c 1 , c 2 )-fractal-fractional integral with the Mittag-Leffler-type kernel is defined by if the integral is finite-valued, where c 1 , c 2 > 0 [43].
3. Description of (c 1 , c 2 )-fractal-fractional SIR-SI-model In 2019, Kumar et al. [52] designed a new model of malaria with the operators involving the exponential law. In fact, they were motivated by the standard system of differential equations given by where the total population of the humans and the total population of mosquitoes are divided into three categories and two categories, respectively. Kumar et al. [52] generalized the standard model (3.1) to a form of the fractional SIRS-SI model with the Caputo-Fabrizio derivative, which involves memory effects, as In the above model, the authors introduced the categories H S (t), H I (t) and H R (t) as the number of susceptible, infected and recovered persons, respectively, and also introduced two categories M S (t) and M I (t) as the number of susceptible and infected mosquitoes, respectively, at the time t ∈ I := [0, τ], (τ > 0). To achieve more accurate numerical results in an efficient manner, due to the important role of two parameters of fractional order and dimension order in fractal-fractional operators for exact simulations, and motivated by both the standard and fractional models (3.1) and (3.2), we design and give a mathematical five-compartmental (c 1 , c 2 )-fractal-fractional SIR-SI-model of malaria disease under antimalarial treatments with the generalized Mittag-Leffler-type kernel between two populations of humans (H) and mosquitoes (M) (shortly, (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model of malaria), which takes a form , M S (t) and M I (t) are similar to those introduced in the Caputo-Fabrizio model (3.2). Also, FFML D (c 1 ,c 2 ) 0,t is the (c 1 , c 2 )-fractal-fractional derivation operator equipped with the fractional order c 1 and fractal order c 2 so that c 1 , c 2 ∈ (0, 1]; and, the kernel of the operator is of the generalized Mittag-Leffler type. In the mentioned (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model The symbol a 1 interprets the probability of transmission of the disease to susceptible individuals through infected individuals, a 2 is the probability of transmission of the disease from infected mosquitoes to susceptible peoples and a 3 stands for the probability of transmission of the disease to susceptible mosquitoes through the infected population. The rate of personal healing is denoted by γ, and the symbol β denotes the power of antimalarial medicines.
In this model, we see that the people belonging to the susceptible category H S are transferred to the infected categories H I and M I by way of the blood exchanging via the rate ηa 1 , or due to an infected mosquito bite via the rate λa 2 for each unit of time, respectively. Also, due to vaccination at the rate p for each unit of time, the people belonging to the susceptible category H S transfer into the recovered category H R . The people belonging to the susceptible category H S die at the rate f H . Moreover, the individuals belonging to the infected category H I die at the rate f H and expire due to the malaria disease at the rate ω for each unit of time.
The people belonging to the recovered category H R also die at the rate f H for each unit of time. The susceptible and infected mosquitoes die due to the exercise of spraying at the rate α for each unit of time. The infected mosquitoes belonging to M I die at the rate f M for each unit of time.
The newly born children are infected by malaria disease through the mother at the rate ϖ for each unit of time. The individuals in the infected category H I can join the recovered category as a result of using antimalarial drugs at the rate γβ for each unit of time. The mosquitoes belonging to the susceptible category M S transfer to the infected category M I by being bitten by an infected mosquito at the rate κa 3 for each unit of time, or they die at the rate f M for each unit of time. The symbol δ is used for the mean value per capita rate of loss of immunity for each unit of time.

Existence results
In this section, we get help from the well-known theorems of fixed-point theory to investigate the existence property. Take the Banach space X = F 5 , and then assume F = C(I, R) with the supremum norm Further, for simplicity, the right-hand side of the (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model (3.3) of malaria can be rewritten as   Here, take into account the system (4.2), and reconstruct it as a compact initial value problem, like by assuming and X t, (4.5) The non-singular Atangana-Baleanu-Reimann-Liouville fractional derivative changes (4.3) to The Atangana-Baleanu fractal-fractional integral on (4.6) gives The following extensions of the base system of fractal-fractional integral equations are given as A new map, to make a fixed-point problem, is defined by T : X → X, which has been formulated as To study the existence of solutions of the (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model (3.3) of malaria, we use the following theorem: Theorem 4.2. Let X ∈ C(I × X, X), and we have the following: (g1) There are F ∈ L 1 (I, R + ) and a nondecreasing function Ψ ∈ C([0, ∞), (0, ∞)) such that, for each t ∈ I and K ∈ X, with F * 0 = sup t∈I |F (t)|. Then, for the FF-system Proof. Consider T : X → X defined by (4.8) and The continuity of X implies the same property for T . The existing inequality in (g1) gives Then, and T is equicontinuous and compact on N R by the Arzelá-Ascoli thoerem. By Theorem 4.1, either (hy1) or (hy2) is to be held. From (g2), take With the help of (g1) and by (4.10), we estimate By taking into account the existence of K ∈ ∂Φ and 0 < µ < 1 with K = µT (K), for these choices of K and µ, and by (4.12), we have which is impossible. Thus, the item (hy2) is not fulfilled and T has a fixed point inΦ (from Then, the functions X 1 , X 2 , X 3 , X 4 and X 5 defined by (4.1) are Lipschitz if L 1 , L 2 , L 3 , L 4 , L 5 > 0, with , Proof. For the function X 1 , we choose H S , H S * ∈ F := C(I, R) arbitrarily. Then, This states that X 1 is Lipschitz with respect to H S with the Lipschitz constant L 1 > 0. Regarding X 2 , for each H I , H I * ∈ F := C(I, R), we have This means that X 2 is Lipschitz with respect to H I with the Lipschitz constant L 2 > 0. Now, for each Thus, X 3 is Lipschitz with respect to H R with the Lipschitz constant L 3 > 0. For each M S , M S * ∈ F := C(I, R), we have which implies that X 4 is Lipschitz with respect to M S with the Lipschitz constant L 4 > 0. Lastly, for each M I , M I * ∈ F := C(I, R), we have Therefore, X 5 is Lipschitz with respect to M I with the Lipschitz constant L 5 > 0; the proof is completed. □ where L ȷ is introduced in (5.1).
Proof. We consider this fact that the theorem is not true. Hence, there exists another solution for the given (H, Then, by (4.8), we have The above inequality holds when ∥H S − H S * ∥ = 0 or H S = H S * . From the inequality This is true when ∥H I − H I * ∥ = 0 or H I = H I * . Moreover, the inequality Hence, H I = H I * . Accordingly, we get M S = M S * and M I = M I * . In consequence, we get Notice that Definition 6.2 is obtained from Definition 6.1.
The following lemmas are useful for our main theorems.
and 4) and 6) and via Remark 1, we are allowed to select G 1 (t) such that and |G 1 (t)| ≤ R 1 . It follows that Then, we estimate It is found that the inequality (6.3) is obtained. We get the inequalities (6.4) and (6.5) similarly. □ To prove the next result, we regard the following: (G4) There are increasing functions χ ȷ ∈ C([0, τ], R + ), ( ȷ ∈ {1, . . . , 5}) and there is ∆ χ ȷ > 0 such that Proof. Let R 1 > 0 be arbitrary. Since H S * ∈ F satisfies by Remark 2, we are allowed to select a function G 1 (t) so that and |G 1 (t)| ≤ R 1 χ 1 (t). It follows that Then, we estimate Similarly, we can obtain the remaining inequalities. Proof. Let R 1 > 0 and H S * ∈ F be an arbitrary solution of (6.1). By Theorem 5.2, we take H S ∈ F as a unique solution of the (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model (3.3) of malaria. Then, H S (t) is defined as Via the triangle inequality, and by Lemma 6.5, estimate Hence, we get Hence, the Ulam-Hyers stability of the (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model (3.3) of malaria is fulfilled. Next, by assuming with Q X ȷ (0) = 0, clearly, the generalized Ulam -Hyers stability is confirmed. □ The Ulam-Hyers-Rassias stability is checked for the (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SImodel (3.3) of malaria in the next theorem.
Via the triangle inequality, and by Lemma 6.6, estimate Accordingly, it gives If we let . Similarly, we have

Numerical scheme via the Newton polynomial method
In this section, we give a numerical scheme for solutions of our (H, M)-(c 1 , c 2 )-fractal-fractional SIR-SI-model (3.3) of malaria which was presented by Atangana and Araz in their book [54] in 2021. For this purpose, we need the compact form of the initial value problem (4.3) again under the conditions (4.4). Thus, Take X * (t, K(t)) = c 2 t c 2 −1 X(t, K(t)). Then, By discretizing the above equation at t = t k+1 = (k + 1)h, we get K(t k )).
If we approximate the above integral, then it becomes In this step, the function X * (t, K(t)) is approximated by the Newton polynomial as . Substitute (7.2) into (7.1): We simplify the above relations, and we get In consequence, By putting (7.4)-(7.6) in (7.3), we obtain Finally, we replace X * (t, K(t)) = c 2 t c 2 −1 X(t, K(t)) in (7.7), and we get Based on the numerical scheme obtained in (7.8), and based on (4.1), by assuming and and and M I k+1 = κ 5 + where the constantsÂ ȷ (k, ℓ, c 1 ) are introduced in (7.9) for ȷ = 1, 2, 3.

Simulations
We discuss the behavior of the model on different plotted simulations by assuming the numerical data for the parameters computed in [52]. According to this source, we take Θ H = 0.027, δ = 1/730, η = 0.038, a 1 = 0.02, λ = 0. 13 Figure 1a shows that, when the fractal-fractional order is 1, the number of susceptible humans decreases rapidly, but not when the fractal-fractional order is 0.5 or less. It also suggests that, when the fractal-fractional order is 0.95, a change in dynamical behavior in the infected and recovered individuals reduces the number of recoveries after week 5, resulting in an increase in the total number of infected humans after week 25. Similarly, we notice in Figure 1b that the susceptibility of mosquitoes decreases rapidly when the fractal and fractional order is 1, as opposed to when the fractal and fractional order is 0.95. In contrast, the fractal-fractional dynamical behavior of the infected mosquitoes increases above the integer order after week 17.  Figure 2a through 2e depict the fractal-fractional behaviors of susceptible humans, infected humans, recovered humans, susceptible mosquitoes and infected mosquitoes when the fractal-fractional derivative is 1 and the fractal-fractional order is 0.98, 0.96, 0.94, 0.92 or 0.90, respectively. In Figure 2a, 2d, you see that the number of susceptible humans and susceptible mosquitoes increases as the fractalfractional order reduces. In Figure 2b, 2e, you see that the number of infected humans decreases as the fractal-fractional order decreases for 13 weeks; it then surpasses the integer order from week 14 onwards. In Figure 2c, you see that the number of recovered individuals reduces as the fractalfractional order reduces. Figure 3a through 3e show the effect of antimalarial drugs on the human population, and that on the mosquito population. Figure 3b, 3e show that antimalarial drugs have a greater effect on the number of infected humans and mosquitos than the other compartmental classes. Figure 3a shows that an increase in the efficacy of antimalarial drug increases the number of susceptible humans after week 8 when the fractal-fractional order is 0.99. Figure 3c shows that an increase in the efficacy of antimalarial drugs increases the number of recovered humans after week 5 when the fractalfractional order is 0.99. Figure 3d shows that an increase in the efficacy of antimalarial drugs increases the number of susceptible mosquitoes after week 9 when the fractal-fractional order is 0.99.  Figure 4b, 4e show that vaccination has a weaker effect on the numbers of infected humans and mosquitos than the other compartmental classes as compared to antimalarial drugs. Figure 4a shows that an increase in vaccination increases the number of susceptible humans for 20 weeks more than antimalarial drugs when the fractal-fractional order is 0.99. Still, after 20 weeks, the antimalarial drugs increase the number of susceptible humans more than vaccination. Figure 4c shows that an increase in vaccination increases the number of recovered humans after week 2 when the fractal-fractional order is 0.99. Figure  4d shows that an increase in vaccination does not produce any significant change in the number of susceptible mosquitoes when the fractal-fractional order is 0.99. Figure 5a through 5e show the effects of spraying on all five state functions.      Figure 5b, 5e show that spraying has a more significant effect on the number of infected mosquitoes than that of infected humans; hence, it suggests that an increase in spraying potentially reduces the number of malarial incidence more significantly than antimalarial drugs. Figure 5a shows that an increase in spraying increases the number of susceptible humans more significantly than antimalarial drugs and vaccination when the fractal-fractional order is 0.99. Figure 5c shows that an increase in spraying produces fewer recovered humans as compared to antimalarial drugs and vaccination when the fractal-fractional order is 0.99. Figure 5d shows that an increase in spraying reduces the number of susceptible mosquitoes when the fractal-fractional order is 0.99. The relative importance of fractional order only and fractal dimension only on the epidemic model is shown in Figures 6 and 7, with the corresponding numerical values shown in Table 1.

Conclusions
In this paper, we analyzed an SIR-SI-model of malaria disease analytically and numerically in the context of a five-dimensional system of the Atangana-Baleanu (c 1 , c 2 )-fractal-fractional differential equations. We introduced all parameters of the model and then derived an equivalent compact fractalfractional IVP. Then, in order, we examined some properties of the solutions of this system in detail, including the existence, Lipschitz property and uniqueness criterion. Also, stable solutions were defined and proved in the sense of Hyers-Ulam and Hyers-Ulam-Rassias. The Newton polynomials were applied for the first time to derive numerical solutions to the given system in the context of the fractal-fractional version of the malaria disease. Using the simulations, we have studied the role and impact of the fractal dimension c 2 and the fractional order c 1 on the behavior of the system. The effects of some parameters and fractal-fractional orders on the vaccination rates, antimalaria drugs and spraying were analyzed in all graphs. Therefore, if we can consider these processes, then the rate of disease outbreaks will be largely controlled. In subsequent studies, we can implement other simulations with the help of different newly defined numerical methods and compare the results together.