# Fractional-Order Ebola-Malaria Coinfection Model with a Focus on Detection and Treatment Rate

Authors: Lingling Zhang; Emmanuel Addai; Joseph Ackora-Prah; Yarhands Dissou Arthur; Joshua Kiddy K. Asamoah
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

Coinfection of Ebola virus and malaria is widespread, particularly in impoverished areas where malaria is already ubiquitous. Epidemics of Ebola virus disease arise on a sporadic basis in African nations with a high malaria burden. An observational study discovered that patients in Sierra Leone’s Ebola treatment centers were routinely infected with malaria parasites, increasing the risk of death. In this paper, we study Ebola-malaria coinfections under the generalized Mittag-Leffler kernel fractional derivative. The Banach fixed point theorem and the Krasnoselskii type are used to analyse the model’s existence and uniqueness. We discuss the model stability using the Hyers-Ulam functional analysis. The numerical scheme for the Ebola-malaria coinfections using Lagrange interpolation is presented. The numerical trajectories show that the prevalence of Ebola-malaria coinfections ranged from low to moderate depending on memory. This means that controlling the disease requires adequate knowledge of the past history of the dynamics of both malaria and Ebola. The graphical dynamics of the detection rate indicate that a variation in the detection rate only affects the following compartments: individuals that are latently infected with the Ebola, Ebola virus afflicted people who went unnoticed, individuals who have been infected with the Ebola virus and have been diagnosed with the disease, and persons undergoing Ebola virus therapy.


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## Body

## 1. Introduction

Malaria is a dangerous and occasionally deadly disease that can cause altered body posture, irregular eye movements, paralysis of eye movements, and coma. The World Health Organization estimates that millions of people worldwide have contracted malaria and thousands have died as a result of it, the majority of whom are youngsters in Africa. Commuters returning from places of the world where malaria transmission occurs, such as sub-Saharan Africa, make up the great majority of cases. Malaria is a potentially fatal disease, yet it is frequently preventable. According to estimates, malaria costs sub-Saharan Africa billion of dollars every year [1, 2]. Ebola virus disease outbreaks occur on a rare basis in African countries where malaria is already a major problem. The majority of Ebola virus disease outbreaks have been minor in the past, with case counts typically under 100 people [3].Epidemiological modeling of infectious diseases using integer-order differential equations to explore and investigate epidemic transmission dynamics has been in existence for many years. The advancement of fractional calculus has revealed important information about disease transmission patterns or dynamical behaviors. In the study of biological and engineering systems, fractional order differential equations have proved themselves as powerful and effective mathematical modeling tools. This is because most often differential operators that are found in these equations or models are associated with memory dynamics, which can be seen in biological and engineering systems [4]. The Mittag-Leffler kernel derivative has recently been utilised to mimic a variety of real-world occurrences, for example [5, 6], using the three fractional derivatives, the authors of [7] analysed the dynamics of the Q fever epidemic. From their research, they deduced that, unlike the integer order, the trajectories of some fractional orders converge to the same endemic equilibrium point. In conclusion, it was found that the Atangana-Baleanu fractional differential operator captures more susceptibilities while allowing for a smaller number of infections. Existence-uniqueness, stability, and simulated solutions to the HIV/AIDS infection model were analysed using the Mittag-Leffler kernel by the authors of [8]. Okyere et al. [9] studied an SIR model using the Caputo derivative. Using the same operator, the work in [10] studied the dynamics of COVID-19 and presented the usefulness of memory in the transmission of COVID-19. Erturk et al. [11] presented a study to describe motion of beam on nanowire. As the order of the fraction increases toward unity, their findings show that the fractional responses become increasingly similar to the classical ones. The fractional Euler-Lagrange equation also provides a flexible model with more information than the classical description, which allows for a much more accurate assessment of the system’s hidden features. Jajarmi et al. [12], applied fractional-order to study capacitor microphone. Results show that, in contrast to the previous mathematical formalism, the freedom to choose the kernel allows for the discovery of new properties of the capacitor microphone under investigation. Baleanu et al. [13] studied the relative importance of memory on cholera outbreak. The work in [14] presented some applications of a regularized Ψ-Hilfer fractional derivative.The 2014 Ebola virus epidemic in three sub-Saharan African countries, namely Guinea, Liberia, and Sierra Leone, was considered to be significant, with approximately 28,616 suspected and confirmed cases and over 11,310 deaths in these three majorally affected countries in sub-Saharan Africa. To examine the spread of Ebola virus disease transmission in Sub-Saharan African countries, Berge et al. [15] developed a vulnerable infected-recovered-death model, with natural mortality in susceptible-infected-recovered (SIR) compartments, it was assumed that recovered individuals lost immunity and became susceptible again. Chowell and Nishiura [16] studied the transmission dynamics and control of Ebola virus disease. Omeloye and Adewale [17] presented a mathematical analysis on Ebola-malaria transmission dynamics, demonstrating that if the detection rate of infected undiscovered persons is high enough, isolation can lead to Ebola eradication in the population. Furthermore, Omeloye and Adewale [18] created an optimal control in the Ebola-malaria coinfection model. They studied the disease-free equilibrium of each model. Their co-infections were shown to be locally and globally asymptotically stable whenever the basic reproduction number is less than unity or endemic otherwise. Thus, prior mathematical investigation on Ebola-malaria coinfections has not taken into account the fractional derivative. As a result, our research add up to the dynamic analysis of Ebola, malaria, and Ebola-malaria coinfections. First and foremost, we guaranteed solutions of the existence and uniqueness by the use of the Krasnoselskii type and Banach fixed point theorem. And also, Hyers-Ulam stability guaranteed the model stability. Motivated by the work in [18] the current work contributes the following:
(i)
A new fractional mathematical model for the co-dynamics of Ebola and malaria is considered and studied using the Atangana-Baleanu derivative [19](ii)
The existence and uniqueness of the solution of the proposed model employing the Banach fixed point theorem and the Krasnoselskii type are shown(iii)
Using the generalized Mittag-Leffler kernel, we exhibited the rich dynamics of this disease when memory of past history of the disease is taken into consideration through simulations(iv)
We highlight the impact of detection rate and treatment rate on the dynamics of coinfection of Ebola and malaria when the fractional order is 0.99, unlike the integer order of 1The remainder of this paper is organized as follows: some critical concepts, basic definitions, and preliminary results are all briefly introduced in Section2. In section 3 we restate the model formulation of the Ebola-malaria coinfection model and briefly describe all the parameters as in [18], and then impose the Mittag-Leffler kernel fractional derivative on the model. Section 4 is devoted to the mathematical analysis of the existence-uniqueness of Ebola-malaria coinfection model. The stability results of the Ebola-malaria coinfections model are presented and discussed in Section 5. The numerical scheme and simulations are discussed in Section 6 and Section 7, respectively. The paper ends with a conclusion in Section 8.


## 2. Preliminaries

Now, we recall some critical ideas, lemmas, and definitions to study the system (11).Definition 2.1 (see [20, 21]).
The ABC-fractional differential operator onΘ∈H1a,b, for ω∈0,1 is
(1)DABCσωΘσ=∇ω1−ω∫0σΘ′sEω−ωσ−sω1−ωds,where ∇ω is the normalization constant that satisfies the property ∇1=∇0=1. And Eω is the Mittag-Leffler function, which can be defined as
(2)Eωy=∑k=0∞ykΓωk+1.Definition 2.2 (see [8]).
ForΘ∈H1a,b and for ω∈0,1, the ABC-fractional integral is given by;
(3)IABCσωΘσ=1−ω∇ωΘσ+ω∇ωΓω∫0σΘsσ−sω−1ds,assuming that the integral on the right converges.Lemma 2.1 (see [4]).
From the ABC-fractional derivative and its integral of the functionΘ, hold for the Newton-Leibniz formula:
(4)IABCσωDABCσωΘσ=Θσ−Θω.Lemma 2.2 (see [8]).
Suppose thatyσ∈Lp0,η, then the solution of fractional differential equation.
(5)DABCσωΘσ=yσ,σ∈0,η,Θ0=Θ0,is given by
(6)Θσ=Θ0+1−ω∇ωΘσ+ω∇ωΓω∫0σΘsσ−sω−1ds.
Now, we letB=C0,1 be a Banach space with the following norm
(7)Θ=maxσ∈0,ηΘ,∀Θ∈B.Lemma 2.3 (see [22]).
From the Krassnoselskii’s fixed point theorem if we assume thatM⊂B, be a closed convex non-empty subset of B and ∃ and two operators Ω1 and Ω2, then we will have the following:
(i)
Ω1Θ+Ω2Θ∈B,∀Θ∈B(ii)
Ω1 is contraction and Ω2 is continuous and compact. Then there exist at least one solution Θ∈B such that(8)Ω1Θ+Ω2Θ=Θ.


## 3. Model Formulation

In this section, we formulate and explain the entire epidemiological compartments related to the human population and vector population at timet. The susceptible individuals is denoted by SHt, LEt is individuals that are latently infected with the Ebola virus, IUt is Ebola virus afflicted people who went unnoticed, IDt is indicated as individuals who have been infected with the Ebola virus and have been diagnosed with the disease, ITt is persons undergoing Ebola virus therapy, Jt denotes isolated Ebola individuals, malaria-exposed population is denoted as EMt, IMt denotes malaria infected individuals, RMt represents people who have recovered from malaria, EEMt represents individuals who are infected with the Ebola virus and at the risk of contracting malaria, and IEMt denotes persons infected with Ebola and Malaria. The vector population is landmarked as follows: SVt represents susceptible to mosquitoes, EVt denotes exposed to mosquitoes, and IVt denotes infected with mosquitoes. NHt is the total human population and NV is the total vector population. Considering the interrelationship with the compartments as referenced in [18] the following nonlinear ordinary differential equations represents the model formulation:
(9)dSHdt=πH−λESH−λMSH−λEMSH−μSH+ϕ1RM+αθJ,dLEdt=ε1λESH−KE+σ1+μLE+ϕ2IT+1−αθJ−1−ρϕ3IEM,dIUdt=1−ε1λESH+ω1KELE−γUE+μ+δUEIU,dIDdt=1−ω1KELE−τ1+μ+δED+σ2ID+γUEIU+τ4EEM,dITdt=τ1ID−ϕ2+μIT,dJdt=σ1LE+σ2ID−μ+δjJ−θJ,dEMdt=ε2λMSH−KM+μEM−τ2EM+ρϕ3IEM,dIMdt=1−ε2λMSH+KMEM−τ3+r+δIM+μIM,dRMdt=τ2EM+τ3IM+rIM−ϕ1+μRM,dEEMdt=ε3λEMSH+KEM+δIEM+μEEM−τ4EEM,dIEMdt=1−ε3λEMSH+KEMEEM−ϕ3+δIEM+μIEM,dSVdt=πV−λVSV−μVSV,dEVdt=λVSV−σV+μVEV,dIVdt=σVEV−μVIV,where λE, λEM, λM, and λV is defined as follows: λE=βEIU+ηDID+ηjJ/NH, λEM=βEMEEM+ηEMIEM/NH,λV=βVbIM+η1EEM+η2IEM/NH, and λM=βMbIV/NV.The total population is given as;(10)NHt=SHt+LEt+IUt+IDt+ITt+Jt+EMt+IMt+RMt+EEMt+IEMt,and NVt=SVt+EVt+IVt.The associated parameters considered in model (9) along with detailed descriptions are given as πH and πV are the recruitment rate of human and vectors, respectively, λM is the force of infection for malaria transmission, λE is the force of infection for the Ebola virus, λEM is the force of infection in IEM, μ is the human death rate, μV is the vector (mosquitoes) death rate, τ1 is the treatment rate for Ebola, τ2 is the malaria infected rate, τ3 denotes malaria treatment rate, τ4 is the exposed rate, and ε1 and ε2 are the Ebola and malaria low immunity rate, respectively. ε3 is the Ebola-malaria low immunity rate, γUE is the detection rate of unknown Ebola virus, δEM is the malaria induced death rate for EM, δIM is the malaria induced death rate for IM, σ1 and σ2 are the isolation rate for LH and ID, respectively. KE, KM, and KEM are the progression rate for malaria, Ebola, and Ebola-malaria, respectively, δUE and δDE are the Ebola induced death rate for IU.and ID, respectively, δj, δEM, and δIEM are the Ebola induced death rate for J, EEM, and IEM, respectively, σV is the progression rate vectors, and ϕ1 is the rate of loss of immunity. βE and βEM are the effective contact rate for Ebola virus and Ebola-malaria, r is the recovery rate of malaria, λM and λV are the force of infection from vector-human and human-mosquito, respectively, ϕ3 is the active rate of Ebola-malaria after treatment, βM is the transmission rate from mosquito to human, βV is the transmission rate from human to mosquito, ϕ2 is the progression rate from IT to the latent stage, b is the number of vector bites per unit time, ω1 is the rate at which latent infected moves to Ebola undetected class, ρ is the rate at which treated Ebola-malaria individuals move to EM, ηD is the modification parameter of ID in relation to LE. ηT is the modification parameter of IT, ηJ is the modification parameter of J, η1 and η2 are the modification parameters of EEM and IEM, respectively, ηEM is the modification parameter of IEM, and θ is the rate at which J individuals are discharged from the treatment centers.


### 3.1. Fractional Model

To capture the memory in the predictions of the Ebola-malaria coinfection model and also to check that both sides of the fractional equations have the exact dimensions, the coefficient1/α1−σ, comprised with the auxiliary parameter α [23, 24] is imposed on model (9). Hence, we suggest the following fractional-order model for the Ebola-malaria coinfection model under the ABC-fractional derivative:
(11)1α1−σABCD0+σSHσ=πH−λESH−λMSH−λEMSH−μSH+ϕ1RM+αθJ,1α1−σABCD0+σLEσ=ε1λESH−KE+σ1+μLE+ϕ2IT+1−αθJ−1−ρϕ3IEM,1α1−σABCD0+σIUσ=1−ε1λESH+ω1KELE−γUE+μ+δUEIU,1α1−σABCD0+σIDσ=1−ω1KELE−τ1+μ+δED+σ2ID+γUEIU+τ4EEM,1α1−σABCD0+σITσ=τ1ID−ϕ2+μIT,1α1−σABCD0+σJσ=σ1LE+σ2ID−μ+δjJ−θJ,1α1−σABCD0+σEMσ=ε2λMSH−KM+μEM−τ2EM+ρϕ3IEM,1α1−σABCD0+σIMσ=1−ε2λMSH+KMEM−τ3+r+δIM+μIM,1α1−σABCD0+σRMσ=τ2EM+τ3IM+rIM−ϕ1+μRM,1α1−σABCD0+σEEMσ=ε3λEMSH+KEM+δIEM+μEEM−τ4EEM,1α1−σABCD0+σIEMσ=1−ε3λEMSH+KEMEEM−ϕ3+δIEM+μIEM,1α1−σABCD0+σSVσ=πV−λVSV−μVSV,1α1−σABCD0+σEVσ=λVSV−σV+μVEV,1α1−σABCD0+σIVσ=σVEV−μVIV,where 0<σ≤1, with the following initial conditions: SH0=SH0≥0,LE0=LE0≥0,IU0=IU0≥0,ID0=ID0≥0,IT0=IT0≥0,J0=J0≥0,EM0=EM0≥0,RM0=RM0≥0,EEM0=EEM0≥0,IEM0=IEM0≥0SV0=SV0≥0,EV0=EV0≥0,IV0=IV0≥0,andIM0=IM0≥0.


## 3.1. Fractional Model

To capture the memory in the predictions of the Ebola-malaria coinfection model and also to check that both sides of the fractional equations have the exact dimensions, the coefficient1/α1−σ, comprised with the auxiliary parameter α [23, 24] is imposed on model (9). Hence, we suggest the following fractional-order model for the Ebola-malaria coinfection model under the ABC-fractional derivative:
(11)1α1−σABCD0+σSHσ=πH−λESH−λMSH−λEMSH−μSH+ϕ1RM+αθJ,1α1−σABCD0+σLEσ=ε1λESH−KE+σ1+μLE+ϕ2IT+1−αθJ−1−ρϕ3IEM,1α1−σABCD0+σIUσ=1−ε1λESH+ω1KELE−γUE+μ+δUEIU,1α1−σABCD0+σIDσ=1−ω1KELE−τ1+μ+δED+σ2ID+γUEIU+τ4EEM,1α1−σABCD0+σITσ=τ1ID−ϕ2+μIT,1α1−σABCD0+σJσ=σ1LE+σ2ID−μ+δjJ−θJ,1α1−σABCD0+σEMσ=ε2λMSH−KM+μEM−τ2EM+ρϕ3IEM,1α1−σABCD0+σIMσ=1−ε2λMSH+KMEM−τ3+r+δIM+μIM,1α1−σABCD0+σRMσ=τ2EM+τ3IM+rIM−ϕ1+μRM,1α1−σABCD0+σEEMσ=ε3λEMSH+KEM+δIEM+μEEM−τ4EEM,1α1−σABCD0+σIEMσ=1−ε3λEMSH+KEMEEM−ϕ3+δIEM+μIEM,1α1−σABCD0+σSVσ=πV−λVSV−μVSV,1α1−σABCD0+σEVσ=λVSV−σV+μVEV,1α1−σABCD0+σIVσ=σVEV−μVIV,where 0<σ≤1, with the following initial conditions: SH0=SH0≥0,LE0=LE0≥0,IU0=IU0≥0,ID0=ID0≥0,IT0=IT0≥0,J0=J0≥0,EM0=EM0≥0,RM0=RM0≥0,EEM0=EEM0≥0,IEM0=IEM0≥0SV0=SV0≥0,EV0=EV0≥0,IV0=IV0≥0,andIM0=IM0≥0.


## 4. Existence and Uniqueness

It is important to determine whether or not such a dynamical problem exists before delving into any type of epidemiological simulations. Fortunately, the fixed point theory provides an ironclad guarantee for this evaluation’s outcome. We attempt to apply the same idea in a perspective of the Banach and Krassnoselskii’s fixed point theory to the stated model (11) to study existence and uniqueness results. In relation to the aforementioned requirement, we reformulate the considered model (11) as follows:
(12)DABC0+σSHσ=ℵ1Δ∗∗,DABC0+σLEσ=ℵ2Δ∗∗,DABC0+σIUσ=ℵ3Δ∗∗,DABC0+σIDσ=ℵ4Δ∗∗,DABC0+σITσ=ℵ5Δ∗∗,DABC0+σJσ=ℵ6Δ∗∗,DABC0+σEMσ=ℵ7Δ∗∗,DABC0+σIMσ=ℵ8Δ∗∗,DABC0+σRMσ=ℵ9Δ∗∗,DABC0+σEEMσ=ℵ10Δ∗∗,DABC0+σIEMσ=ℵ11Δ∗∗,DABC0+σSVσ=ℵ12Δ∗∗,DABC0+σEVσ=ℵ13Δ∗∗,DABC0+σIVσ=ℵ14Δ∗∗,where Δ∗∗=σ,SH,LE,IU,ID,IT,J,EM,IM,RM,EEM,IEM,SV,EV,IV and
(13)ℵ1Δ∗∗=πH−λESH−λMSH−λEMSH−μSH+ϕ1RM+αθJ,ℵ2Δ∗∗=ε1λESH−KE+σ1+μLE+ϕ2IT+1−αθJ−1−ρϕ3IEM,ℵ3Δ∗∗=1−ε1λESH+ω1KELE−γUE+μ+δUEIU,ℵ4Δ∗∗=1−ω1KELE−τ1+μ+δED+σ2ID+γUEIU+τ4EEM,ℵ5Δ∗∗=τ1ID−ϕ2+μIT,ℵ6Δ∗∗=σ1LE+σ2ID−μ+δjJ−θJ,ℵ7Δ∗∗=ε2λMSH−KM+μEM−τ2EM+ρϕ3IEM,ℵ8Δ∗∗=1−ε2λMSH+KMEM−τ3+r+δIM+μIM,ℵ9Δ∗∗=τ2EM+τ3IM+rIM−ϕ1+μRM,ℵ10Δ∗∗=ε3λEMSH+KEM+δIEM+μEEM−τ4EEM,ℵ11Δ∗∗=1−ε3λEMSH+KEMEEM−ϕ3+δIEM+μIEM,ℵ12Δ∗∗=πV−λVSV−μVSV,ℵ13Δ∗∗=λVSV−σV+μVEV,ℵ14Δ∗∗=σVEV−μVIV.For simplicity we write the model (11) in the form;
(14)DABCσωWσ=Ψσ,Wσ,σ∈0,η,W0=W0,where
(15)W=SH,LE,IU,ID,IT,J,EM,IM,RM,EEM,IEM,SV,EV,IVT,W0=T1,T2T,Ψσ,Wσ=ℵiΔ∗∗T,i=1,2,3,⋯,14,where .T presents the transpose of the vectors, T1=SH0,LE0,IU0,ID0,IT0,J0,EM0, and T2=IM0,RM0,EEM0,IEM0,SV0,EV0,IV0T. From Lemma 2.2, the system (14) is equal to the following fractional integral equation;
(16)Wσ=W0+1−ω∇ωΨσ,Wσ+ω∇ωΓω∫0σΨs,Wsσ−sω−1ds.Let us sayB=C0,η is the Banach space, supposing that the following assumptions hold;(F1) There exist a nonnegative constant Y, Z, and a∈0,1 such that
(17)Ψσ,Wσ≤YWa+Z.(F2) There exist a nonnegative constant Lμ>0 for all W,W∗∈B then
(18)Ψσ,Wσ−Ψσ,W∗σ≤LμW−W∗.Also, let us define operatorAp:B⟶B such that
(19)ApWσ=Ω1Wσ+Ω2Wσ,basically, we let
(20)Ω1Wσ=W0+1−ω∇ωΨσ,Wσ,Ω2Wσ=ω∇ωΓω∫0σΨs,Wsσ−sω−1ds.From this knowledge, equation (16) can be written as;
(21)ApWσ=W0+1−ω∇ωΨσ,Wσ+ω∇ωΓω∫0σΨs,Wsσ−sω−1ds.Theorem 4.1.
Suppose that (F1) and (F2) hold, such that, 1−ω/∇ωLμ<1, then the Ebola-malaria co-infection model (14) has at least one solution.Proof.
We divide the proof into two steps:
Step 1. We prove that operatorΩ1 is contraction. Then, let W∗∈Π, where Π=W∈B:W≤ρ,ρ>0 is a close convex set, thus
(22)Ω1W−Ω1W∗=1−ω∇ωmaxω∈0,ηΨσ,Wσ−Ψσ,W∗σ,≤1−ω∇ωLμW−W∗.
HenceΩ1 is contraction.
Step 2. We prove thatΩ2 is compact and also continuous, for all W∈Π, then Ω2 will be continuous as W is continuous; thus,
(23)Ω2W=maxω∈0,ηω∇ωΓω∫0σΨs,Wsσ−sω−1ds,≤ω∇ωΓω∫0ηη−sω−1Ψs,Wsds.≤ηω∇ωΓωYWa+Z.
HenceΩ2 is boundedness. For equicontinuous, let σ1andσ2∈0,η, such that
(24)Ω2Wσ1−Ω2Wσ2=ω∇ωΓω∫0σ1Ψs,Wsσ1−sω−1ds−∫0σ2Ψs,Wsσ2−sω−1ds≤YWa+Z∇ωΓωσ1ω−σ2ω.
Asσ1⟶σ2, then Ω2Wσ1−Ω2Wσ2⟶0 which make operator Ω2 an equicontinuous and compact by the Arzela-Ascoli theorem. Therefore the existence for the Ebola-malaria co-infection model (11) is proved.Theorem 4.2.
Suppose that∃ a nonnegative integer Λ>0 such that
(25)Λ=1−ω∇ωLμ+ηω∇ωΓωLμ<1,then operator Ap has a unique fixed point.Proof.
LetW,W∗∈B, then we say
(26)ApW−ApW∗≤Ω1W−Ω1W∗+Ω2W−Ω2W∗,≤1−ω∇ωmaxω∈0,ηΨσ,Wσ−Ψσ,W∗σ+ω∇ωΓωmaxω∈0,η∫0σΨs,Wsσ−sω−1ds−∫0σΨs,W∗sσ−sω−1ds,≤1−ω∇ωLμ+ηω∇ωΓωLμW−W∗,=ΛW−W∗.
Hence, by the Banach contraction principle,Ap has a unique fixed point. Consequently, the Ebola-malaria co-infection model (11) has unique solution.


## 5. Hyers-Ulam Stability

In the context of differential equations, stability is crucial. The Hyers-Ulam (HU) type of stability has emerged as one of the most intriguing types of stability in recent years. Here, we use HU type stability to examine a theoretical model of Ebola and malaria transmission.Definition 5.1.
The Ebola-malaria coinfection model (11) is HU stable if for δ>0 and letting W∈B be any solution of below inequality
(27)DABCσωWσ−Ψσ,Wσ≤δ,∀σ∈0,η;and with a unique solution W∗ of problem (14) with a positive constant λq>0, such that,
(28)W−W∗≤λqδ,∀σ∈0,η.Definition 5.2.
Given a functionϕ∈C0,η,R, such that ϕ0=0 for any solution W of (27) and W∗ be a unique solution of (14), then
(29)W−W∗≤ϕδ,then the Ebola-malaria co-infection model (14) is generalized HU stable.Remark 5.1.
Supposeχσ∈C0,η,R, we say W∈B satisfies inequality (27) suppose that,
(i)
χσ≤δ, for all σ∈0,η(ii)
DABCσωWσ=Ψσ,Wσ+χσ, ∀σ∈0,η.Now, we consider the resulting perturbation equation of system (14) as follows;
(30)DABCσωWσ=Ψσ,Wσ+χσ,W0=W0.The below Lemma is needed to help us get our results.Lemma 5.1.
From equation (30), we say the following result hold. Thus,
(31)Wσ−ApΨσ,Wσ≤1−ω∇ω+ηω∇ωΓωδ.Proof.
Consider Lemma2.2 relatively, solution for equation (14) is given as;
(32)Wσ=W0+IABCσωΨσ,Wσ+IABCσωχσ.
Now, with the help of equation (21), we deduce that
(33)Wσ−ApΨσ,Wσ≤1−ω∇ωχσ+ηω∇ωΓω∫0σσ−s1−ωχσds≤1−ω∇ω+ηω∇ωΓωδ.Theorem 5.2.
Suppose that the Ebola-malaria co-infection model (14) is Ulam-Hyers stable, if there exist Λ=1−ω/∇ωLμ+ηω/∇ωΓωLμ<1.Proof.
With the help from the Lemma5.1, let W∈B be any solution and W∗∈B be a unique solution for considered problem (14), then
(34)Wσ−W∗σ=Wσ−ApW∗σ≤Wσ−ApWσ+ApWσ−ApW∗σ≤1−ω∇ω+ηω∇ωΓωδ+1−ω∇ωLμ+ηω∇ωΓωLμW−W∗.
Thus,(35)Wσ−W∗σ≤1−ω/∇ω+ηω/∇ωΓω1−1−ω/∇ωLμ+ηω/∇ωΓωLμδ.
Hence, we conclude that, the Ebola-malaria co-infection model (14) is HU stable. Consequently, the Ebola-malaria co-infection model (14) is HU generalized stable.Definition 5.3.
The problem (14) is Hyers-Ulam-Rassias (HUR) stable given that the function ξσ∈C0,1,R, δ>0 and letting W∈B be any solution of the below inequality
(36)DABCωσWσ−Ψσ,Wσ≤ξσδ,∀σ∈0,η;and also ∃ unique solution W∗ of problem (14) with a positive constant λq>0 then,
(37)W−W∗≤λqξσδ,∀σ∈0,η.Definition 5.4.
Given a functionν∈C0,η,R, with λq,ν and δ>0, for all W of equation (36) and W∗ be a unique solution of (14), then
(38)W−W∗≤λq,ννσ,∀σ∈0,η,then system (14) is HUR generalized stable.Remark 5.2.
Supposeμσ∈C0,1,R, we say W∈B satisfies inequality (36), suppose that,
(i)
μσ≤δνσ,∀σ∈0,η.(ii)
DωABCωσWσ=Ψσ,Wσ+μσ, ∀σ∈0,η.Now, we consider the resulting perturbation equation of system (14) as follows:
(39)DABCσωWσ=Ψσ,Wσ+μσ,W0=W0.Lemma 5.3.
From equation (39), we say the following result hold. Thus,
(40)Wσ−ApΨσ,Wσ≤1−ω∇ω+ηω∇ωΓωμσδ.Proof.
Consider Lemma2.2 relatively, solution for equation (39) is given as;
(41)Wσ=W0+IABCσωΨσ,Wσ+IABCσωμσ.
Now, with the help of (21), we deduce that
(42)Wσ−ApΨσ,Wσ≤1−ω∇ωνσ+ηω∇ωΓω∫0σσ−s1−ωμσds≤1−ω∇ω+ηω∇ωΓωμσδ.Theorem 5.4.
Suppose that the Ebola-malaria co-infection (11) is HUR stable if ∃(43)Λ=1−ω∇ωLμ+ηω∇ωΓωLμ<1.Proof.
With the help from the Lemma5.3, let W∈B be any solution and W∗∈B be a unique solution for considered problem (14), then
(44)Wσ−W∗σ=Wσ−ApW∗σ≤Wσ−ApWσ+ApWσ−ApW∗σ≤1−ω∇ω+ηω∇ωΓωνσδ+1−ω∇ωLμ+ηω∇ωΓωLμW−W∗.
Thus,(45)Wσ−W∗σ≤1−ω/∇ω+ηω/∇ωΓω1−1−ω/∇ωLμ+ηω/∇ωΓωLμνσδ.
Hence, we conclude that, the Ebola-malaria coinfection (11) is HUR stable. Consequently, the Ebola-malaria coinfection model (11) is generalized HUR stable.


## 6. Numerical Scheme

Here we provide the numerical schemes for the two-step Lagrange interpolation used in our considered ABC-fractional operator model of the Ebola-malaria coinfection. By using the initial condition and the operatorIABC0ω, we transform the Ebola-malaria co-infection (14) into a system of fractional integral equations, as shown below.
(46)SHσ−SH0=IABC0ωℵ1σ,SHσ,LEσ−LE0=IABC0ωℵ2σ,LEσ,IUσ−IU0=IABC0ωℵ3σ,IUσ,IDσ−ID0=IABC0ωℵ4σ,IDσ,ITσ−IT0=IABC0ωℵ5σ,ITσ,Jσ−J0=IABC0ωℵ6σ,Jσ,EMσ−EM0=IABC0ωℵ7σ,EMσ,IMσ−IM0=IABC0ωℵ8σ,IMσ,RMσ−RM0=IABC0ωℵ9σ,RMσ,EEMσ−EEM0=IABC0ωℵ10σ,EEMσ,IEMσ−IEM0=IABC0ωℵ11σ,IEMσ,SVσ−SV0=IABC0ωℵ12σ,SVσ,EVσ−EV0=IABC0ωℵ13σ,EVσ,IVσ−IV0=IABC0ωℵ14σ,IVσ,which we can easily get the following:
(47)SHσ=SH0+1−ω∇ωℵ1σ,SHσ+ω∇ωΓω∫0σℵ1s,SHsσ−sω−1ds,LEσ=LE0+1−ω∇ωℵ2σ,LEσ+ω∇ωΓω∫0σℵ2s,LEsσ−sω−1ds,IUσ=IU0+1−ω∇ωℵ3σ,IUσ+ω∇ωΓω∫0σℵ3s,IUsσ−sω−1ds,IDσ=ID0+1−ω∇ωℵ4σ,IDσ+ω∇ωΓω∫0σℵ4s,IDsσ−sω−1ds,ITσ=IT0+1−ω∇ωℵ5σ,ITσ+ω∇ωΓω∫0σℵ5s,ITsσ−sω−1ds,Jσ=J0+1−ω∇ωℵ6σ,Jσ+ω∇ωΓω∫0σℵ6s,Jsσ−sω−1ds,EMσ=EM0+1−ω∇ωℵ7σ,EMσ+ω∇ωΓω∫0σℵ7s,EMsσ−sω−1ds,IMσ=IM0+1−ω∇ωℵ8σ,IMσ+ω∇ωΓω∫0σℵ8s,IMsσ−sω−1ds,RMσ=RM0+1−ω∇ωℵ9σ,RMσ+ω∇ωΓω∫0σℵ9s,RMsσ−sω−1ds,EEMσ=EEM0+1−ω∇ωℵ10σ,EEMσ+ω∇ωΓω∫0σℵ11s,EEMsσ−sω−1ds,IEMσ=IEM0+1−ω∇ωℵ11σ,IEMσ+ω∇ωΓω∫0σℵ11s,IEMsσ−sω−1ds,SVσ=SV0+1−ω∇ωℵ12σ,SVσ+ω∇ωΓω∫0σℵ12s,SVsσ−sω−1ds,EVσ=EV0+1−ω∇ωℵ13σ,EVσ+ω∇ωΓω∫0σℵ13s,EVsσ−sω−1ds,IVσ=IV0+1−ω∇ωℵ14σ,IVσ+ω∇ωΓω∫0σℵ14s,IVsσ−sω−1ds.Consider the ABC derivative under the Cauchy problem, and the ABC integral of Lemma2.2 can be replicated using the fundamental theory of calculus.
(48)Θσ=Θ0+1−ω∇ωℵσ,Θσ+ω∇ωΓω∫0σℵσ,Θσσ−sω−1ds.Taking the pointσz∗+1=z∗+1h and σz∗=z∗h,z∗=0,1,2,⋯, with h being the time step, we can simply deduce
(49)Θσz∗+1=Θ0+1−ω∇ωℵσ,Θσ+ω∇ωΓω∫0σℵs,Θsσ−sω−1ds,=Θ0+1−ω∇ωℵσz∗,Θσz∗+ω∇ωΓω∫0σz∗+1ℵθ,Θθσz∗+1−θω−1dθ,=Θ0+1−ω∇ωℵσz∗,Θσz∗+ω∇ωΓω∑r∗=0z∗∫σr∗σr∗+1ℵθ,Θθσz∗+1−θω−1dθ.Having the interval ofσz∗,σz∗+1, the two term Lagrange polynomial is given as follows:
(50)γr∗θ=θ−σr∗−1σr∗−σr∗−1ℵσr∗,Θσr∗−θ−σr∗σr∗−σr∗−1ℵσr∗−1,Θσr∗−1,=ℵσr∗,Θσr∗hθ−σr∗−1−ℵσr∗−1,Θσr∗−1hθ−σr∗,≃ℵσr∗,Θr∗hθ−σr∗−1−ℵσr∗−1,Θr∗−1hθ−σr∗.Taking the approximation solution of (50) into (49);
(51)Θσz∗+1=Θ0+1−ω∇ωℵσz∗,Θσz∗+ω∇ωΓω×∑r∗=0z∗ℵσr∗,Θr∗h∫σr∗σr∗+1θ−σr∗−1σz∗+1−σω−1dθ−ℵσr∗−1,Θr∗−1h∫σr∗σr∗+1θ−σr∗σz∗+1−σω−1dθ.Solving the integral equations in the (51), let us take:
(52)Yω,r∗,1=∫σr∗σr∗+1θ−σr∗−1σz∗+1−σω−1dθ,Yω,r∗,2=∫σr∗σr∗+1θ−σr∗σz∗+1−σω−1dθ.Now, we can deduce from (52) as follows by applying integration by substitution:
(53)Yω,r∗,1=hω+1z∗+1−r∗σz∗−r∗+2+σ−z∗−r∗σz∗−r∗+2+2σωω+1,Yω,r∗,2=hω+1n+1−r∗σ+1−z∗−r∗σz∗−r∗+1+σωω+1.Here, knowingYω,r∗,1 and Yω,r∗,2, we simply substituted into (51) which then gives us the following numerical scheme:
(54)Θσz∗+1=Θ0+1−ω∇ωℵσz∗,Θσz∗+ω∇ω×∑r∗=0z∗hωℵσr∗,Θr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵσr∗−1,Θr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω.Hence, we suggest the following fractional-order model for the Ebola-malaria coinfection model under the ABC-fractional derivative:(55)SHz∗+1=SH0+1−ω∇ωℵ1σz∗,SHσz∗+ω∇ω×∑r∗=0z∗hωℵ1σr∗,SHr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ1σr∗−1,SHr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,LEz∗+1=LE0+1−ω∇ωℵ2σz∗,LEσz∗+ω∇ω×∑r∗=0z∗hωℵ2σr∗,LEr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ2σr∗−1,LEr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,IUz∗+1=IU0+1−ω∇ωℵ3σz∗,IUσz∗+ω∇ω×∑r∗=0z∗hωℵ3σr∗,IUr∗Γω+2z∗+1−r∗ωz∗−k+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ3σr∗−1,IUr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,IDz∗+1=ID0+1−ω∇ωℵ4σz∗,IDσz∗+ω∇ω×∑r∗=0z∗hωℵ4σr∗,IDr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ4σr∗−1,IDr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,ITz∗+1=IT0+1−ω∇ωℵ5σz∗,ITσz∗+ω∇ω×∑r∗=0z∗hωℵ5σr∗,ITr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ5σr∗−1,ITr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,Jz∗+1=J0+1−ω∇ωℵ6σz∗,Jσz∗+ω∇ω×∑r∗=0z∗hωℵ6σr∗,Jr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ6σr∗−1,Jr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,EMz∗+1=EM0+1−ω∇ωℵ7σz∗,EMσz∗+ω∇ω×∑r∗=0z∗hωℵ7σr∗,EMr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ7σr∗−1,EMr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,IMz∗+1=IM0+1−ω∇ωℵ8σz∗,IMσz∗+ω∇ω×∑r∗=0z∗hωℵ8σr∗,IMr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ8σr∗−1,IMr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,RMz∗+1=RM0+1−ω∇ωℵ9σz∗,RMσz∗+ω∇ω×∑r∗=0z∗hωℵ9σr∗,RMr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ9σr∗−1,RMr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,EEMz∗+1=EEM0+1−ω∇ωℵ10σz∗,EEMσz∗+ω∇ω×∑r∗=0z∗hωℵ10σr∗,EEMr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ10σr∗−1,EEMr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,IEMz∗+1=IEM0+1−ω∇ωℵ11σz∗,IEMσz∗+ω∇ω×∑r∗=0z∗hωℵ11σr∗,IEMr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ11σr∗−1,IEMr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,SVz∗+1=SV0+1−ω∇ωℵ12σz∗,SVσz∗+ω∇ω×∑r∗=0z∗hωℵ12σr∗,SVr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ12σr∗−1,SVr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω,EVz∗+1=EV0+1−ω∇ωℵ13σz∗,EVσn+ω∇ω×∑r∗=0z∗hωℵ13σr∗,EVr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ13σr∗−1,EVr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ωIVz∗+1=IV0+1−ω∇ωℵ14σz∗,IVσz∗+ω∇ω×∑r∗=0z∗hωℵ14σr∗,IVr∗Γω+2z∗+1−r∗ωz∗−r∗+2+ω−z∗−r∗ωz∗−r∗+2+2ω−ω∇ω∑r∗=0z∗hωℵ14σr∗−1,IVr∗−1Γω+2z∗+1−r∗ω+1−z∗−r∗ωz∗−r∗+1+ω.


## 7. Numerical Results and Discussion

We illustrate the analytical results of this study by carrying out numerical simulations of the models using parameter values in Table1 with initial values SH0=1000, LE0=270, IU0=210, ID0=300, ID0=300, IT0=320,J0=200, EM0=150, IM0=800, RM0=500, EEM0=800, IEM0=200,SV0=650, EV0=420, and IV0=350. For the given sets of parameters in Table 1, we show the approximate solutions obtained using the considered iterative approaches against different fractional orders for each compartment, as seen in Figures 1–3. The illustrative graphs in Figures 1(a), 1(c), 1(d), 1(f), 2(b), 2(d), 2(e), 2(f), and 3(b) show no crossover effect but Figures 1(b), 1(e), 2(a), 2(c), and 3(a) show a crossover effect when the fractional order is changed. Figure 4 shows the fractional dynamics when one varies the malaria treatment rate with a fractional order of σ=0.90. It shows that an increasing treatment rate reduces the number of infected individuals with malaria but does not affect the number of infected individuals with Ebola. Figure 5 shows the dynamics of the disease when one increases the rate of treatment of individuals infected with Ebola. Figure 6 shows the fractional dynamics when one varies the Ebola detection rate with a fractional order of 0.9. In a nutshell, we notice that the variation in the treatment rate for malaria does not affect susceptible individuals. SH, individuals that are latently infected with the Ebola virus LEt, Ebola virus afflicted people who went unnoticed IU, individuals who have been infected with the Ebola virus and have been diagnosed with the disease ID, persons undergoing Ebola virus therapy IT, isolated Ebola individuals J, malaria-exposed population EM, individuals who are infected with the Ebola virus and at the risk of contracting malaria EEM, persons infected with Ebola and malaria IEM, and the vector population NV=SV+EV+IV. Similarly, when one varies the treatment rate for Ebola, we notice that the following compartment is not affected; thus, susceptible individuals SH, Ebola virus afflicted people who went unnoticed IU, malaria-exposed population EM, individuals who are infected with the Ebola virus and at the risk of contracting malaria EEM, individuals infected with Ebola and malaria IEM, individuals affected with malaria only IM, recovered individuals from malaria RM and the vector population NV=SV+EV+IV. The graphical dynamics of the variation in the detection rate indicate that the variation in the detection rate only affects the following compartments: individuals that are latently infected with the Ebola virus LEt, Ebola virus afflicted people who went unnoticed IU, individuals who have been infected with the Ebola virus and have been diagnosed with the disease ID, and people undergoing Ebola virus therapy IT.Table 1
Parameter values in the model.
ParametersValuesSourcesParametersValuesSourcesπH800Assumedr,γUE0.05, 0.003AssumedπV500Assumedθ0.028Assumedμ1/65×365ForecastedηEM0.08Assumedτ10.009AssumedηD0.01Assumedτ30.013Assumedρ0.0095Assumedε1,ε20.0001, 0.001Assumedσ1,σ20.001, 0.001[16]δIM0.003Assumedb0.0031AssumedKE0.0008AssumedβV0.008AssumedδUE,δDE0.0027, 0.008[17, 18]ϕ10.018AssumedσV0.1[17]ϕ30.0012[16]βE,βEM0.080, 0.080AssumedKM,KEM0.008, 0.008Assumedη1,η20.034, 0.067Assumedδj,δEM,δIEM0.008, 0.003,0.008AssumedβM0.04Assumedε30.082Assumedϕ20.028Assumedτ40.0069[18]ω10.021Assumedτ20.0018AssumedηT0.0018AssumedμV0.004AssumedηJ0.01AssumeFigure 1
Fractional dynamics of different classes at different fractional orderσ.
(a)(b)(c)(d)(e)(f)Figure 2
Fractional dynamics of different classes at different fractional orderσ.
(a)(b)(c)(d)(e)(f)Figure 3
Fractional dynamics of different classes at different fractional orderσ.
(a)(b)Figure 4
Fractional dynamics when one varies malaria treatment rate with fractional orderσ=0.90.
(a)(b)Figure 5
Fractional dynamics when one varies Ebola treatment rate with fractional orderσ=0.90.
(a)(b)(c)(d)Figure 6
Fractional dynamics when one varies Ebola detection rate with fractional orderσ=0.90.
(a)(b)(c)(d)


## 8. Conclusion

This paper considers Ebola-malaria coinfection under the Mittag-Leffler kernel fractional derivative. We have determined epidemiological, computational, and theoretical inferences to understand better how to prevent the Ebola, malaria, and Ebola-malaria coinfections simultaneously in the human population. In a prior mathematical investigation into Ebola-malaria coinfections, the fractional derivative was not taken into account. As a result, our research adds up to the dynamic analysis of Ebola, malaria, and Ebola-malaria coinfections. First and foremost, we guaranteed solutions’ existence and uniqueness by using the Krasnoselskii’s type and the Banach fixed point theorem. HU stability ensured the model’s stability. The simulation has been given with the help of the Lagrange interpolation to solve the considered problem analytically. Our results reveal that the prevalence of the Ebola, malaria and Ebola-malaria coinfections varied from low to moderate depending on the fractional operators. In addition, we observed from our solutions that there was no significant difference in the Ebola-malaria coinfections of the immune response. Moreover, Ebola-malaria coinfection-related mortality varied from moderate to high depending on the fractional operators. Hence, we conclude that the global nature of ABC-fractional order dynamics excellently explains the coinfection model characteristics. Thus, the concept in this paper has crucial implications for biological models and their problems, and it is helpful for Ebola-malaria coinfection analysis and control strategy. In future work, different fractional order derivatives and their theoretical and numerical stability can be investigated with other control measures.



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