Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R0=1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.


Introduction
COVID-19 (coronavirus disease 2019) is a significant worldwide health hazard that has affected and killed millions of people worldwide. Wuhan, China, marking the beginning of Coronavirus outbreak, in December 2019 [1,2]. The World Health Organization named the disease a pandemic, on 11 March 2020 [3]. To limit the spread of this dangerous disease, several control measures are being used, including social distance, case isolation, family quarantine, university and school shutdown [4,5]. In reaction to China's containment methods, other governments have used extensive community quarantines or lockdowns as a form of control. (see details; [6,7]). In Ethiopia, the first COVID-19 case was reported by the Ethiopian Ministry of Health and Ethiopian Public Health Institute on March 13, 2020, in Addis Ababa [8]. The prevention and spread control of COVID-19 in Ethiopia is complicated by a number of reasons. Some of the reasons are poor physical distancing and the use of face mask, late confirmation of COVID-19 cases from the dead body investigation, inadequacy of quarantine cites, religious and cultural activities, insufficient health workers in treatment centre's, social instability, a lack of public awareness of the infection, mass transportation system, attending of funeral ceremonies, and poor tradition of using electronic transaction [9]. All of these variables allow infectious and noninfectious people to come into contact.
Around the world, several mathematical models have been constructed to better understand the new virus's transmission dynamics and intervention techniques. For example, in [10][11][12] COVID-19 dynamics were examined using the SEIR model, which included the environment and social distancing. They found that in the absence of effective control measures, the basic reproduction number R 0 ¼ 2:03 indicated that the pandemic would persist in the human population. A fractional mathematical model of COVID-19 epidemics in Pakistan with treatment is discussed in [13]. The authors constructed the Atangana-Baleanu type fractional model and numerically solved it. In [14], a fresh dynamical modeling SEIR was proposed with global analysis incorporated to the real data of spreading COVID-19 in Saudi Arabia. The reproduction number and extensive stability analysis were used to present the dynamics of the suggested model. They focused on the best methods, controls, and techniques for significantly lowering the outbreak quickly (see also [15][16][17]).
The fractional-order derivative (FOD), which is described as an extension of the integer derivative to a non-integer order (arbitrary order) operator, has been used to model a variety of storage and latency phenomena during the last decade, notably pandemic behavior [37]. FOD models are a potential methods for modeling complex systems because, in addition to properly capturing the memory and delay involved, they also allow for greater flexibility in fitting the data exactly than integer-order models [38]. In this paper, we investigate how a fractional version of a model can be used to forecast the dynamics of the COVID-19 outbreak in Ethiopia.
In this paper, the COVID-19 epidemic transmission dynamics in Ethiopia is represented by entirely new fractional-order mathematical model. In Caputo sense, the model is presented as system of fractional order differential equations. The solutions stability analysis, positivity, and boundedness, as well as their existence and uniqueness, are all analyzed. The basic reproduction number is calculated using the values of model parameters and its importance in target setting to limit the spread of the pandemic is discussed using the sensitivity of basic reproductive number with respect to the key model parameters. The Legendre spectral collocation method is used to convert the fractional system to an algebraic system. Numerical results of the model are given in graphical forms.
Preliminaries and Notations. This section covers the Laplace transform (LT) notation, the Mittag-Leffler (M-L) function, the fractional derivative, and a few key Legendre polynomial characteristics.
Definition 1.1. The M-L function is characterized as [39]: The two-parameter M-L function is defined by [40] as: From [41], The LT of the function is s dÀw s d Àg ; for g > 0: for n 2 N 0 and n < d d e;

Cðnþ1Þ
Cðnþ1ÀdÞ t nÀd ; for n 2 N 0 and n P d d e; ( ð1:6Þ where N 0 ¼ 0; 1; 2; . . . f gand the ceiling function d d e denote the smallest integer greater than or equal to d.
The LT of Caputo fractional derivative of order d is provided by: k¼0 s dÀkÀ1 f ðkÞ 0 ð Þ; ðn P d > n À 1 2 NÞ: with F s ð Þ is the Laplace transform of f t ð Þ. Legendre polynomials: On the interval ½À1; 1, Legendre polynomials are defined and can be generated with the aid of the following recurrence formula [44]: with starting polynomials L 0 ðxÞ ¼ 1 and L 1 ðxÞ ¼ x. To apply these polynomials to the interval t 2 ½0; 1, we utilize the socalled shifted Legendre polynomials (SLP), which involve changing the variable x ¼ 2t À 1. Let the SLP L m ð2t À 1Þ be denoted by P m ðtÞ, then it is possible to get them as follows: with starting polynomials P 0 ðtÞ ¼ 1 and P 1 ðtÞ ¼ 2t À 1. The analytic form of SLP P m t ð Þ of degree m given by: ðÀ1Þ mþk ðm þ kÞ! ðm À kÞ!ðk!Þ 2 t k : ð1:10Þ In terms of the SLP, a function yðtÞ that is square integral in ½0; 1 may be represented as: where the coefficients v i are given by The natural death rate l and the death rate x caused by COVID-19 are the same in all subgroups. Natural death or disease-induced death will lead some individuals to be removed from the population. Individuals are recruited into susceptible classes at a constant rate # and get infected by Coronavirus infectious individuals with the force of infection b mAþLþFþI ð Þ N . The exposed cases acquire population from Coronavirus infection at a rate b mA þ L þ F þ I ð Þ S=N. Exposed population progress to symptomatically infected when individuals show symptoms of coronavirus and tested positive at a rate h and the remaining exposed individuals will be removed at the natural death rate l and enter the asymptomatic class at a rate d 1 À h ð Þ. Asymptomatic infected individuals develop symptoms and become symptomatically infected at a rate k, enter a lockdown class at a rate n 1 À k ð Þ, attending the funeral ceremony at a rate s 1 À n ð Þ 1 À k ð Þ and recovered from COVID-19 disease at a rate 1 1 À s ð Þ 1 À n ð Þ 1 À k ð Þ, while the remaining individuals died naturally or as a result of COVID-19 disease at a rate of l þ x. Lockdown individuals participate in funerals during the pandemic time at a rate of e, and they join the recovery class at a rate of qð1 À eÞ, with the remaining individuals from this class being removed from the population by natural death or coronavirus induced death. Individuals who are symptomatically infected are quarantined at rate of g and transfer into treatment centers at a rate of uð1 À gÞ. The parameter p represents a rate at which quarantined infected individuals are moved to hospitals (treatment centers) for better care and recover at a rate of rð1 À pÞ. At the rates of / and cð1 À /Þ, infected individuals identified from funeral participants are quarantined and recovered, respectively. w is the recovery rate of hospitalized individuals, b is the transmission rate (contact rate), and m is the transmission rate from asymptomatically infected to the susceptible individuals. Infectious individuals will be removed from the population at a rate of l þ x, and non-infectious individuals might be removed at rate of l because of natural death and will not be assigned to any class. This model assumes that recovered individuals may not go back to the susceptible class (no reinfection). Positive values are assumed for model parameters. All susceptible individuals have an equal chance of being infected.

Existence and uniqueness of solutions
We simplify the proposed model (2.2) in the following form: where z is the vector presented as S; E; A; L; F; I; Q; H; R ð Þ . Thus, model (2.4) has the following form under the condition that Problem (2.5) has an integral representation is given by Þof a continuous function with the norm defined by Proof. For variable S, we have As a result, f 1 ðt; SÞ fulfills the Lipschitz condition with Lipschitz constant w 1 . Furthermore, if 0 6 w 1 < 1, then f 1 ðt; SÞ is a contraction. Also, we can show existence of Lipschitz constants w i , and a contraction for f i ðt; zÞ; i ¼ 2; 3; . . . ; 9. Now for t ¼ t k ; k ¼ 1; 2; . . ., defined the following recursive form of (2.7): The difference between successive terms in the above equation and taking the norm on both sides of the resulting equation, we get: Taking z ¼ S, (2.11) can also be reduced to the following form: As a result, we have : ð2:12Þ In the same way, for the rest of expressions, (2.11) can be reduced to the following form: Assume that the following condition is met: ð2:17Þ Repeating the process recursively leads to Taking t ¼ f yields The combination of Taking the norm both sides, we get Cðaþ1Þ > 0, we obtain SðtÞ À S 1 ðtÞ k k¼ 0. Thus, we

Solution positivity and boundedness
The system (2.2) depicts the human population, it is important to demonstrate that all results of the system with nonnegative beginning values will stay positive for all t > 0 and are bound. The subsequent lemma and theorem will establish this. Proof. The dynamics of the total population can be gained by adding each equation of the model (2.2), given by ð2:24Þ Applying the Laplace transform on (2.24) and the idea of (1.3) we get: Since, 0 6 E a;1 Àlt a ð Þ6 1, we have N t ð Þ 6 N 0 ð Þ þ # l and hence, the region W is a positively invariant set for the system (2.2). h Theorem 2.4. If the initial solutions satisfy Sð0Þ P 0; Eð0Þ P 0; Að0Þ P 0; Lð0Þ P 0; Fð0Þ P 0; Ið0Þ P 0; Qð0Þ P 0; Hð0Þ P 0; Rð0Þ P 0, then the solutions SðtÞ; EðtÞ; AðtÞ; LðtÞ; FðtÞ; IðtÞ; QðtÞ; HðtÞ; RðtÞ of the system (2.2) are positive 8t P 0.
Proof. Starting with the first equation of the system (2.2) whereis a constant equal to bK 0 N þ l. Applying the LT in (2.27), we arrive at SðtÞ P E a; 1 À-t a ð ÞSð0Þ: ð2:28Þ Since E a; 1 À-t a ð ÞP 0 and Sð0Þ P 0, then the solution SðtÞ is positive. Following the above process, from the remaining equations of (2.2), we can readily demonstrate model's other state variables remain positive 8t P 0. The disease-free equilibrium is attained when there are no infections in the population. The infectious compartments have all been reset to zero. Also, in our model equation, we set the fractional derivatives of non-infectious compartments to zero. As a result, the disease-free equilibrium point is

The basic reproduction number
When a typical infective enters a susceptible individual, the average number of secondary infections is defined as the basic reproduction number. The next-generation matrix technique [46] can be used to calculate this value. We pick epidemiologically valid matrices and algorithms to generate the expression. The spectral radius of the next generation matrix provides the required effective reproduction number.
Proof. To determine the local stability of the disease-free equilibrium, we examine the behavior of our model population near this equilibrium solution. Now, the Jacobian matrix of (2.2) at E 0 is: In this Jacobian matrix, four of the eigenvalues are negative, that is m 1 ¼ m 2 ¼ Àl; m 3 ¼ Àd 6 , and m 4 ¼ Àd 7 . The remaining eigenvalues can be obtained from the characteristic equation: In the above expressions, the coefficient d 5 is positive when R 0 < 1, and all the other coefficients are positive. Further, the Routh-Hurwitz criteria for fifth-order polynomials are d i > 0, for i ¼ 1; 2; 3; 4; 5; can be easily satisfied by using the above coefficients. So, the model (2.2) at E 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1. h

Endemic equilibria
Adjusting the model equation to zero and solve simultaneously, we get the endemic equilibrium point R 0 that satisfies the quadratic equation . . . ; 7 are given in Eq. (2.31). A positive endemic equilibrium point exists only for R 0 > 1 and no endemic equilibrium exists whenever R 0 < 1.

Global stability analysis of equilibrium points
Using the Lyapunov function method, we showed the global asymptotic stability of both equilibrium points in this subsection.
i. Global stability of disease-free equilibrium point COVID-19 can be controlled in Ethiopia when R 0 < 1 if the initial sizes of the cases in each of compartment or society are very close to the disease free equilibrium point; the initial size is in the DFE's basin of attraction, E 0 ¼ # l ; 0; 0; 0; 0; 0; 0; 0; 0 , as previously demonstrated by the local asymptotic stability of the disease free equilibrium. It is incredibly important to verify that the elimi- nation of COVID-19 is independent of the starting size of the number of cases in society by showing that the DFE is globally asymptotically stable if R 0 < 1. The following theorem is used to explore this situation.
Theorem 2.6. If R 0 6 1, then the disease-free equilibrium point E 0 in (2.29) is globally asymptotically stable on a positively invariant region W.

Proof. Consider the Lyapunov function
FðtÞ ¼ c 1 EðtÞ þ c 2 AðtÞ þ c 3 LðtÞ þ c 4 FðtÞ þ c 5 IðtÞ: Differentiate the Lyapunov function FðtÞ with respect to time t using Caputo fractional derivative of order a, we get Substituting the appropriate values from (2.2) and using (2.31), we get Theorem 2.7. If R 0 > 1, then endemic equilibrium point given by is globally asymptotically stable in the region W.
Proof. Assume that the effective reproduction number R 0 > 1, which indicates that the endemic equilibrium point exists.
We'll look at the Lyapunov function defined by: Then, time derivative of the Lyapunov function in Caputo sense and following [47], we get; Substituting the appropriate values from (2.2) for each fractional derivative and using (2.31), we arrive at Thus, by LaSalle's invariance principle the EEP is globally asymptotically stable. h Theorem 2.7 shows that, independent of the starting size of infectious people in the community, COVID-19 will establish itself in society when R 0 > 1.

The Legendre spectral collocation method for the solution of COVID-19 model
To apply the Legendre collocation method for the model Eq.
We utilize roots of the shifted Legendre polynomial P mþ1À a d e ðtÞ to find appropriate collocation points. We can also obtain nine equations by replacing Eq. (3.1) in the beginning conditions (2.3).  Fig. 1 The flow rate of the parameters of the model.
Eq. (3.2), together with the equations of the initial conditions (3.3), give 9m þ 9 equations that can be solved using Newton's iterative methods, for the unknowns a i ; b i ; c i ; d i ; e i ; f i ; g i ; h i ; and p i ; i ¼ 0; 1; . . . ; m.

Input parameters and assumptions
The parameters are estimated using real-world data from COVID-19 confirmed cases in Ethiopia from 22 June 2020 to 28 February 2021. Data available at http://www.ephi.gov.et and https://www.worldometers.info [48]. The following parameters are derived from the data: Ethiopia has a total population of roughly N ¼ 1:14 Â 10 8 , and the life expectancy in 2020 is 66.95 years. As a result, the natural death rate is computed appropriately. Further, from 29,424 tests performed in the first week of this study (June 22-28, 2020), 1,157 new confirmed COVID-19 cases in Ethiopia were reported. Of these, 736 were asymptomatic infected cases. In the first week of this work, there were 919 newly recovered COVID-19 cases. 3,160 cases on treatment, 7,079 cases on follow-up, 120 cases on HBIC (Home Based Isolation and Care), and assuming the initial number of funeral cases to be 87. The initial population is assumed Nð0Þ ¼ 114; 041; 946.
All the graphical depictions sketched in the below sections are based on the parametric values listed on Table 1.
Sensitivity analysis of R 0 : The sensitivity of R 0 to a parameter } can be computed using the following formula The sensitive parameters and their sensitivity index are shown in the table above. On the next figures, we have also shown the effect of the most sensitive factors for R 0 (see Fig. 1).
The forecast of our model generated by the system of nonlinear fractional differential Eqs. (2.2) with real data from the Ethiopian Ministry of Health and the Ethiopian Public Health Institute incorporated for 36 weeks is shown in Fig. 2. Furthermore, the numerical finding is immediately recognizable as being close to the real data (see Fig. 3 and 4).
The graphical interpretations of Figs. 5-13 show the influence of the fractional order a on each compartment of the model.

Discussion
In this study, a fractional mathematical model for COVID-19 transmission dynamics in Ethiopia was developed, and its many properties, such as existence and uniqueness, local and global stability analysis of equilibrium points were investi-gated. Based on the real data that was made accessible in [48], the fitted parameters were constructed using leastsquare curve fitting in Matlab R2020a. The basic reproduction number is then computed to be R 0 ¼ 1:4575 > 1, showing that the endemic equilibrium point is both locally and globally asymptotically stable and the infection free equilibrium point is both locally and globally asymptotically unstable. Table 2 also discusses the sensitivity of the basic reproduction number R 0 , pointing out that it is more sensitive to the quarantine and hospitalization rates of symptomatically  Fig. 14 Behaviors of susceptible and exposed cases when the value of the parameter g is increasing for a ¼ 0:7.      Behaviors of susceptible and exposed cases when the value of the parameter u is increasing for a ¼ 0:2.
infected people, as well as the lockdown rate of people attending funerals and the transmission rate. b has positive imapct, which means decreasing this parameter leads to a reduction in the value of R 0 . g; u and e have negative impacts. That is, increasing the value of these parameters leads to a decrease in R 0 . When b 6 0:1299; g P 0:3776; u P 0:2420 and e P 0:0075 and the remaining parameters in Table 1 are unchanged, the stable infection free equilibrium point for Eq. (2.2) is obtained. Other parameters change within their suitable range, making all its significant effects, even the value of the reproduction number R 0 greater than or less than one (see Figs. 14-22).   The behavior of approximate solutions of a model Eq. (2.2) for different values of fractional order a is shown in Figs. 5-13. The fractional order a has its own impact on a model's numerical solutions. We observed a significant reduction in the number of infected individuals in a fractional order a ¼ 0:5. Fig. 23 shows that the highest value of the basic reproduction number R 0 shows the highest number of asymptomatic and symptomatic cases, and as the number of infectious cases increases, the susceptible population decreases. For R 0 ¼ 2:4125, we have an evidence that in this simulation, most of the infected individuals are rapidly increasing. This indicates that one infected person will cause more than two new infections, and virus transmission will become uncontrollable. When the basic reproduction number R 0 ¼ 0:7519 is used, the curves reveal that infected individuals tend to be few. This indicates that the number of newly infected individuals is decreasing, that society is safe from the virus, and that the infection will eventually die out from the Ethiopia.

Conclusion
The transmission dynamics of the Covid-19 epidemic are studied using a mathematical model developed in this work. In the Caputo sense, the model is presented as a system of fractional order differential equations, with the numerical solution achieved using the Legendre collocation method. The findings of the proposed model are found to be quite close to the real data. The model considers the impact of various control techniques on disease transmission. These strategies have been shown to result in considerable changes to reduce the danger of disease transmission, as simulated above. Ethiopia's government must take the required steps to make control tactics a must throughout the epidemic. If used properly, control techniques such as lockdown, quarantine, and hospitalization can considerably reduce the disruptive effects of COVID-19 while also protecting the nation. The government may also direct its control strategy toward reducing the R 0 via the sensitive parameters.

Data Availability
Data from COVID-19 confirmed cases in Ethiopia from 22 June 2020 to 28 February 2021 were utilized to support this investigation.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.