ON THE FRACTIONAL-ORDER MODELING OF COVID-19 DYNAMICS IN A POPULATION WITH LIMITED RESOURCES

. Resource availability plays a pivotal role in the ﬁght against emerging infections such as COVID-19. In the event where there are limited resources the control of an epidemic disease tends to be slow and the disease spread faster in the human population. In this paper, we are motivated to formulate and investigate a mathematical model via the Caputo derivative which incorporates the impact of limited resources on COVID-19 transmission dynamics in the population. We analyze the fractional model by computing the equilibrium points, and basic reproduction number, ( R 0 ) , and also used the Banach-ﬁxed point theorem to prove the existence and uniqueness of the solution of the model. The impact of each parameter on the dynamical spread of COVID-19 was examined by the help of Sensitivity analysis. Results from mathematical analyses depict that the disease-free equilibrium is stable if R 0 < 1 and unstable otherwise. Numerical simulations were carried out at different fractional order derivatives to understand the impact of several model parameters on the dynamics of the infection which can be used to establish the inﬂuential parameter driving the epidemic transmission path. Our numerical results show that an increase in the recovery rate of hospitalization increases the number of infected individuals. The results of this work can help policymakers to devise strategies to reduce the COVID-19 infection.


Severe acute respiratory syndrome (SARS) first occurred in the year 2002-2003 in Guangdong
Province, China [1]. In 2015, South Korea experienced a similar outbreak known as the Middle East respiratory syndrome (MERS) [2,3]. On the other hand, in Wuhan, China, a disease called SARS Coronavirus 2 (SARS-CoV-2) which is a new type of variant coronavirus emerged in December 2019 [4]. The world health organization (WHO) later declared it a pandemic on 11 March 2020 [5]. This disease can transmit directly from human to human or through contact with contaminated surfaces, coughing or sneezing [6,7] while the asymptomatic stage of the infection can last for up to 14 days before becoming infectious. The infected individuals at this stage show passive symptoms ranging from respiratory infection like coughing and wheezing which can later to an inability to smell and loss of taste [7]. While the elderly and those with underlining illness such as diabetes, obesity, and hypertension easily succumb to the disease [8,9,10,11]. Currently, the world has recorded a total of 612,076,308 coronavirus cases with about 6,509,597 deaths and 589,858,670 recovered individuals until date [12].
The alarming rate of COVID-19 cases and mortality has caused a global health concern.
Several control measures such as social distancing, wearing masks, regular hand washing, the use of hand sanitizers, social distancing, isolation of infected people, bans on air travel, and social gatherings in different areas were implemented at the early stage of the pandemic to reduce the total number of infections. Additionally, various countries and regions of the world enforced lockdowns in the most affected areas to help control the spread of the virus and halt the chain of transmission caused by the infectious individuals in the whole susceptible population [13].
Mathematical modeling has played a vital role in predicting several ways that enable most world governments to mitigate the spread of the disease. Most of these models included using a statistical approach, agent-based modeling, and ordinary differential equations consisting of integer and non-integer order models to quantify the COVID-19 transmission and beneficial ways to mitigate the infection. See the following published literature which has implemented the integer order model [14,15,16,17,18,19,20,21]. Fractional calculus has shown to have a memory effect that gives more accurate results in forecasting physical systems including mathematical models [22]. This has led to new advancements in developing new fractionalorder operators namely Riemann-Liouville, Caputo, Caputo-Fabrizio, and Atangana-Baleanu utilized in solving both integer and non-integer orders systems of differential equations arising from real-world problems such as applications to integrodifferential equations [23,24], mathematical epidemiology [25,26], economic and financial [27,28] and many other fields. On the other hand, fractional order, which consists of Caputo fractional and Atangana-Balenu (ABC) derivatives, has also been applied in modeling the spread of COVID-19 as in [29,30,31].
We now focus on reviewing some of the above-mentioned literature. The work in [30]  Furthermore, Ahmad et al. [32] used an ABC fractional derivative to study the impact of quarantine and social distancing on the COVID-19 epidemic. It is found that the best method to stop the spread of the virus is to social distance in the form of staying at home which can be ahead through lockdown implementation, in contrast, the infected individuals should isolate to avoid an increase in disease transmission. Also, Shah et al. [8] formulated a non-integer order model for COVID-19 dynamics. The reproduction number was calculated and the asymptotic stability of the proposed model was examined. It was found that the numerical simulation is in good agreement with the theoretical results. The fractional order differential equations were used by Atangana [24] to study the spread of COVID-19.
Cakan [16] used an SEIR model to study the impact COVID-19 in the presence of limited resources considering the hospital setting. Their findings suggest that higher contact rates between susceptible and infected individuals lead to higher hospitalized individuals, which can grow beyond the carrying capacity of the present hospitals. Authors believe that the lack of adequate resources to control such kind of virus contributed to a large number of cases of the COVID-19 pandemic across the globe [12]. In this study, a new mathematical model for the COVID-19 outbreak by incorporating limited resources is extended into a fractional-order model in the sense of the Caputo fractional derivative. We modified the basic epidemic model with limited public health resources as extended by [33,34]. Using the idea of the fractional ordinary differential equation, we are thinking that the fractional-order model will accommodate the real phenomenon of the spread of COVID-19. Similarly, the results of this model will help policymakers to devise strategies to reduce the COVID-19 infection. In the next section, we present preliminaries for the fractional Caputo derivative.

BASICS OF FRACTIONAL CALCULUS
This section presents brief essential definitions regarding fractional calculus in Caputo sense.
Definition 1 (see [35]). Consider y ∈ C n be function, then Caputo derivative having fractional order α in (n − 1,n) where n ∈ N is defined as: with Γ(.) is the gamma function and C D α t ( f (t)) tends to f (t) as α −→ 1. Definition 2 (see [35]). The Corresponding integral with fractional order α > 0 of the function f:R + −→ R is expressed as follows: Definition 3. Let y * denote the equilibrium of the Caputo fractional model then:

MODEL FORMULATION
This section evaluates the formulation of an epidemic model of COVID-19 disease in a community with insufficient aids. Thus, the community denoted as N(t), is distributed into six mutually exclusive components, i.e., susceptible class, S(t) (individuals who are likely to contract COVID-19), exposed class, E(t) (those who are infected but not infectious yet), quarantined class, Q(t) (those who are in isolation center or self-isolation), infectious class, I(t) (those who display the symptoms and are capable of the disease spread), H(t) (those who are infectious and admitted to a health care facility) and recovered class, R(t) (those who have recovered from COVID-19 infection). Thus, In order to understand the effect of the capacity and inadequate comprehend the effect of the capability and inefficient health structures, the recovery rate for people in class H integrating the effect of capacity and inefficient health facilities is developed as a nonlinear function as where b > 0 represents the hospital bed-population ratio, σ 0 is the minimum per capita recovery rate owing to the function of the basic healthcare system, and σ 1 accounts for the maximum per capita recovery rate corresponding to the sufficient clinical resources and few hospitalized humans. A similar function to the ones given in (5) was previously suggested in [33,34].
Consequently, the model governing the COVID-19 dynamics in the population according to the flowchart illustrated in Figure 1 is derived as The explanation of the parameters used in model (6) is provided in Table 1, while the scheme of the model is as shown in Figure 1.   Proof. Assume thatt=sup{t > 0 : Thust > 0, and it follows from the first equation of model (6) that resulting to, In a similar fashion it can also be exhibited that E(0) > 0, Q(0) > 0, I(0) > 0, H(0) > 0 and R(0) > 0 for all t > 0, and this completes the proof.

Invariant region.
This Section present the point at which the model (6) will be positively invariant. Proof. We attend the proof presented in [36,37]. Based on the equation (6), the time derivative of N(t) is assigned by From (7) we have dN dt ≤ 0 which implies that Φ is a positively invariant set. We also note that by solving (7) we have where N(0) is the initial condition of N(t). Thus, 0 ≤ N(t) ≤ Λ d 0 as t −→ ∞ and hence Φ is an attractive set.

FRACTIONAL-ORDER MODEL
This section evaluates a fractional-order COVID-19 model (6). The fractional model appropriating to the system (6) is illustrated below: where α in 0 < α ≤ 1 represents the order of the fractional derivative. The fractional derivative of the model (8) is in the significance of Caputo. The Caputo method is mainly utilized in actual applications. The major benefits of the Caputo approach are the introductory values for the fractional differential equations putting up with the same establishment as for integer order differential equations [35]. The Caputo fractional derivative is interpreted below.
Proof. The followings were obtained after using Caputo integral on system (8) Arising from definition (2), the following were obtained, The kernels are obtained as follows, All the equations in (11) satisfy the Lipschitz conditions with all the compartments possessing the upper limit. Suppose the functions S(t) and S * (t) are considered, applying a similar approach for other functions, which gives rise to . Using a similar approach, the remaining equations are obtained as follows where g 1 , g 2 , g 3 , g 4 , g 5 and g 6 denote the Lipschitz constants respectively which is corresponding to all the six kernels and by this, the Lipschitz condition is concerned satisfied. The algorithm of equations in (10) can be presented as follows: The form presented below is the successive terms along the initial conditions of the model with their corresponding differences; . Now, we consider that: We obtained form (15), Hence, all the model variables depict the bounded functions and the representation of the kernels satisfies the Lipschitz condition.
Theorem 3. The Caputo COVID-19 model stated by system (8) has a unique solution for all Proof. From the above theorem, one will notice that all the model variables depict the bounded functions and the expressions of the kernels satisfy the Lipschitz condition. Applying recursive principle on equation (17) with respect to system (8), the system below is obtained: Hence, the progression exist and satisfy the conditions describes Φ S,n (t) −→ 0, Φ E,n (t) So by hypothesis, 1 Γ(α) g jm < 1, and S n , E n , Q n , I n , H n , R n is the Cauchy progression. By this, the complete desired result for the model (8) is obtained.

Iterative solution and stability analysis. The subsequent hypothesis and solutions have
been furnished to stabilize the outcome of this proposed model (8) Theorem 4 Let (B, . ) indicate a Banach space, and X * defined a self map on B. Further, z n+1 = x(X * , z n ) exhibits the recursive expression while C (X * ) implies the fixed point set upon X * indicating the fixed point set upon X * . Also, by defining such that y * n+1 − x(X * , y * n ) such that {y * n ⊆ B}. Then, in the iterative approach, y n+1 = x(X * , y n ), X * is stable if lim n−→∞ C n = 0, that is lim n−→∞ C * n = p * for z n+1 = X * where n is put up with as the Picard's iteration then X * iteration is stable. The hypothesis can be condensed as below. Let (B, . ) define a Banach space and X * be a self-map upon B, then for all x, y ∈ B, we have that Suppose Z is defined as a self map, and the following results are obtained: Z is stable if the following conditions are satisfied The following equations were obtained when evaluated around the map Z which is a fixed point, Taking the norm of both sides of the above equation Further simplification of (21), leads; Using these assumptions, after putting the above relation, leads Since the progressions S n (t), E m (t), Q m (t), I m (t), H m (t), R m (t) are convergent and bounded, their exist six different constant S 1 > 0, S 2 > 0, S 3 > 0, S 4 > 0, S 5 > 0 and S 6 > 0 for all t.
Through the relation, we can attain Hence, the proof is complete.

4.3.
Disease-free equilibrium. The fractional model has a disease-free equilibrium presented by (26) an event portraying a society free of infection. The basic reproduction number, R 0 , inferred as the anticipated number of secondary instances produced by a sole contagious person in a totally susceptible community throughout its contagious time is a boundary variable that enables us to foresee if the infection will stop or prevail [38]. Typically, R 0 < 1 implies that the disease (COVID-19) cannot overrun the community, R 0 > 1 implies that each affected person develops more than one secondary infected person, and R 0 = 1 respects additional scrutiny. The resolution of R 0 is performed utilizing the next-generation matrix method [38,39]. Utilizing this procedure, we retain . At the disease-free equilibrium, D 0 , σ (b, H) = σ 1 . Thus, the R 0 of model (8) is given by The COVID-19 disease can be eradicated from the population (R 0 < 1) if the initial sizes of the population of the model are in the basin of attraction of the disease-free equilibrium. This will be established by the following theorem.
Theorem 5 For any two positive integers n * 1 , n * 2 with gcd(n * 1 , n * 2 ) = 1 where α = n * 1 n * 2 and M = n * 2 . Thus, the system (8) is locally asymptotically stable if | arg(λ ) |> π 2M for every root of λ that is represented by the following equation, Proof. The Jacobian of system (8) at D 0 is The characteristics equation (31) The replicated two eigenvalues are −d 0 , −d 0 , have a negative real part. We employ the polynomial in (32) and estimated the residing eigenvalues by finding the coefficients provided below: if R 0 < 1. Moreover, we can determine an argument less than π 2M for R 0 > 1.
Thus disease-free equilibrium is locally asymptotically stable for R 0 < 1.

Lemma 1
The disease-free equilibrium D 0 is locally asymptotically stable whenever R 0 < 1 and is unstable if R 0 > 1.
The epidemiological significance of Lemma (4.3) is that it is possible to curb COVID-19 (or alumina it) from a community (when R 0 < 1) if the preliminary quantities of the sub-populations of the fractional model are in the bay of interest of the disease-free equilibrium (D 0 ).

Global stability of disease-free equilibrium.
To guarantee active management (or elimination) of COVID-19 in a community when it is autonomous of the preliminary quantity of the sub-populations of the fractional model (8), it is crucial to exhibit that the D 0 is globally asymptotically stable (GAS). This will be performed by assigning the Lyapunov function procedure.

Theorem 6
The disease-free equilibrium (D 0 ) of the fractional model (8), given by (26), in the shortage of the progression rate of people in class S to class Q (µ = 0) is GAS if R 0 ≤ 1.
Proof. Assess the fractional model (8) with µ = 0 and the subsequent Lyapunov functional defined by Relating the Caputo-fractional derivative on V , concurrently with the practice of the model (8), we obtain, The time derivative of the Lyapunov function (34), which is represented by C D α t V , along the solution path of system (8) with µ = 0, is given by Using the limiting value N = Λ d 0 and since S ≤ Λ d 0 in the positively-invariant region Φ, it follows, by serious rigorous simplification of (35), results into which finally yields this It can be followed from the results given in [36], the solution of system (8) with non-negative initial conditions tends to D 0 whenever t −→ ∞ in Φ. So, the system (8) at D 0 is globally asymptotically stable (GAS).
Therefore, solving the system (8) at equilibrium is given by Substituting (42) into (41), and after simplification, we get If R 0 > 1, then a unique endemic equilibrium exist.
The qualitative behavior of solutions with respect to Theorem (4.4) has shown in Fig (2(a)) where solutions at different initial conditions tends to the disease-free equilibrium point asymptotically. This result implies that the disease elimination is possible regardless of the number of infectious individuals present in the population as long as the basic reproduction number of the disease is less than unity. Similarly, Fig (2(b)) shows the population of infected individuals at different initial conditions converge to endemic equilibrium point when the basic reproduction number is greater than unity. The implication is that the disease will persist in the population regardless of the number of infected individuals in the populations as long as the basic reproduction number of the disease is greater than unity.

Sensitivity analysis.
In this section, the sensitivity analysis of the formulated model (6) is studied. In other to achieve this, the method used in the following literature [11,19,21,36] were adopt. The purpose of this is to know the parameters of the model that contribute majorly to the spread of COVID-19 in the population. Thus, to do this the COVID-19 threshold quantity, the basic reproduction number R 0 given in (29), as the response function with respect to the model parameters (6), the normalized forward-sensitivity index of R 0 that depends on a parameter p is defined as Given the explicit formula (29) for the basic reproduction number R 0 , the analytical expressions for the sensitivity of R 0 in respect of the parameters defining it are computed. In particular, the analytical expression for the sensitivity of R 0 with respect to β in view of (45) is a constant value, while that of ε is a complex expression, and are both obtained as In a similar manner, the analytical sensitivity indices of R 0 for the other parameters are computed. But the results are omitted due to their complexity. However, the sensitivity index (SI) of R 0 for all the parameters comprising it are evaluated at the baseline parameter values given in Table 3. The signs and values of SI are presented in Table 2.  Table 3 Parameter Sign of SI  Table 2, it is observed that the sign of SI is positive for some parameters (ψ, ε, h 2 , β ), while it is negative for the others (δ , h 1 , d 0 , ρ, σ 1 , d 1 ). From the set of parameters with positive SI sign, β , ε and ψ are most positive. Whereas, σ 1 , δ and h 1 are most negative from the set of parameters with negative SI signs. The epidemiological insight from the positive sign of SI of the COVID-19 threshold quantity, R 0 , is that increasing or decreasing the value of any of the parameters in this category will generate an increase or decrease in the threshold R 0 of COVID-19. The negative sign of SI on the contrary suggests that increasing the value of each of the parameter set in this category will lead to a decrease in the R 0 value, and vice-versa.
For example, S R 0 β = +1 indicates that increasing the effective transmission rate of COVID-19 by 10% will increase the basic reproduction number, R 0 , of the disease by 10%, and the other way round. Similarly, S R 0 σ 1 ≈ −0.44 means that increasing (or decreasing) σ 1 by 100% always decreases (or increases) R 0 by 44%. Consequently, in view of the results of sensitivity analysis in Table 2, considerations of any control strategies that reduce the effective transmission rate of COVID-19 (β ), modification parameter relative to the infectiousness of hospitalized (ε), progression rate from exposed to become symptomatic (ψ), and increase the maxima per capital recovery rate associated with the sufficient healthcare resources and few hospitalized humans (σ 1 ), detection rate for exposed self-quarantine (δ ) and hospitalization rate for symptomatic infectious (h 1 ) are needed to ensure an effective control of COVID-19 transmission and spread in the population. The Fig 3 below is the sensitivity value of COVID-19 model parameters.

NUMERICAL SIMULATIONS AND DISCUSSION
The simulation for the Caputo COVID-19 model (8) is presented in this segment. The natural variable's significance for the numerical simulation is provided in Table 3. The proposed fractional model is unraveled numerically employing the notion explained in detail in [40,41].   Table 3. Below we give the graphs obtained from the numerical simulation of our Caputo model.
In Fig.(4), several values of α have been examined on the model compartments graphically.
The findings demonstrate that as the order of Caputo operator α rises, the burden of the syn-       Also, Figure 6 shows the effect of the detection rate for exposed individuals to become quarantined δ . In Figure 6(a)-6(b) the susceptible individual increases as the detection rate for exposed individuals to become quarantined increases, while in Figure 6(c)-6(d) the exposed individual increases as the detection rate for exposed individuals to become quarantined decrease.
In Figure 6(e) the quarantined individual increases as the detection rate for exposed individuals to become quarantined decreases but in Figure 6(f) it convergence to endemic over time. In Figure 6(g)-6(h) the infected individual increases as the detection rate for exposed individuals to become quarantined decreases, while in Figure 6(i)-6(j) the hospitalized individual decreases as the detection rate for exposed individuals to become quarantined increase. The impact of hospitalized rates for infectious h 1 on the population dynamics of COVID-19 is depicted in Figure 7. By increasing the value of the hospitalized rate, one can observe the decrease in the hospitalized and recovered individuals when α = 0.5 as shown in Figure 7(a)-7(c) but when α = 0.9 similar things occur for the first 36 days, as h1 approaches its optimal, the hospitalized and recovered individuals increases sporadically as shown in Figure 7 Similarly, Figure 9 shows the effect of disease progression rate from exposed class (ψ) on the population dynamics of COVID-19. In Figure 9(a)-9(b) as ψ increases also infected community increased. In Figure 10 as the effect of progression rate of individuals in class S to class Q increases quarantined, infected, and hospitalized class increase for both α with time but α = 0.9 it convergence at a point for quarantined and infected class and reaches a peck at maximum for hospitalized class at 30 days.

CONCLUSION
In this paper, we formulate and analyze a Caputo fractional model of COVID-19 incorporating the potential impact of limited resources on the human population. Mathematical analyses of the model were done, which included determining the well-posedness of the system, equilibrium point, basic reproduction number R 0 and the existence and uniqueness of solutions, as well as the global stabilities of the model equilibria, were established based on the R 0 . Our model basic reproduction number was calculated analytically and then evaluated numerically to be R 0 = 1.4592 which indicates the high transmissibility of the COVID-19 virus in the population induced by the disease. Sensitivity analysis was also carried out to know the contributory effect of each parameter on the dynamical spread of COVID-19 in the community. Importantly, the qualitative analysis of the model shows that an increase in the fractional order parameter leads to an increase in the number of humans infected with COVID-19. Furthermore, in the absence of resources such as masks, limited hospital beds, and various other non-pharmaceutical interventions, our model predicts an increase in the number of infectious individuals as more people get exposed to the virus. Similarly, the results of this model will help policymakers to devise strategies to reduce the COVID-19 infection. We acknowledge that the modeling presented in this paper has a limitation; the model is not fitted to the COVID-19 epidemiological dataset for any specific region. Rather we parameterize our model using the available parameter values from published literature which conforms with the actual attributes of COVID-19 dynamics. The work presented in this manuscript can be extended by incorporating fractional optimal control parameters, which may investigate the types of interventions required to reduce COVID-19 viral transmission in an event for a population with inadequate resources.

ACKNOWLEDGMENTS
The authors will like to acknowledge the respective universities for the production of this manuscript

CONFLICT OF INTERESTS
The authors declare that there is no conflict of interests.