key: cord-0966949-klk3z5jq
authors: Inc, Mustafa; Acay, Bahar; Berhe, Hailay Weldegiorgis; Yusuf, Abdullahi; Khan, Amir; Yao, Shao-Wen
title: Analysis of novel fractional COVID-19 model with real-life data application
date: 2021-02-26
journal: Results Phys
DOI: 10.1016/j.rinp.2021.103968
sha: 50c3a19b9f7f9a2982f47fc73a96169295ef9edc
doc_id: 966949
cord_uid: klk3z5jq

The current work is of interest to introduce a detailed analysis of the novel fractional COVID-19 model. Non-local fractional operators are one of the most efficient tools in order to understand the dynamics of the disease spread. For this purpose, we intend as an attempt at investigating the fractional COVID-19 model through Caputo operator with order [Formula: see text]. Employing the fixed point theorem, it is shown that the solutions of the proposed fractional model are determined to satisfy the existence and uniqueness conditions under the Caputo derivative. On the other hand, its iterative solutions are indicated by making use of the Laplace transform of the Caputo fractional operator. Also, we establish the stability criteria for the fractional COVID-19 model via the fixed point theorem. The invariant region in which all solutions of the fractional model under investigation are positive is determined as the non-negative hyperoctant [Formula: see text]. Moreover, we perform the parameter estimation of the COVID-19 model by utilizing the non-linear least squares curve fitting method. The sensitivity analysis of the basic reproduction number [Formula: see text] is carried out to determine the effects of the proposed fractional model’s parameters on the spread of the disease. Numerical simulations show that all results are in good agreement with real data and all theoretical calculations about the disease.

The epidemiology of infectious diseases examines how the disease occurs under ordinary conditions and so properties of location, environment, and factor are evaluated. The cycle formed by taking these features into consideration is called the infection chain in which there are relationships between the agent, the mode of transmission, and the host. When one of these relationships is prevented, it is not possible to see that infectious disease in the community. Throughout history, epidemics such as cholera, chickenpox, malaria, HIV, and especially plague have deeply affected states and people. Infectious diseases were more easily spread due to unfavorable living conditions, and until the 17th century, all epidemics were called plague. Plague, a deadly disease, continued its effect until the 18th century, and in particular, the plague, known as the black death in Europe, destroyed a third of the European population in the 14th century, causing the death of many people. The lives of the survivors have never been the same after this hard period, and people had to continue their lives by suffering severe trauma and fear. In the 19th century, outbreaks of typhoid, cholera, and typhus began to appear. The infectious disease cholera occurred in India in the late 18th and early 19th centuries and was as effective as the plague. Besides, smallpox, another disease affecting the world, was generally seen in children, causing them to bear the traces of the disease on their faces throughout their lives. To cope with the deaths caused by this disease, the Turks applied the vaccine to children. Although the malaria epidemic caused many deaths like other outbreaks, it did not have devastating effects such as plague and cholera. While typhoid and cholera can be transmitted from person to person with body wastes and secretions, typhus is transmitted to the person when a lice biting the sick person bites a non-sick person, that is, it needs a vector. It is possible to be protected from these diseases by paying attention to hygiene rules, using wastewater infrastructure, providing clean water, and cleaning food and beverages. In the 19th century, when diseases were not yet known to spread with pathogens (i.e. disease-causing microorganisms and viruses), doctors could cause pathogens to spread from one patient to another, causing their patients to become infected and even to die from the infection. The view that illnesses could be transmitted through touch, especially through doctors who were used to heal, was not easily accepted at that time. It should be emphasized that the determination of human beings to RL I α φ (t) = 1 Γ(α) t a φ (τ) (t − τ) 1−α dτ, Re(α) > 0;

(1) RL D α φ (t) = d n dx n RL I n−α φ (t), n = ⌊Re(α)⌋ + 1, Re(α) ≥ 0. (2) Although the definition of Riemann-Liouville has a very important place in fractional calculus, it has some shortcomings in the application due to initial conditions. To overcome these shortcomings and gain an advantage in the application, the Caputo derivative has been defined by making a modification to the Riemann-Liouville operator. This favorable fractional derivative, which is often preferred to solve real-world problems, is defined as follows:

where Re(α) ≥ 0 and n = ⌊Re(α)⌋ + 1 [6] . In addition to these definitions we have mentioned, many more fractional operator definitions and various applications have been presented in the literature as can be seen in [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] . Some of these derivatives are obtained by iteration of local derivatives, while others are generalized versions of existing fractional derivatives. On the other hand, some fractional operators have also been obtained using singular or non-singular kernels. One of the most important features of all these various fractional derivatives and integrals is that they have a memory effect. Hence, we feel inspired to analyze the fractional version of a disease model called COVID-19 in [27] with the efficient fractional operator of Caputo. In order to capture certain properties of the dynamical model of COVID-19 in detail, we benefit from the virtue of the fractional derivative that allows us to achieve a better fit with real data of Turkey. The authors in [27] divide the population into seven classes to forecast the impact of the home quarantine which is one of the most crucial measures for preventing the transmission of COVID-19.

The remaining parts of this study are designed as follows. In Section 2, the formulation of the COVID-19 model is presented by means of Caputo fractional operator. The real incidence data is given for Turkey in Section 3. We perform such detailed analysis as the existence and uniqueness of the solutions, stability analysis, iterative solutions, the positiveness of the solutions, basic reproduction number, sensitivity analysis of the fractional model under investigation in Section 4. In Section 5, The parameters' estimation of the COVID-19 model is presented by using real data of Turkey. Finally, in Section 6, we introduce the numerical simulation and discussions to grasp all results of the current work.

The extended version of classical SEIR model is provided a description for the spread of COVID-19 under the quarantine strategy applied by the government [27] . Some theoretically necessary assumptions required for simplification of the current problem can be listed as follows:

• In this model, once a patient has been treated well, the probability of getting infected a second time is not considered.

• Since it is a short-term model, the natural birth and death rate is ignored.

• All coefficients of the model are positive constants.

Under the above assumptions and quarantine strategy, the dynamical model of the transmission of COVID-19 is presented by

As we see in the model (4), the population consists of seven compartments, where

• S(t)= susceptible individuals at time t,

• E(t)= exposed individuals who carry the coronavirus at low-level (there is no infectiousness),

• I(t)= infected individuals with certain symptoms of COVID-19,

• R(t)= recovered individuals at time t,

• Q(t)= individuals who are in the quarantine process at time t (they do not contact with the infected people),

• A(t)= asymptomatic infected individuals at time t,

• D(t)= the number of medically verified cases in which patients are in quarantine treatment at time t,

• α=contact rate to individuals,

• δ = the rate of spread among infectious people with symptoms of COVID-19 and asymptomatic infected class is different. Hence, the parameter δ ∈ (0, 1) represents this main difference,

• κ= the quarantine rate of individuals,

• γ= the release rate of quarantined class Q,

• ψ= the transmission rate of exposed people to infected class,

• η= after infected, the proportion of getting symptomatic is denoted by η, and it is 1 − η when getting asymptomatic,

• µ A = the diagnostic rate of asymptomatic infectious individuals,

• µ I = the diagnostic rate of symptomatic infectious individuals,

• 1/ϕ A = the mean recovery period of compartment A,

• 1/ϕ I = the mean recovery period of compartment I,

• 1/ϕ D = the mean recovery period of compartment D.

• ρ I = the disease-induced death rate of compartment I.

• ρ D = the disease-induced death rate of compartment D.

After introducing above essential information on the model (4), we wish to propose the corresponding fractional type model employing Caputo derivative as below:

In order to make use of the advantages of the fractional calculus, we investigate some crucial theoretical and numerical properties of the dynamical system (5) in the following sections. Therefore, we obtain more sensitive results for the spread of COVID-19 and the effect of quarantine on the transmission of disease thanks to the non-integer order operator called Caputo derivative.

The first case of COVID-19 identified by the Ministry of Health in Turkey announced on March 10, 2020, and the first virus-related death in the country occurred on March 15, 2020. Also, he announced that cases of COVID-19 spread all over Turkey On April 1, 2020. As of June 21, 2020, in Turkey, it was declared that 4,950 people lost their lives owing to COVID-19 and the total number of cases reached 187,685. Moreover, the head of the Turkish Ministry of Health expressed that the spread of COVID-19 in Turkey reached its peak in the fourth week on April 14, 2020. According to the published data on the official website, we have collected these data from April 1 to May 1, 2020, in order to perform parameter estimation. The period we determine is 31 days Turkey faced with a large number of COVID-19 cases. Table 1 shows these reported cases of COVID-19 across Turkey. In the light of these data, it is aimed to carry out the most appropriate estimates for the parameters in the model to obtain much better numerical results. 

,

If we assume that

then we can express the system (6) by the Caputo operator as

It should be stressed that

and R(t) have an upper bound. Let S(t) and S * (t) be couple functions, then we get as follows

For

accordingly, when

Thus, it can be concluded that the Lipschitz condition is satisfied for K 1 , K 2 , K 3 , K 4 , K 5 , K 6 and K 7 .

Recursively, (8) can be written as follows

. By subtracting the successive terms, we attain

If we consider as below

and utilizing the equations (10), (11) and considering Ψ S,n−1

, it can be obtained the following relations

Consequently, we can prove the theorem below:

Theorem 1. The fractional COVID-19 model (5) has a unique solution under the condition that

when t ∈ [0, r].

Proof. As we showed above, the functions S(t), Q(t), E(t), A(t), I(t), D(t) and R(t) are bounded and K 1 , K 2 , K 3 , K 4 , K 5 , K 6 , K 7 satisfy the Lipschitz condition. So, with the help of the recursive principle and (15), we reach

On the other hand, employing the triangle inequality and the system (17) for any p, one can get

such that k i = M (χ) χ r χ ν i < 1. Thus, S n , Q n , E n , A n , I n , D n , R n are Cauchy sequences in B(J). For this reason, it can be said that they are uniformly convergent. Through the limit theorem, the limit of the sequences (12) is the unique solution of the fractional system (5).

Here, iterative solutions are introduced by utilizing the Laplace transform of Caputo fractional derivative. On the other hand, stability criteria for the fractional COVID-19 model is presented with the help of fixed point theorem. Let (B, ||.||) be a Banach space and Q * be a self-map of B. Moreover, let us take into account the recursive procedure in the form of the y n+1 = h(Q * , y n ) and G(Q * ) be a fixed point set of non-empty Q * . It should be expressed the sequence y n converges to the point of q * ∈ G(Q * ). We define ||z * n+1 − h(Q * , z * n )|| where {z * n ⊆ B}. The iterative approach, y n+1 = h(Q * , y n ) is Q * -stable if lim n→∞ c n = 0, that is, lim n→∞ c * n = p * . For the sequence z n to be convergent, it must have an upper limit. If all the conditions mentioned above are satisfied for y n+1 = Q * where n is considered as Picard's iteration as in [24] , then the iteration is Q * -stable. Hence, we can express the theorem below: Theorem 2. Let (B, ||.||) be a Banach space and Q * be a self-map on B, then for all x, y ∈ B, the following inequality is satisfied

where K ≥ 0, k ∈ [0, 1). If we assume that Q * is Picard Q * -stable, then recursive formula can be given as follows

Theorem 3. Let F be a self-map, then it is defined as follows

that is F -stable in the space of L 1 (a, b) , if the following conditions are satisfied

[1 + α χ (P + Q) f 1 (ρ) + α χ δ χ (P + R)g 1 (ρ) − ψ χ k 3 (ρ)] < 1,

Proof. It is clear that F is a fixed point. Thus, we can write the following iterations for all (m, n) ∈ N × N.

After taking the norm of both sides of the first equation in (23), we attain F (S n (t)) − F (S m (t)) = S n (t) − S m (t)

and by utilizing the triangular inequality, one can write

Thorough some necessary simplifications, (25) takes the form of F (S n (t)) − F (S m (t)) ≤ S n (t) − S m (t)

Due to the same behavior of functions inside the fractional system, we suppose that

Inserting (27) into the relation (26), one can have

Owing to the fact that the sequences S n (t), I m (t) and A m (t) are convergent and bounded, there are three different constants P > 0, Q > 0 and R > 0 for all t. Thereby, we get

From the relations (28) and (29), we reach

where f 1 , g 1 and k 1 are the functions acquired by the inverse Laplace transform in (28) . In a similar way, we achieve

where the condition (22) is valid. So, it can be expressed that F has a fixed-point. To prove that F satisfies the conditions of Theorem 3.3, we presume that (31) and (32) hold and also the following relations are satisfied

Hence, we reach the intended result.

Let us determine the invariant region and indicate that all solutions of the fractional COVID-19 system (5) are positive for all t ≥ 0. The main objective is to introduce the convenience of the solutions for the model analyzed by observing whether they enter the invariant region ϒ. Employing the advantages of the Caputo fractional derivative, we assume that ϒ = (S, Q, E, A, I, D, R) ∈ R 7 + ,

be any solution of the model (5) with non-negative initial conditions. Moreover, we have n = (S(t), Q(t), E(t), A(t), I(t), D(t), R(t)) T . We must also demonstrate that the vector field points to R 7 + upon each hyperplane which is bounded by the non-negative hyperoctant. So, one can write

(39) So, we give the convenient region as follows

Thereby, the fractional COVID-19 model (5) is biologically appropriate and mathematically well-defined in the region ϒ when t > 0. Additionally, this region is positively invariant, that is, solutions of the underlying system (5) are positive for all t.

The basic reproduction number denoted by R 0 = ρ(FV −1 ) where ρ(.) is the spectral radius of the matrix FV −1 can be obtained by the next-generation matrix approach. The matrix F of transmission and matrix V of transformation for the fractional COVID-19 model (5) are given by

Thus, we obtain the reproduction ratio for the fractional model of COVID-19 as below

(43) Table 2 : Sensitivity indices of the reproduction number R 0 against mentioned parameters.

The sensitivity analysis of R 0 has drawn a lot of attention in various scientific areas. As the parameters of a dynamical model are estimated, it is possible to have some uncertainty about their values utilized to draw conclusions about the proposed system. In order to decrease the spread of the infectious disease, it can be carried out the sensitivity analysis by determining the parameters. Sensitivity analysis is a crucial part of the disease model analysis although computation of it can become exhaustive for complex dynamical systems. For this reason, it is very important to evaluate the effects of each parameter on the spread of the disease and thus find the parameters that have the most important effect on the reduction and spread of the outbreak. Here, we perform the sensitivity analysis by means of the sensitivity index for the parameters of the COVID-19 model. This technique helps to measure the most sensitive parameters inside the system for the reproduction number R 0 . The following formula is employed to calculate the sensitivity index of the reproduction number R 0 of the fractional COVID-19 model presented by the Caputo derivative.

It can be used three methods to compute the sensitivity indices: Direct differentiation, Latin hypercube sampling method, and linearizing system, and then it is solved the obtained set of linear algebraic equations. We utilize the direct differentiation method because it provides analytical expressions for the indices. The indices not only gives us the effect of some aspects associated with the spreading of infectious disease but also gives us crucial information about the comparative change between R 0 and other parameters. So, it helps to reach the control strategies. Table 1 shows that the parameters α, δ and η have a positive influence on the reproduction number R 0 , which describe that the growth or decay of these parameters say by 10 percent will increase or decrease the reproduction number by 10 percent, 0.37 percent and 6.89 percent, respectively. However, on the other hand, the index for parameters µ A , µ I , ρ I , ϕ I and ϕ A show that increasing their values by 10 percent will decrease the values of basic reproduction number R 0 by 2.16 percent, 0.14 percent, 0.227 percent, 0.223 percent and 2.16 percent, respectively. 

Here, we perform the estimation of parameters in the COVID-19 model with the help of the non-linear least squares curve fitting method. All parameters of the underlying model estimated from the reported data in Table 1 will be employed for the numerical simulations in the next section of this paper. The confirmed cases of COVID-19 infected individuals handled in the current study represent those who are tested positive by health care workers. In order to furnish a better fit to the real-life data for the aforementioned system, we obtain the more convenient parameter values by utilizing the least square curve fitting. While the non-linear function fitted contains solving ordinary differential equations employing a numerical integration scheme, the problem is set as a classical non-linear least-squares problem. To carry out the parameters' estimation, the following steps are applied:

• The model of COVID-19 is solved by means of the ODE45 function and Euler's method by guessing initial parameters.

• The solution of the system (5) is compared with the real-life data and an optimization algorithm can be utilized to get the estimated parameter values having a much better fit to the real data.

• The model of COVID-19 is solved by using the new parameter values and the results are compared with real-life data.

The initial values of population employed for parameters' estimation of the model (4) is 414225100, 3207, 595, 563, 227, 3) and also the initial parameter values is p 0 = (α, δ , κ, γ, ψ, η, µ A , µ I , ϕ A , ϕ I , ϕ D , ρ I , ρ D ) = (5.5010e − 9, 0.1000, 1/3, 1/90, 1/7, 0.8800, 1/5, 1/4, 0.1496, 0.0998, 0.1496, 0.0046, 0.0031). The estimated values of parameters is given in Table 3 . We simulated each of the S(t), Q(t), E(t), A(t), I(t), D(t) and R(t) state variables, as shown in Figures 11-14 for χ = 1, 0.95, 0.9, 0.85. These figures show that a shift in the χ value has an effect on the epidemic dynamics. Examining the depicted figures one can see that the outbreak rate is broader and lower for low χ values. This outlook is significant considering the epidemiological viewpoint as its definition indicates a longer period over which the health system may be affected by contaminated individuals.

Herein, the fractional variant of the model under consideration via Caputo fractional operator is numerically simulated with the help of first order convergent numerical techniques as can be seen in [28] [29] [30] . These numerical techniques are accurate, conditionally stable, and convergent for solving arbitrary order linear and non-linear system of ordinary differential equations.

To begin the simulation we go as follows: Consider a general Cauchy problem of fractional order with autonomous nature

where y = (a, b, c, w) ∈ R 4 + is a real-valued continuous vector function satisfying the Lipchitz condition presented by

where M is a positive real Lipchitz constant.

Employing the fractional integral operators, one can get

where J Ω 0,t is the Riemann-Liouville fractional integral. Considering an equi-spaced integration intervals over [0, T ] with the fixed step size h (= 10 −2 for simulation) = T n , n ∈ N. Also, let us soppose that y q be the approximation of y(t) at t = t q for q = 0, 1, . . . n.

The numerical technique for the governing model under Caputo fractional derivative operator takes the form

Now we discuss the obtained numerical outcomes of the governing model in respect of the approximate solutions. To this aim, we employed the effective Euler method under the Caputo fractional operator to do the job. The initial conditions and the estimated parameter values are stated in the immediate section. It is popularly known that the most confusing and standstill affair in the limelight is the subtle characteristics of the COVID-19 pandemic. The deceptive nature of the virus causes scientists, researchers, and medical professionals to constantly analyze the attitudes and attributes to execute the real truth about the virus. Here, by means of some effective numerical scheme, we numerically simulated the model under consideration and physically see how it behaves depending on the scenario.

In figure 3 , the profile for the behavior of the infectious class with (a) Transition rate of exposure to infected class φ and (b) quarantined rate κ under the proposed model with Caputo derivative while considering the increasing and decreasing values of each, have been depicted. While, figure 4 shows the profiles for the behavior of the infectious class with (a) the disease-induced death rate ρ I and (b) the difference for the symptoms and asymptomatic infectious class θ under the proposed model with Caputo derivative while considering the increasing and decreasing values of each, respectively. In both figures 3 and 4, one can see the interacting changes for the infectious class depending on the sensitivity of the parameters. The behavior of each state variable for the Caputo version of the fractional model using the values of the parameters has been depicted in figure 5. One can easily see the decreasing-creasing character of each state variable.

To clearly see the dynamical characteristics with respect to Caputo fractional derivative, we vary the values of α = 1, 0.999.0.988 on each of the state variables. Figure 6 (a) depicts the dynamical behavior for S(t)(susceptible individuals at time t) and 6(b) depicts the dynamical behavior of Q(t)(individuals who are in the quarantine process at time t). One can see that S(t) is strongly decreasing while Q(t) is creasing. In figure 7(a) , we illustrate the dynamic for A(t)(asymptomatic infected individuals at time t) and figure 7(b) shows E(t)(exposed individuals who carry the coronavirus at low-level). In this case, both A(t) and E(t) have a monotonic behavior. In figure 8(a) , we present the dynamic of I(t)(infected individuals with certain symptoms of COVID-19) and figure 8(b) depicts the dynamic of D(t)(the number of medically verified cases in which patients are in quarantine treatment at time t). Monotonic behavior can be seen in this case. Figure 9 represents the outlook of R(t)(recovered individuals at time t). It has an increasing behavior. Figure 19 : The dynamics of E(t) and A(t) for different χ. 

As a summary of the analyzes in this study, the following conclusions can be obtained:

• In this study, we have presented an analysis of the fractional COVID-19 model in detail. In addition to the theoretical calculations for the proposed model, the advantages of the fractional operator have been realized by real data by carrying out numerical simulations.

• In order to analyze the dynamics of the underlying fractional model, we have suggested the fractional-order Caputo operator. Making use of this efficient and advantageous fractional derivative, the existence and uniqueness of the solutions for the fractional COVID-19 model have been explored by means of the fixed point theory and the positiveness of the solutions has been shown.

• We have estimated the parameters of the aforementioned model by using real statistics of the reported cases of the COVID-19 model in Turkey, 2020. We have observed the effect of these estimated parameters on the fractional COVID-19 model by simulating for different values of χ. On the other hand, the model fitting under the estimated parameters has been performed.

• To determine the most sensitive parameters of the proposed fractional model, we have carried out the sensitivity analysis of the basic reproduction number. In this way, the impact of the parameters inside the model under investigation has been evaluated.

• Moreover, iterative solutions of the fractional COVID-19 model have been presented with the help of the Laplace transform of the Caputo fractional derivative. Also, we have furnished the stability criteria for this fractional model under the fixed point theory.

• Finally, the variant of the fractional model under consideration has been simulated by utilizing efficient numerical techniques via Caputo operator in order to observe and grasp the advantages of the novel COVID-19 model with arbitrary order χ. Therefore, the current study indicates that non-local fractional-order models studied in different fields are beneficial because they have the non-integer order derivative that makes the analysis performed stronger especially owing to having the memoryeffect.

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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.