Analysis of fractional COVID-19 epidemic model under Caputo operator.

The article deals with the analysis of the fractional COVID-19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease-free equilibrium using the method of Castillo-Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first-order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.

INTRODUCTION

In December 2019, Wuhan City (China) gains worldwide attention, as an unknown virus starts infected human, resulting in a huge number of casualties coupled with unparalleled financial loss and social disruption. The virus was then recognized as a coronavirus (virus) and is then referred to as an extreme virus (SARS‐CoV‐2). The government tried their best to control the spread of the virus, but it took the world by storm and it got perforated to other countries by different means. Initial emergency control measures were put in place, in addition to the emphasis of the World Health Organization (WHO) on research funding to investigate its causes, get accurate diagnostics, and find a cure for the disease. These factors are still in the evolving stage, and very recently, vaccination efforts have born fruit after successful tests, which lead to the vaccination of chosen groups of population in a developed nation.

The duration of the virus is 11 to 14 days as per initial medical studies. When the comorbid disorders are noticed in patients, aged individuals are particularly susceptible to COVID‐19. The rate of transmission of COVID‐19 is very high, and between 2.2 and 3.58 is the basic reproduction number. That is why it is quickly spreading across the globe and impacting 213 countries. Thus, COVID‐19 was announced as one of the global pandemics by WHO on January 30, 2020.

After China, the country with highly exposed individuals were Iran and Italy. Iran is Pakistan's neighboring country, and thousands of people visit Iran for religious and commercial reasons every year. Some of these individuals were infected while coming back from Iran, and they became the cause of the spread of the virus in Pakistan. In spite of the border closure, the first case in Pakistan was officially stated on February 26, 2020, after the infected person returned from Iran from Karachi. As a result, the government acted swiftly by imposing smart lock down quarantine of the infected persons at home. Mathematical modeling is regarded as an efficient method to explain the infection complex behavior. , , , , For decades, mathematicians have often used mathematical modeling methods for various natural calamities. The study of infectious diseases for the purposes of disease planning and control has been vague. , , , , ,

It is not practical to use a single model to characterize the actual process distribution. Increasing complexity, but at the expense of complicated mathematics, will make models more precise and important. Simple models such as the one described in the current study are also useful where there is no formal vaccine or careful monitoring of treatment.

The generalization of classical calculus is fractional calculus. Fractional derivative mathematical models provide more insights into the disease under consideration. , , , , , In Srivastava et al and Singh, separate fractional operators with singular and nonsingular kernels were proposed. Latest literature and references to previous works , , can be found in the implementations of these fractional operators. Very few COVID‐19 models have recently been proposed based on fractional‐order operators. Bonyah and Zarin found a fractional derivative mathematical model for cancer and hepatitis co‐dynamics and tested its effects. Currently, in this work, inspired by the above discussion, we work the dynamic analysis of the COVID‐19 model given in Mandal et al.

The remainder of the organization of the paper is as follows: Section 2 introduces the model formulation for the fractional‐order derivative. Section 3 includes, for the Caputo model, the presence and uniqueness of solutions, the basic reproductive number, the invariant region, and the points of equilibrium. Section 4 presents the local stability of the proposed model at the corona‐free and corona‐present equilibrium points of the fractional‐order derivative. In Section 5, we review the global stability of the proposed model at the equilibrium points of corona free and corona current. The relevant parameters are calculated in Section 6, and their effect on the basic reproduction number is presented through sensitivity analysis. Section 7 describes the graphical outcomes by biological debate. The work has been summarized in Section 8 with valuable suggestions about the management of diseases in society.

MODEL FORMULATION

Consider the recruitment constant rate to susceptible (S) class be A and to depict the rate of transmission. Now, when S mingles with an exposed (E) ones, the virus gets a significant increase; therefore, one can attain some part of S human with cautious precaution and some part of the E individual with cautious precaution for the transmission of disease.

The portion of the population class E with positive COVID‐19 is also deemed to be contaminated, and they are believed to be hospitalized. Let the parts of the E class which moves to I and Q classes, respectively, be and . Let the parts of communities move to the I level among the Q groups of populations, and the part changes to I.

Surmise and be the R rate of I and E hospitalized infected populations, respectively. Assume d is the normal death rate to all groups of populations and the death rate caused by COVID‐19 is . It is also statistically observed that an individual who recovered from COVID‐19, has a little risk of becoming infected again. We therefore presume that no part of R is going back to the disadvantaged class. We also recognize that the individuals of the hospitalized infected class will not transmit the disease or spread an eligible amount of disease in the formulation of a statistical model since they are separated entirely from the S individuals. Further, sufficient governmental initiatives were introduced by the numerous governmental and nongovernmental organizations to monitor the COVID‐19 pandemic. This policy may therefore be regarded as one of the most powerful control methods, and this policy will primarily support the vulnerable population of COVID‐19 cases.

Following the aforementioned assumptions and using the steps described, the Caputo variant of the model is as comes next: with

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR THE CAPUTO MODEL

We consider and where

Summarizing that and are couple functions, we reach

Taking into account one reaches

Continuing in the same way, one gets where Then, we get together with and . Thus, we attain

It is vital to observe that . Additionally, by using Equations (3.5) and (3.6) and considering that , we reach

If

then (2.1) has a unique solution for .

The , and satisfy the Lipchitz condition. Thus, we obtain

Hence, as . Therefore, we have with by hypothesis. Thus, uniform convergent is achieved. Applying the theorem on limit in Equation (3.8) as affirms the solution of uniqueness (2.1). Hence, the unique solution for Equation (2.1) exists on the basis of (3.10).

Basic reproductive number

Surmise that represents the disease‐free equilibrium point for (2.1) so that

Let ( be the infected compartment. Then, we obtain

Then, we reach Now decomposing the matrix J in terms of F and V, that is , yields and is given by

Invariant region

We define Then, we get When , we reach . We study (2.1) as

Endemic equilibrium point

The endemic equilibrium point for (2.1) is denoted by for which the disease is endemic in the population (i.e., at least one of , and is nonzero). The equations of (2.1) are reset to attain , and . This gives

For , the positivity of the above equilibrium point (3.15) is assured.

LOCAL STABILITY

We establish the local stability of the (2.1) in this section at corona‐free point as well as at corona‐present equilibrium point .

Corona‐free equilibrium local stability

The corona‐free equilibrium (CFE) point of the system (2.1) is locally asymptotically stable if .

Jacobian matrix of the system (2.1) at is where

Then, we obtain where

Therefore, the virus‐free equilibrium point is asymptotically stable.

At corona‐present equilibrium point

Now we study the local asymptotic stability of the endemic equilibrium .

If   , then the corona‐present equilibrium of the system (2.1) is locally asymptotically stable.

Suppose ; then the existence of the endemic equilibrium point is assured. The Jacobian matrix of the system (2.1) at is where

The two eigenvalues of are negative; that is, and . Additionally, we can have Then, we get where

It is observed that here and are all positive for any parametric value. Therefore, (2.1) is locally asymptotically stable around its endemic equilibrium point .

GLOBAL ASYMPTOTIC STABILITY

We use the method of Castillo‐Chavez et al to establish the global stability for disease‐free equilibrium, whereas for the global stability of endemic equilibrium, the generalization of Lyapunov theory is used.

Corona‐free equilibrium global stability

According to Castillo‐Chavez et al, decompose (2.1) into two subsystems and put the fractional parameter equal to one i.e where and are the number of uninfected and infected individuals, respectively. Therefore, and . For disease‐free equilibrium, the existence of global stability depends on the following:

If is globally asymptotically stable.

where for (

where is an M‐matrix having the positive off‐diagonal entries and Δ represents the feasible region. Thus, the following statement holds.

The equilibrium point of the system (2.1) is globally asymptotically stable, if the above conditions are satisfied and .

At corona‐free equilibrium , the model (2.1) is globally asymptotically stable if and unstable otherwise.

We take into consideration , and describe , where

Then, we obtain

For , , and , we get

From Equation (5.4) as , . Thus, is globally asymptotically stable.

To show the second condition, that is,

where for (

we have where . Since from second condition or and

then we can calculate

Thus, is positive definite.

Matrix B is given by

As from the model (2.1), the total population is bounded by , that is, , so which implies that is positive definite. Also from equation (5.8), it is clear that the matrix B is M‐matrix that is the off diagonal element are non‐negative. Thus, Conditions 1 and 2 are satisfied; so by Lemma 1, the disease‐free equilibrium point is globally asymptotically stable.

Endemic equilibrium (global stability)

According to Li and Muldowney, take into account where is connected simply and is a function that . Surmising that depict the solution of (5.9) and to , we consider the following:

A compact absorbing set exists.

Equation (5.9) has a unique equilibrium.

Now for asymptotically global and local stability, consider the following lemma.

Assume that the Bendixson criterion for (5.9) and Conditions 3 and 4 are satisfied, then is globally asymptotically stable in U only if it is stable.

Take into consideration the following:

Equation (5.10) is a matrix valued function on U. Suppose that exist and is continuous for Now describe a quantity , such that where is the second additive compound matrix of J; that is, and . Let be the Lozinski measure of the matrix B with respect to the norm ‖.‖ in presented as follows:

Let U be connected simply and Conditions 3 and 4 are held, then the unique equilibrium of Equation (5.9) is globally asymptotically stable in U, if .

If , then the model (2.1) is globally asymptotically stable at endemic equilibrium and unstable otherwise.

: Consider (2.1), so that

The Jacobian matrix of system (5.13) is where . The second additive compound matrix is The function , and then So, it follows Matrix can be expressed in the matrix form where Suppose ( is a vector in , with the norm ‖.‖ described by and let be the Lozinski measure with respect to this norm; we choose where and | are matrix norms with respect to the vector norm, and refers to the Lozinski measure with respect to this norm; then and Hence, we have

From (2.1), we get Thus, we have

By using we get

Moreover, we get Then which implies . Therefore, the Bendixson criterion is verified. We prove that positive equilibrium ( is globally asymptotically stable.

Examine the subsystem of system (2.1) we rewrite the system in the form

The integrating factors for the system is and , which are used to solve the system. So for , and , which is sufficient to prove that the endemic equilibrium point is globally asymptotically stable.

SENSITIVITY ANALYSIS

Following the methods in Khan et al, the elasticity index along with parameter values have been obtained as shown in Table 1. The sensitivity of various parameters with is depicted in Figures 1, 2, 3, 4, 5.

TABLE 1

Parameters and elasticity indices for

Parameter	S index	Value	Parameter	S index	Value
β	S β	1.00000000	d	S d	−0.4999885937
σ	S σ	−0.7142914285	b 2	S_b_2	−0.3711929652
A	S A	1.00000000	ρ 1	S_ρ_1	−0.7213098196
ρ 2	S_ρ_1	−0.4285717142	α	S α	0.31632629344
M	S M	0.1245906249	p	S p	−0.2446571428

FIGURE 1

M and A versus [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

and d versus [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3

and d versus [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4

M and d versus [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5

and A versus [Colour figure can be viewed at wileyonlinelibrary.com]

NUMERICAL SIMULATIONS AND DISCUSSION

We present the numerical simulations in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 by using Table 2. Consider a general Cauchy problem of fractional order having autonomous nature where is a real‐valued continuous vector function which satisfies the Lipchitz criterion given as where M is a positive real Lipchitz constant.

FIGURE 6

Susceptible individuals [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7

Exposed individuals [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 8

Quarantined individuals [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 9

Infected individuals [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 10

Recovered individuals [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 11

The dynamics of susceptible class for different values [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 12

The dynamics of exposed class for different values [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 13

The dynamics of quarantined class for different values [Colour figure can be viewed at wileyonlinelibrary.com]

TABLE 2

Parameters used in model (2.1) and their values

Parameter	Description	Values/range
A	Total recruitment	50
β	Disease transmission rate	[0.5, 2.3]
ρ 1	Portion of S contact with E	(0,1)
ρ 2	Portion of E contact with S	(0,1)
d	Natural death rate	0.2
b 1	The rate that Q becomes S	0.25
b 2	The rate that E becomes quarantine	0.8
α	The rate that E becomes I	0.3
η	The rate that I becomes R naturally	0.25
σ	The rate that E becomes R naturally	0.2
c	The rate that Q becomes I	0.12
δ	The mortality rate for I	0.25
M	Policy parameter	0.8
p	Implementation rate of policy	0.78

Using the fractional‐order integral operators, one obtains where is the fractional‐order integral operator in Riemann–Liouville. Consider an equispaced integration intervals over [0,  with the fixed step size . Suppose that is the approximation of at for Consequently, our model becomes

The initial conditions and the parameters' values are used as described in the table above. The physical perspective of the model's individual state variables under the Caputo fractional operator is shown in Figures 6, 7, 8, 9, 10. Since the symptomatic individuals are assumed to be more infectious than the asymptomatic, the transmission of COVID‐19 through contacts in households, workplaces, schools, from foodstuffs, or during commute rises. This leads to a surge of the virus in environments such as workplace, school, foodstuffs, and public transport, and consequently, more cases of the coronavirus are confirmed. In Figures 11, 12, 13, 14, 15, it can be noticed that the fractional‐order is varied for 1, 0.888, 0.666. One can easily see the robust nature of the Caputo operator than the integer variant of the model. For decreasing varying values of (disease transmission rate) as shown in Figures 16 and 17, is also increasing. Similarly, in Figures 18 and 19, the effect of mortality rate on has been shown. For increasing values of as shown in Figure 18, an increasing pattern in is noticed. Similarly, for decreasing values of as shown in Figure 19, a decreasing pattern in is noticed. Increasing–decreasing patterns are shown in this case. The model shows that quarantined policy is needed to avoid a large COVID‐19 epidemic. There is a growing concern that this disease could continue to ravage the human population globally since many aspects of the COVID‐19 are yet to be discovered, which also poses a challenge to the long‐term mathematical modeling of the disease.

FIGURE 14

The dynamics of infected class for different values [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 15

The dynamics of recovered class for different values [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 16

Behavior of the Infectious class for decreasing values of [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 17

Behavior of the Infectious class for increasing values of [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 18

Behavior of the infectious class for increasing values of [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 19

Behavior of the infectious class for decreasing values of [Colour figure can be viewed at wileyonlinelibrary.com]

CONCLUSION

The fractional derivative in Caputo sense is deployed to analyze the COVID‐19 epidemic model. Local and global stability of the model along with the existence and positivity of the solution is studied. The sensitivity of different parameters corresponding to basic reproduction number has been highlighted graphically. The present finding shows that the effects of our model's predictions are in line with those reported in the literature and are therefore estimated to be more acceptable values of the parameters. At different stages, we varied the related parameters to their baseline values and the graphical results for the separate values of the fractional parameter of the Caputo derivative. The decrease in successful contacts (up to 35 %) and an increase in contact‐tracing policy to quarantine the exposed individuals (up to 45%) to their baseline value have been found to significantly decrease the peak of contaminated curves. Furthermore, for smaller values of the fractional parameter , the reduction in the infected population becomes more important. Thus, we conclude that the findings obtained are accurate, practical, and more biologically feasible for the fractional scenario.