Modeling and Dynamics of the Fractional Order SARS-CoV-2 Epidemiological Model

Khan, Tahir; Ullah, Roman; Yousef, Ali; Zaman, Gul; Al-Mdallal, Qasem M.; Alraey, Yasser

doi:https://doi.org/10.1155/2022/3846904

Complexity

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Abstract Introduction Numerical Simulation Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Applications of Delay Differential Equations in Biological Systems 2021

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Research Article | Open Access

Volume 2022 | Article ID 3846904 | https://doi.org/10.1155/2022/3846904

Modeling and Dynamics of the Fractional Order SARS-CoV-2 Epidemiological Model

Tahir Khan,^1,2Roman Ullah,²Ali Yousef,³Gul Zaman,⁴Qasem M. Al-Mdallal,⁵and Yasser Alraey⁶

Academic Editor: M. De Aguiar

Received12 Feb 2022

Revised03 Jul 2022

Accepted26 Jul 2022

Published27 Sept 2022

Abstract

We propose a theoretical study to investigate the spread of the SARS-CoV-2 virus, reported in Wuhan, China. We develop a mathematical model based on the characteristic of the disease and then use fractional calculus to fractionalize it. We use the Caputo-Fabrizio operator for this purpose. We prove that the considered model has positive and bounded solutions. We calculate the threshold quantity of the proposed model and discuss its sensitivity analysis to find the role of every epidemic parameter and the relative impact on disease transmission. The threshold quantity (reproductive number) is used to discuss the steady states of the proposed model and to find that the proposed epidemic model is stable asymptotically under some constraints. Both the global and local properties of the proposed model will be performed with the help of the mean value theorem, Barbalat’s lemma, and linearization. To support our analytical findings, we draw some numerical simulations to verify with graphical representations.

1. Introduction

The coronavirus family causes infections in humans, beginning with a common cold and progressing to SARS. Two coronavirus epidemics have been recorded in the preceding twenty years [1–3]. SARS was one of them, and it created a large-scale outbreak in several nations. Approximately, 8000 people were affected by this outbreak, with 800 of them dying. In December 2019, a serious respiratory sickness outbreak began in Wuhan, China [4]. In early January 2020, the causal agent, a new coronavirus, was identified and isolated from a single patient (COVID-19). According to scientific evidence, animals were the earliest source of virus transmission, although the majority of cases are caused by infected humans contacting susceptible humans. The spread of this virus is a hot topic that has touched practically every corner of the globe and has been reported in over 200 countries. According to current records, there have been over 401,288,380 confirmed cases, with 5,783,182 deaths occurring till February 9th 2022. According to the WHO, this is a public health emergency. The World Health Organization (WHO) has designated it a public health emergency of worldwide concern due to the severity of the condition. This virus appears to be highly contagious, spreading rapidly to nearly every country on the planet, prompting the declaration of a global pandemic. It signifies that it is a highly major public health threat, with symptoms such as cough, fever, lethargy, and breathing problems after infection.

Fractional computing is an emerging area of mathematics and attracted the attention of researchers. Because of the wide applications to express the axioms of heritage and recall different physical situations that occur in various fields of applied science. Many classical models have been shown with less accuracy in prediction about the temporal dynamics of the disease, while on the other hand, models with noninteger order provide better information in allocating and preserving data for large-scale analysis [5–7]. Moreover, the derivative of integer order does not find the dynamics between two various points [8, 9]. Furthermore, the comparison of integer and noninteger order epidemic models reveals that models with noninteger order are the generalization of integer order and provide more and accurate dynamics rather than the classical order, (for detail see [9–11]). A model with noninteger order demonstrating the complex dynamics of a biological system has been proposed by Asma et al. [12]. A fractional-order epidemic model has been investigated to explore the dynamics of toxoplasmosis in feline and human populations [13]. Another study has been reported and studied the stability analysis of pests in tea with fractional order [14]. Similarly, many authors studied dynamics of infectious diseases with fractional-order derivatives, for e.g., Hadamard and Caputo and Rieman and Liouville [15–19]. For the solution of Caputofractional order, epidemiological models of many iterative and numerical methods have been developed; however, the complications of singular kernel arise. So, Caputo and Fabrizio presented an idea based on the nonsingular kernel to overcome the limitation that arises in the above fractional-order derivatives [20].

Coronavirus disease 2019 (COVID-19) is one of the top infectious diseases among other ones and therefore has been recognized as a global threat by World Health Organization (WHO). Due to novel characteristics of the coronavirus disease various researchers have taken a keen interest. Several researchers formulated various epidemiological models to study the dynamics of communicable diseases (see for instance [21–25]). The current pandemic of novel coronavirus disease is also a burning issue and many biologists and mathematicians reported different studies. For example, Wu et al. introduced a model to describe the transmission of the disease based on reported data from 31.12.2019 to 28.01.2020 [26]. Imai et al. studied the transmission of the disease with the help of computational modeling to estimate the disease outbreak in Wuhan, whose main focus was on the human-to-human transmission [27]. Another study has been investigated by Zhu et al. [28] to analyze the infectivity of the novel coronavirus. The dynamical analysis of the novel disease of COVID-19 under the effect of the carrier with environmental contamination has been performed by Hattaf et al. [29]. All the reported studies indicate that bats and minks may be two animal hosts of the novel coronavirus. Similarly, many more studies have been reported on the dynamics of a novel coronavirus, for instance, see [30, 31]. Nevertheless, the literature reveals that the work proposed is an excellent contribution, however, it could be possible to improve further by incorporating some interesting and important factors related to the novel coronavirus disease.

The spreading of coronavirus disease globally rises from the human-to-human transmission, while the initial source of the disease was an animal/reservoir. The characteristic of SARS-CoV-2 confirms that various infection phases are significant and affect the transmission. The role of asymptomatic is notable because with no symptoms it becomes the major source of transmission of the infection. So, a small number of this population leads to a big disaster. We develop a mathematical model according to the novel disease of coronavirus and keeping in view the aesthetic of the virus. To do this, first, we formulate the model and then fractionalize it to perform the fractional type analysis of the proposed model. The fractional derivative used in this study is a particular case of the new generalized Hattaf fractional (GHF) derivative [32, 33]. We show that the proposed fractional-order epidemiological model is bounded and possesses positive solutions. We also find the steady states of the epidemic problem and discuss asymptotic stabilities. For this, we use the dynamical systems theory. Particularly, we utilize the linearization, mean value theorem, and Barbalat’s Lemma. Moreover, the sensitivity analysis will be performed for the threshold parameter to find the impact of each epidemic parameter involved in the model mechanism. We use the sensitivity index formula for this purpose. We perform the numerical visualization of the analytical results to verify the theocratical part and show the effectiveness of the control strategy. We also show the difference between integer and noninteger order epidemiological cases.

2. Formulation of the Model with Fractional Analysis

The proposed problem is formulated by taking into account the characteristics of the novel coronavirus illness. We divide the total human population into four various compartments and assume that represents the reservoir. In the proposed study, we also consider several transmission routes, such as from human-to-human and from a reservoir-to-human. Before we show the model, we make the following assumption:(i)The parameters and variables involved in the model are positive or non-negative values(ii)The inflow of newborn are susceptible(iii)The novel disease is transmitted by several routes, such as from latent and symptomatic individuals as well as from reservoirs and so accordingly incorporated.(iv)Those who have a strong immune system got natural recovery(v)Two types of recoveries i.e., from latent and symptomatic populations are taken(vi)The death rate due to disease is taken in the symptomatic infected compartment

As a result of combining all the above assumptions, the following system of nonlinear differential equations emerges:

and the initial population sizes are assumed to be as follows:

In the proposed epidemiological model, the parameters described as is the new birth rate, and the disease transmission rates are symbolized by, , , and , which represent the transmission from latent, symptomatic, and reservoir, respectively. We also denote the reduced transmission coefficient by and , while is the moving ratio of latent to infected and denotes the recovery rate. We also denote the recovery rate under treatment by , while is the natural mortality. The disease-induced rate is . Furthermore, and are the two ratios that contribute production of the virus in the seafood market. We denote the removing rate of the virus with .

2.1. Fractional-Order Epidemiological Model

Let be the fractional-order parameter . We will extend the model to its associate fractional order. First, we give some fundamental concepts that will be used in getting our findings.

Definition 1. (see [9]). Let and assume that , if and such that , then the derivative in the sense of Caputo as well as the Caputo-Fabrizio with order are given as follows:where CF and C are used for the representation of Caputo-Fabrizio and Caputo, respectively, while and are the normalization function, and .

Definition 2. see [9]). (If and varies with time , then the integral is described as follows:The above integral is known as the Riemann–Liouville integral.The integral defined by Equation (5) is said to be the Caputo-Fabrizio-Caputo (CF) integral.
Since is the fractional order, and using the notion and for the shake of simplicity, therefore the fractional order model looks like the following equation:We show that the proposed fractional-order epidemic model as reported by the above system is both biologically and mathematically feasible. For this, we discuss the positivity and boundedness of the model (6), which proves that the underconsidered problem is well-possed. We also investigate that the dynamics of the proposed model are confined to a certain region invariant positively. The following Lemmas is established for this purpose.

Lemma 1. Since are the proposed model (6) solutions and let us consider that it possessing non-negative initial sizes of population, then are non-negative for all .

Proof. Since, is the fractional order and assuming that represents the fractional operator with order , then system (1) leads toThis implies thatwhere are in and , respectively. Following the methodology proposed in [34] and consequently used by Qureshi et al. [35], we reach to the conclusion that the solutions are non-negative for all non-negative .

Lemma 2. Let us assume that the is the feasible region of the model (6), then within it, the model that is under consideration is invariant and the feasible region is given by

Proof. Let represent the total human population, then the use of the proposed fractional model leads to the assertion is given by the following equation:Solving equation (10), we get the following equation:It could be also noted that , so the last equation of the fractional model (6) looks like the following equation:The solution of (12) leads to the following equation:In (11) and (13), denotes the Mittag-Leffler function and . Furthermore, it is obvious that when times grows without bound then (11) and (13) gives that and . Thus, if and , then and for every , while if and , then and contained in and will never leave. So, the dynamics of the fractional epidemic model can be investigated in feasible region . □

3. Stability Analysis

In this section, we will examine the stability of fractional epidemiological model (6). We find the steady states first and threshold parameter (basic reproductive number) of the fractional model to investigate the stability conditions. We use the notion for disease-free equilibrium calculating at steady state with . It is easily stated that the component of looks like and . We now use the disease-free state and find the threshold parameter. This quantity represents the maximum epidemic potential of a pathogen, which describes what would happen if an infectious agent were to enter a susceptible community, and therefore is an estimate based on an idealized scenario. The effective threshold quantity depends on the nature of the population’s current susceptibility. This measure the potential transmission, which is likely lower than the basic reproduction number, depends on various factors e.g., whether some individuals have immunity due to prior exposure to the pathogen or whether some individuals are vaccinated against the disease. Therefore, this quantity is effective and changes over time and is an estimate based on a more realistic situation within the population. We calculate this quantity i.e., the threshold quantity of the proposed model by following [36], therefore following the next generation matrix approach, we calculate the associated matrices i.e., and as given by the following equation:

The associated threshold quantity of model (6) is the spectral radius of the matrix is as , where

It could be noted that the threshold quantity consists of three parts that describe various transmission routes. One may observe, that there is a transmission from an infected human, while the other from reservoirs.

Similarly, we use the above quantity and assume that is the endemic equilibrium of the fractional order model, then the components are calculated by solving system (6) simultaneously at steady state. We also set , , , and for the sake of convenience, then the corresponding endemic equilibrium leads to , whoso components are defined by the following equation:

The endemic equilibrium reveals that exists only if . For this, we sate the following result.

Lemma 3. The endemic equilibrium for the proposed problem (6) exists only whenever, the threshold quantity is greater than unity.
We use the linear stability analysis to discuss the temporal dynamics of the fractional model (6) around and . So we have the following results.

Theorem 1. If the threshold quantity is less than unity, then the local, as well as global dynamics of the problem, is asymptotically stable around .

Proof. Following Theorem 3 reported in [37] to obtain the required results. Since it is clear that all other compartments of the proposed model do not depend explicitly on the recovered class, so we study the dynamics of the model for only the three-compartment, which will be enough for the whole model. Let be the Jacobian matrix of the proposed model (6) around ; then,The calculation shows that , obviously has two negative eigenvalue i.e., and . To find the nature of the remaining, we take the matrix given by the following equation:It is sufficient for the Routh–Hurwitz criteria that : , and holds. We calculate the and , such thatIt can be noted from the above equations (19)–(20) that trace(A(X1))<0 and det(A(X1))>0, if R1+R2<1.. So the Routh–Hurwitz criteria are satisfied if . It proves the conclusion that the local dynamics of the model (6) is asymptotically stable, if . □
The application of linear stability analysis is utilized to find the dynamics of the proposed model (6) around its associated endemic equilibrium (16). For this, we describe the result as follows.

Theorem 2. If the threshold quantity is greater than unity i.e., , then the local as well as the global dynamics of the endemic equilibrium, is asymptotically stable.

Proof. Using the theory of a dynamical system, we discuss the local dynamics of the proposed system around the endemic equilibrium. Let be the Jacobian matrix of system (6) around ; then,We find the characteristic polynomial of the matrix (21), such thatwhereEigenvalues (roots) of (22) are negative or having negative real parts, if : , and holds, whereIt could be noted, from the above equations, that obviously holds, if . Thus, we conclude that the local dynamics of the proposed model (6) at endemic equilibrium is asymptotically stable, if the threshold quantity is greater than unity.

Theorem 3. If , then the point of the model (6) is globally asymptotically stable, while the same holds for whenever .

Proof. To perform the global analysis of the proposed fractional order epidemiological model, first we assume that and if , then it has finite limit, therefore by following the result 3.1 in [38], then from first equation of the model (7), we may write the following equation:Since for every , so by following the result Theorem 1 in [38] with the application of mean value theorem, the above (25) may leads to the following equation:where , and . Consequently, we may derive the following equation:Similarly, of , , , and can be proved. We also assume thatThus, in the light of mean value theorem, there exists constants , such thatSo, Theorem 2.1 and 1 in [39] implies that is uniformly continuous. Thus, the application of Barbalat’s Lemma (see for detail, [40]) gives the following equation:Consequently,Therefore, is the equilibrium of model (6) and by the similar procedure as adopted in [41], we reach to the following equation:So, it could be concluded that the disease endemic state does not exists if , and so whenever approaches and if then , and as approaches , while on the other hand whenever , then exists and thus as tend to .

4. Numerical Simulation

In this section, we present the numerical simulation of the proposed epidemic problem. We divided the section into two subsections in which we discuss the sensitivity analysis of every epidemic parameter and its relative impact on disease transmission. We also discuss the temporal dynamics for the long run and present the significance of fractional parameters.

4.1. Sensitivity Analysis

We discuss the local sensitivity analysis of the model parameters to define the relation between threshold quantity and the epidemic parameters. This allows us to measure the relative impact of every epidemic parameter on disease transmission. We follow the work presented in [42] to perform the sensitivity analysis. Using the sensitivity index formula, we get the sensitivity indices as given in Table 1. It may be noted from the sensitivity indices that the set of parameters has a direct relation with threshold quantity, which means that increase in the value of these parameters causes an increase in the value of the threshold quantity of the model. On the other hand, there is an inverse relationship between the set of parameters and so an increase in these parameters will cause a decrease in the value of threshold quantity. The highest sensitivity index parameter is having the sensitivity index 0.999999, which means that an increase in the value of this parameter say by 10% would increase the value of the threshold quantity by 9.99999%, as shown in Figure 1(a). Similarly the collectively impact of other parameters of is approximately 9.89%, if their values increase or decrease by 10%, as shown in Figures 1(a) and 1(b). Moreover, the parameters of have a negative relation and therefore its collectively impact is 9.94% whenever the value of the parameters given in S2 increases or decreases by 10 % (see Figures 1(c) and Figures 1(d)). More precisely, if the value of the parameters of are increased by 10%, the basic reproductive number will be decreased by 9.94%, while if one decreases the value of the parameters, the threshold quantity will be increased by 9.94%.

(a)

(b)

(c)

(d)

4.2. Verification of Stability Results

We find out the numerical simulation to verify the theocratical work carried out for the fractional-order SARS-CoV-2 transmission epidemiological model (6). To show the validity of the analytical findings we present the large-scale simulation. There are not many choices like the traditional numerical methods to choose various schemes for the numerical simulation of fractional-order models [43], therefore extensive attention is required to formulate new and convenient techniques for the simulation of fractional models. We follow a numerical scheme formulated in [18, 44]. We assume the time step for integration with the simulation interval , and . We also assume that , therefore the discretization for the proposed model looks like the following equation:

Furthermore, we chose the value of epidemic parameters biologically while the initial population sizes are assumed to be non-negative values . We use the MATLAB software package to execute the model for numerical simulations. We justify the stabilities results to show the dynamics of the disease-free and endemic states as given in Figures 2 and 3. This investigates the graphical verification of the dynamics of the considered problem around disease-free state . Besides from a mathematical point of view, the biological interpretation reveals that whenever the value of the threshold parameter is less than unity, each solution curve of will tend to its equilibrium position as shown in Figure 2(a). This shows that there will be always a susceptible population. Moreover, the dynamics of the other compartments around the disease-free state are depicted in Figures 2(b)–2(d), and which describe that the solution curves will tend to the associated equilibrium position and remain stable. So it could be noted that the elimination of the contagious disease of the novel coronavirus from the community depends on the value of , and the disease could be easily eliminated if . Furthermore, the dynamics of the fractional-order model around endemic equilibrium are shown in Figures 3(a)–3(e), which respectively show the temporal dynamics of susceptible, latent, infected, recovered, and reservoirs. From these results, we observed that if proper control measures are not adopted the disease will attain the endemic position. It is clear that the susceptible population decreases from the beginning and then has no effect after some time and so becomes stable as shown in Figure 3(a). The dynamics of the latent population state that there will be a sudden increase in the initial period of infection, while then decreases after some unit of time and become stable, as shown in Figure 3(b), which verifies that there will be always latent population. Similarly, the dynamics of the infected population are shown in Figure 3(c). This reveals that the infected ratio increases day by day and reaches its endemic position in a few units of time. Furthermore, the simulation of the model for recovered population and reservoir is given in Figures 3(d)–3(e). All these results suggest that if no proper control measure is implemented, the disease will attain its endemic position whenever the value of the threshold quantity is greater than unity.

(a)

(b)

(c)

(d)

(e)

(a)

(b)

(c)

(d)

(e)

We also show the significance of the fractional-order via disease transmission as shown in Figures 4–8, which respectively visualizes the temporal dynamics of , , , , and . Particularly, the temporal dynamics of the susceptible are shown in Figure 4. We noted a significant impact of the fractional order on the transmission dynamics of susceptible individuals that if the fractional parameter increases than the number of susceptible individuals decreases as shown in Figure 4. The long run of the latent, infected, and the recovered population for various orders of fractional order are presented in Figures 5–7. We noted that there is a strong influence of on disease transmission. The temporal dynamics of the reservoir are presented in Figure 8. Thus, we investigate that the CF model gives more accurate dynamics of the disease and provides valuable outputs instead of classical models.

5. Conclusion

We investigated the dynamics of SARS-CoV-2 with asymptomatic, symptomatic, and quarantined individuals using an epidemic model. First, the formulation of the model is proposed, and then consequently fractionalized due to the increasing development in fractional calculus. Particularly, we used the well-known Caputo-Fabrizio operator for the said purposes. Both the biological and mathematical feasibilities are discussed in detail for the proposed model and proved that the problem is well-passed. We also calculated the threshold parameter and performed stabilities of the fractional model. The detailed sensitivity is also discussed and quantified the role of every epidemic parameter and its relative impact on the disease transmission. We showed that the proposed model is stable in both local and global sense. Finally, we gave some graphical representations and showed the validations of the obtained results. We also presented the relative impact of the fractional parameter on the various groups of the compartmental populations graphically and proved that the major outcome of the reported work is that the fractional-order CF epidemic models are more appropriate and the best choice rather than the classical order. We believe that the findings of this work will be helpful for the audience working in the field of mathematical epidemiology.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to acknowledge and express their gratitude to the United Arab Emirates University, Al Ain, UAE, for providing financial support with Grant No. 12S086. In addition, the authors are thankful for the financial support of King Khalid University, Institute for Research and Counseling Studies Administration of Contracts, under grant 24-S-2020.

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Copyright © 2022 Tahir Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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