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Computers, Materials & Continua
DOI:10.32604/cmc.2020.012060
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Article

Qualitative Analysis of a Fractional Pandemic Spread Model of the Novel Coronavirus (COVID-19)

Ali Yousef1,*, Fatma Bozkurt1,2 and Thabet Abdeljawad3,4,5

1Department of Mathematics, Kuwait College of Science and Technology, 27235, Kuwait
2Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey
3Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
4Department of Medical Research, China Medical University, Taichung, 40402, Taiwan
5Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
*Corresponding Author: Ali Yousef. Email: a.yousef@kcst.edu.kw
Received: 12 June 2020; Accepted: 03 September 2020

Abstract: In this study, we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal (natural host) to humans. We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one, and from the intermediate one to the human host. At the same time, we focus on the potential spillover of bat-borne coronaviruses. We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria. Moreover, we analyze the existence and uniqueness of the constructed initial value problem. We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t. Furthermore, the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions. Finally, we conduct a series of numerical simulations to enhance the theoretical findings.

Keywords: Allee Effect; coronavirus; fractional-order differential equations; local stability; Neimark–Sacker bifurcation

1  Introduction

In the last few months, nature has showed its laws in establishing the environment of the 21st century. It is out of our primary objective whether the coronavirus (COVID-19) is used as a biological weapon or not. The main point is now that humans are fighting against something to survive that has a genome size of 27 to 34 kilobases. Coronaviruses are members of the sub-family coronavirinae in the family coronaviridae and the order Nidovirales [1,2]. They show four genera, which are given in Tab. 1.

Table 1: Genera of COVID-19 and the pathogenic class

images

The natural host of SARS-CoV, MERS-CoV, HCoV-NL63, and HCoV-229e are bats, while HCoV-OC43 and HKU1 have originated from rodents [3,4]. In the spread of transmission, domestic animals have only intermediate host role from the natural host to the human one. Covid-19 was not considered as highly pathogenic, until the outbreak of SARS-CoV in 2002 and MERS-CoV in 2012. The spread of SARS-CoV in China (Guangdong) showed a COVID-19 that was transmitted from bats to an intermediate host, like market civets from which the transmission spreads to the human host. At the same time, the outbreak of MERS-CoV in the Middle East Countries also came from bats to dromedary camels as an intermediate host, and from the dromedary camels to humans [58]. These viruses cause respiratory and intestinal infections, with symptoms including fever, dizziness, and cough. In December 2019, a novel Coronaviridae was reported in China (Wuhan). The outbreak was associated again with intermediate hosts like reptilians, while the natural host was assumed as bats. This virus was designated later as Covid-19 by the WHO.

Covid-19 was characterized by two members of β-coronavirus; the human-origin coronavirus (SARS-CoV Tor2) and bat-origin coronavirus (bat-SL-CoVZC45). Intensive studies show that it was most closely related to the bat-origin coronavirus [9]. Thus, the primary assumption formed was that the natural host of Covid-19 spreads by infected bats of genus Rhinolophus that are mainly in the area of Shatan River Valley.

Domestic animals, like snakes in that area, were hunted for the food market in Wuhan, which played an intermediate host role in the transmission. Finally, this virus spillover from the intermediate hosts to cause several diseases in human. A virus that started with an endemic pathogenic behavior in China (Wuhan) reaches somehow to a pandemic point worldwide with the infection from human-to-human.

2  The Model Description

It has been realized that the dynamics of many biological and medical phenomena can be characterized via mathematical models. Over the years, many models are formulated mathematically to analyze events in biological and medicine such as infections, treatments, or environmental phenomena [1013]. The study of these phenomena has been restricted to models of integer-order differential equations (IDEs). However, it is seen that many problems in biology, as well as in other fields like engineering, finance, and economics, can be successfully formulated by the so-called fractional-order differential equations (FDEs); see, for instance, the papers [1420]. The nonlocal property of models of FDEs is not only depending on the current state but also provides an adequate description for the historical ones. It is evidenced that FDEs can model certain phenomena that cannot be modeled by IDEs. Thus, FDEs are mainly used on biological models since they are relevant to systems with memory and hereditary [2127].

In this paper, we establish a model that describes the pandemic infection, which occurs when the virus is transmitted from the human body to the intermediate host and continues to spread from human-to-human. The model consists of five fractional differential equations. The first three equations show an SI (susceptible-infected) model to explain the transmission from human-to-human, where images is the susceptible class, images is the infected type that does not know they are infected because of the late occurred symptoms of COVID-19 and images shows the infected class that knows they are infected. The spillover from the intermediate infected class images to the human host imagesdenotes a predator-prey mathematical model, while for the transmission from the natural host images, which is the bat population, to intermediate host images is a host-parasite model of Holling Type II.

Indeed, the mathematical model of this biological phenomena has the form:

images

where

images

represents the Holling type II function and all the parameters of the model (1) belong to images and images.

The susceptible images is composed of individuals that have not contacted the infection but can get infected through contacts from the human that does not know they are infected and from the intermediate hosts. The parameter images is the population growth rate of the susceptible population and images denotes the logistic rate.images is a rate of the susceptible population per year. The susceptible lost their class following contacts with infectives images and the intermediate host images at a rate images and images, respectively. images links the parameter of the interaction between the hunted images class and the predator images population.

The images class does not know that they have COVID-19. In this equation, images is the population growth rate of the class, while images is the logistic rate. The population of this class decreases after screening at a rate images and be aware of the infection. Another possibility is that after the S-images contact, the symptoms occur in early stages so that both classes noticed that they are infected, which is given with the rate images. The intermediate host infected group could also show early symptoms to be aware of the infection, which is provided by a rate of images The logistic rate of images is denoted as images.

images is the domestic animal as an intermediate class in the corona transmission spread. images is the intrinsic growth rate of the population, while images is the logistic rate. images shows the effect on the hunted images during the interaction between the intermediate host and susceptible class. images denotes the predation rate in the host-parasite scheme.

images represents the natural host (bat population) of COVID-19 in this dynamic system. images is the intrinsic growth rate and images is the logistic rate of the population. images shows the conversion factor of the natural host. images is the attack rate of the bat population to infect the images, while images represents the fraction of the potential infectivity of the natural host. images is the rate of average time spend on infecting the domestic intermediate class, which is also known as the handling time.

Tab. 2 shows description of the parameters that are given in system (1).

Table 2: Description of the parameters

images

images

Definition 2.1 Podlubny [25] The fractional integral of order images of a function images is given by

images

defined on images

Definition 2.2. Podlubny [25] Let images be a continuous function. The Caputo fractional derivative of order images is given by

images

Definition 2.3. Podlubny [25] The function

images

with images being the set of complex numbers is called the Mittag–Leffler function of one parameter.

3  Stability Analysis of the Co-Existing Critical Point

Consider the model

images

To analyze the stability of model (6), we perturb the equilibrium point by adding images that is,

images

Thus, we have

and

Thus, we obtain a linearized system about the equilibrium point of the form

images

where images. Moreover, J is the Jacobian matrix at the equilibrium:

images

where the co-existing equilibrium point is images. Then, we have images, where C is given by

images

and images are the eigenvalues and B the eigenvectors of J. Therefore, we get

images

whose solutions are given by Mittag–Leffler functions

images

images

images

images

and

images

By using the result of [28], if images then images are decreasing and therefore we conclude that images are decreasing. Let images be the solution of Eq. (8). If the solution of Eq. (8) is increasing, then images is unstable and if images is decreasing, then images is locally asymptotically stable.

Evaluating the Jacobian matrix (9) for the co-existing equilibrium point images we obtain

images

where

images, images, images images,

images, images,

images, images, images

images, images,

and

images, images.

The characteristic equation of the matrix (17) is given as

images

and

images

if

images

From Eq. (18), we have two quadratic equations, which are

images or

images

and

images or

images

where images and images images is the basic reproduction number, which represents the transmission potential of images class, while images shows the transmission potential of the intermediate-natural host classes images For the following theorems in this section, we consider the case, where both images and images which hold for the following statements:

(i) images,

(ii) images,

(iii) images

(iv) images and images

(v) images and images.

Theorem 3.1. Let images be the co-existing critical point of system (6) and assume that (i)–(iv) hold such that images and images Moreover, let images and images. If

images and images,

where

images,

then all roots of Eq. (18) are real or complex conjugates with negative real parts and images is equivalent to the Routh–Hurwitz criteria. This implies that images is locally asymptotically stable.

Proof. Let us consider the case for images to have eigenvalues with negative real parts. Thus, we have

images

and

images

From (ii) and Eq. (24), we obtain

images

if

images

where images.

In considering both (iii) and Eq. (26), we get

images

where images Moreover, the discriminant of Eq. (21) is, in this case, positive.

Let us consider now the case for images to have eigenvalues with negative real parts. Thus, from

images

we obtain

images

and

images

From (v) and Eqs. (28)(29), we obtain

images for images

and

images for images

Since the discriminant of Eq. (22) is positive, the proof is complete.

Remark 3.1. Theorem 3.1. shows that among the human hosts, those who do not know they are infected, are the control class in the spread. In contrast, between the animal hosts, the intermediate class plays a dominant role, since that one has the essential role in transmitting from animal to human. The transmission potential for both images and images are images and images. Moreover, the susceptible class and the images class is stable based on two parameters, which are the awareness of the symptoms and the screening rate.

Theorem 3.2. Let images be the co-existing critical point of system (6) and assume that (i)–(iv) hold such that images and images Furthermore, let images and images If

images and images and the ratio between the susceptible and intermediate host is given by images, where

and

Then all roots of Eq. (18) are complex conjugates with positive real parts, which implies that images is locally asymptotically stable.

Proof. Let us consider the case for images to have eigenvalues with positive real parts. This holds if

images

and

images

From (ii) and Eq. (32) we obtain

images

if

images

where images. In considering both (iii) and Eq. (34), we obtain

images ,

where

images

Additionally, we get images since images where

Similarly, let us consider the case for images to have eigenvalues with positive real parts. From

images

we obtain

images

and

images

From (v) and Eqs. (37)–(38) we have

images for images

and

images for images

Moreover, we get images since images where

images

This completes the proof.images

Remark 3.2. In Theorem 3.2., we emphasize that class images should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the images class to stop the outbreak. For the susceptible class, it is more important to keep the population rate per year non-infected. The transmission of the virus to the offspring would reach an uncontrollable phenomenon worldwide.

Theorem 3.3. Let images be the co-existing critical point of system (6) and assume that (i)–(iv) hold such that images and images (i) Let images, images, images and

images.

If

imagesand images

where

then the images class represents real or complex conjugates with negative real parts, while the images class shows complex conjugates with positive real parts.

(ii) Let images, images and

images .

If

images, images and the ratio between the susceptible and intermediate host is given by images where

then the images class represents complex conjugates with positive real parts, while the images class shows real or complex conjugates with negative real parts.

Example. In this part, we present numerical simulations that are in good agreement with our theoretical results. We assume the initial conditions of the system (1) as images and images.

In Fig. 1 the blue graph denotes the susceptible class images and the red graph shows images who does not know they are infected. Fig. 1 represents the transmission of the infection that occurs as an epidemic case in some areas, but it spreads intensively to a pandemic case and covers almost the susceptible class. Here we want to emphasize the point of screening, where we assume that about images do testing in the hospitals before the symptoms appear. Additionally, we consider that the symptoms appear late, and thus the awareness of the infection is also at images. This changes the endemic spread from epidemic to an uncontrolled pandemic form.

images

Figure 1: Spread of the images class and effect on the susceptible images class, where images and images = images

In Fig. 2, we keep the screening parameter as images, while we consider the case that the people become aware of the virus and the symptoms of it through media and health organizations. An organized and constant information flood from media might increase the awareness up to images = images.

images

Figure 2: Spread of the images class and effect on the susceptible images class, where images and images = images

This awareness of the people through media and health organizations let them go to hospitals for screening so that the class who does not know they are infected decreases. Fig. 3 shows the effect of the testing when it reaches to %5. The spread is under control and returns to an epidemic form.

images

Figure 3: Spread of the imagesclass and effect on the susceptible images class, where images and images = images

We considered in these examples the infection from human-to-human since the pandemic case reaches from the human transmission. We want to emphasize the strong coordination between health organizations and the media which is an essential tool for two critical parameters, which are images and images (images)

The design of nature keeps the natural host and intermediate host in a stable dynamical system in the habitat. The intermediate host had only a transmission role from animal to human, while the main spread happens through human to human from the images class who does not know they are infected.

4  Existence and Uniqueness of the Initial Value Fractional-Order Problem

Considering system (6) with the initial conditions images and images, the initial value problem can be written in matrix form as

images

for images where images and images.

Let us assume that images and images, images,

when images In this case, the following definitions can be adopted to the main theorems in this section.

Definition 4.1. Let images be the class of continuous column vector images whose components images are the class of continuous functions on the interval images The norm of images is given by

images

when images we write images and images.

Definition 4.2. Let the initial value problem Eq. (39) has a solution given by images. If

(i) images where images and

images

(ii)images satisfies Eq. (39).

Theorem 4.1. The initial value problem Eq. (39) has a unique solution images

Proof. Because of Eq. (39), we have

images

Operating images on Eq. (40), we obtain

images

Define the operator images by

images

It follows that

images

images

images

This implies that images If we choose W such that images then we obtain images. Therefore, using the Banach fixed point theorem, we conclude that the operator images given by Eq. (42) has a unique fixed point. Consequently, Eq. (41) has a unique solution images From Eq. (41), we have

and

which implies

from which we can deduce that images Thus, we have

It follows that

images

which implies

images

and thus

Therefore, this IVP is equivalent to Eq. (39), which completes the proof.

5  The Case of Extinction via Strong Allee Effect

In 1838, Pierre Verhulst [29] considered the logistic growth function to explain mono-species growth. Later on, it is demonstrated that the logistic equation needs modifications to explain the growth of the population in low density-size, which is known as the Allee effect.

The Allee effect can be divided into two main types:

(i) strong Allee effect and

(ii) weak Allee effect.

A population with a strong Allee effect will have a critical population size, which is the threshold of the population, and any size that is less than the threshold will go to extinction without any further aid. However, a population with a weak Allee effect will reduce the per capita growth rate at lower population density or size [3034].

Let us incorporate an Allee function to the images class at time t such as

images

where

images

is a function of Holling Type II and images is an Allee function at time images

Let

images

where we obtain images, if

images

and

images

where

images

Remark 5.1 The susceptible class and the classes who do not know they are infected are the main populations that affect the Allee function in stabilizing the spread of transmission. While it is essential to keep human non-infected, the other essential aim is to detect the infected class before the symptoms occur.

The characteristic equation of system (43) is given by

images

and

images

where

images, images

From Eq. (48), we have two quadratic equations, which are

images

and

images

where images and images is the basic reproduction number, which represents the transmission potential of the images class in the case of early detection, while imagesshows the transmission potential of the intermediate-natural host classes. This indicates that the reproduction numbers are not dependent on the Allee function.

For a strong Allee effect, let us assume that the Allee function is given by

images

where images represents the Allee threshold of the infected class, that do not know they are infected.

The following Theorem is given without proof since it is similar to the stability analysis of Section 3.

Theorem 5.1. Let images be the co-existing critical point of system (43) and assume that (i)–(iv) hold with Eqs. (45)(47) such that images and images

(i) Let images, images and images If

images and images,

where

images,

then all the roots of the system are real or complex conjugates with negative real parts.

(ii) Let images, images and

images

If

images and images

and the ratio between the susceptible and intermediate host is given by images, where

and

Thus, all roots of the system are complex conjugates with positive real parts.

(iii) Let images, images,

images and

images.

If

images and images

where

then the images class represents real or complex conjugates with negative real parts, while the images class shows complex conjugates with positive real parts.

(iv) Let images, images, images and

images .

If

images, images and the ratio between the susceptible and intermediate host is given by images where

then the images class represents complex conjugates with positive real parts, while the images class shows real or complex conjugates with negative real parts.□ images

6  Neimark–Sacker Bifurcation of the Dynamical Behavior with Discretization

In this section, we consider the discretization process to analyze Neimark–Sacker bifurcation. We will modify our system in (1) in considering the discrete-time effect on the model. The discretization of system (1) is as follows:

images

where

images

The solution of system (53) for images is given by

If we repeat the discretization process n times, we get

For images and images, while images, we have

images

The Jacobian matrix of (55) around the co-existing equilibrium point images is

images

where

images, images, images

images, images, imagesimages, images, imagesimages,

images, images

We obtain the characteristic equation of the matrix such as

images

and

images

where (i)-(v) hold and

images

To analyze the conditions for Neimark-Sacker Bifurcation, we use the following Theorem.

Theorem 6.1. [35] For a quadratic polynomial images such as

images

a pair of complex conjugate roots of (1) lie on the unit circle if and only if

(a) images

(b) images

(c) images

(d) images

Theorem 6.2. Let images be the co-existing critical point of system (55) and assume that (i)–(v) hold. If

where images, then the images class undergoes a Neimark-Sacker bifurcation. Additionally, if

where images and images 72for images and images then the images classes shows also a dynamical behavior of Neimark–Sacker bifurcation.

Proof. Let us first consider the statements in Theorem 6.1 for Eq. (57). Thus, from (a)-(c) together with (i) we have

which holds for

images

where images.

Finally, from (d) we obtain

which gives

images

where

In considering both Eqs. (61) and (62), we get

which completes the proof of the images class.

The characteristic equation Eq. (58) holds for Theorem 5.1./(a)–(c), if

then

images

images

and

images

Finally, from (d) we get

images

which holds for

where

images

images

This completes the proof.

7  Conclusion

In this paper, we classified the coronaviruses and their spread from the natural host to the human host. We proposed a model of the novel coronavirus, which is known as COVID-19, as a system of fractional-order differential equations. We divided the system into five sub-classes:

the susceptible class images the infected class images, that does not know they are infected since specific symptoms did not appear,

the infected class images that knows they are infected because of some symptoms such as respiratory and intestinal infections, including fever, dizziness, and cough, appeared.

the intermediate domestic host images that has a transmission role from the natural host to the human host

the natural host images that are bats of genus Rhinolophus.

We consider the pandemic infection case; animal to human and human to human. Therefore, the first three equations in the constructed model show human to human transmission. The spillover from the intermediate infected class to the human host denotes a predator-prey mathematical model, and the transmission from the natural host to intermediate host images is a host-parasite model of Holling Type II.

In Sections 3 and 4, we analyzed the local stability of the co-existing equilibrium point by using the Routh–Hurwitz Criteria. We proved the existence and the uniqueness of the initial value problem.

Theorem 3.1., shows that among the human hosts, those who do not know they are infected are the control class in the spread. While between the animal hosts, the intermediate class plays a dominant role in the spread since that class has an essential role in transmitting the virus from animal to human. The transmission potential for both images and images is images and images respectively. Also, the susceptible class and the images class is stable based on two parameters, which is the awareness of the symptoms and the screening rate.

In Theorem 3.2., we emphasized that images class should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the images class to stop the outbreak. For the susceptible class, it is more important to keep the population rate per year non-infected. The transmission of the virus to the offspring would reach an uncontrollable phenomenon worldwide.

In Section 5, we incorporate the Allee function at time images. The strong Allee effect is analyzed so that the screening for possible inflectional cases is an essential control parameter to support the Allee function in stabilizing the effect of the spread.

In Section 6, we deduced that the system demonstrates a Neimark–Sacker bifurcation under specific conditions.

Availability of Data and Material: All data generated or analyzed during this study are included in this published article.

Authors’ Contributions: Yousef and Bozkurt conceived the study and was in charge of overall direction and planning. Bozkurt and Yousef designed the mathematical model and set up the main parts of the study. They proved the theorems. Bozkurt, Yousef, and Abdeljawad collected the data and analyzed them. All authors interpreted the data and carried out this implementation. Bozkurt and Yousef conducted the simulation results using MATLAB 2019. All the authors are involved in writing and editing the manuscript. There is no Ghost-writing.

Acknowledgement: F. B. acknowledges the support of Erciyes University for the research study.

Funding Statement: The author(s) received no specific funding for this study.

Conflict of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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