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

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

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 is the susceptible class, is the infected type that does not know they are infected because of the late occurred symptoms of COVID-19 and shows the infected class that knows they are infected. The spillover from the intermediate infected class to the human host denotes a predator-prey mathematical model, while for the transmission from the natural host , which is the bat population, to intermediate host is a host-parasite model of Holling Type II.

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

where

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

The susceptible 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 is the population growth rate of the susceptible population and denotes the logistic rate. is a rate of the susceptible population per year. The susceptible lost their class following contacts with infectives and the intermediate host at a rate and , respectively. links the parameter of the interaction between the hunted class and the predator population.

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

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

represents the natural host (bat population) of COVID-19 in this dynamic system. is the intrinsic growth rate and is the logistic rate of the population. shows the conversion factor of the natural host. is the attack rate of the bat population to infect the , while represents the fraction of the potential infectivity of the natural host. 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

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

defined on

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

Definition 2.3. Podlubny [25] The function

with 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

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

Thus, we have

and

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

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

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

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

whose solutions are given by Mittag–Leffler functions

and

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

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

where

, , ,

, ,

, ,

, ,

and

, .

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

and

if

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

or

and

or

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

(i) ,

(ii) ,

(iii)

(iv) and

(v) and .

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

and ,

where

,

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

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

and

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

if

where .

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

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

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

we obtain

and

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

for

and

for

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 and are and . Moreover, the susceptible class and the class is stable based on two parameters, which are the awareness of the symptoms and the screening rate.

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

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

and

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

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

and

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

if

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

,

where

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

we obtain

and

From (v) and Eqs. (37)–(38) we have

for

and

for

Moreover, we get since where

This completes the proof.

Remark 3.2. In Theorem 3.2., we emphasize that class should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the 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 be the co-existing critical point of system (6) and assume that (i)–(iv) hold such that and (i) Let , , and

.

If

and

where

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

(ii) Let , and

.

If

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

then the class represents complex conjugates with positive real parts, while the 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 and .

In Fig. 1 the blue graph denotes the susceptible class and the red graph shows 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 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 . This changes the endemic spread from epidemic to an uncontrolled pandemic form.

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

In Fig. 2, we keep the screening parameter as , 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 = .

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

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.

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

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 and ()

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 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 and , the initial value problem can be written in matrix form as

for where and .

Let us assume that and , ,

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

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

when we write and .

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

(i) where and

(ii) satisfies Eq. (39).

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

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

Operating on Eq. (40), we obtain

Define the operator by

It follows that

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

and

which implies

from which we can deduce that Thus, we have

It follows that

which implies

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 class at time t such as

where

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

Let

where we obtain , if

and

where

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

and

where

,

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

and

where and is the basic reproduction number, which represents the transmission potential of the class in the case of early detection, while shows 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

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

(i) Let , and If

and ,

where

,

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

(ii) Let , and

If

and

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

and

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

(iii) Let , ,

and

.

If

and

where

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

(iv) Let , , and

.

If

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

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

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:

where

The solution of system (53) for is given by

If we repeat the discretization process n times, we get

For and , while , we have

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

where

, ,

, , , , ,

,

We obtain the characteristic equation of the matrix such as

and

where (i)-(v) hold and

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

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

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

(a)

(b)

(c)

(d)

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

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

where and 72for and then the 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

where .

Finally, from (d) we obtain

which gives

where

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

which completes the proof of the class.

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

then

and

Finally, from (d) we get

which holds for

where

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

the infected class 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 that has a transmission role from the natural host to the human host

the natural host 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 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 and is and respectively. Also, the susceptible class and the 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 class should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the 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 . 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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