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

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.


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.
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 Indeed, the mathematical model of this biological phenomena has the form: where f t ð Þ ¼ M ðtÞ 1 þ hexM ðtÞ (2) represents the Holling type II function and all the parameters of the model (1) belong to R þ and t 2 0; 1 ½ Þ.
The susceptible S 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 r 1 is the population growth rate of the susceptible population and m 1 denotes the logistic rate. p is a rate of the susceptible population per year. The susceptible lost their class following contacts with infectives C 1 and the intermediate host M at a rate b 1 and b 2 , respectively. The parameter s 1 links the parameter of the interaction between the hunted M class and the predator S population.
The C 1 class does not know that they have COVID- 19. In this equation, r 2 is the population growth rate of the class, while m 2 is the logistic rate. The population of this class decreases after screening at a rate u and be aware of the infection. Another possibility is that after the S-C 1 contact, the symptoms occur in early stages so that both classes noticed that they are infected, which is given with the rate e 1 . The intermediate host infected group could also show early symptoms to be aware of the infection, which is provided by a rate of e 2 : The logistic rate of C 2 is denoted as m 3 .
M is the domestic animal as an intermediate class in the corona transmission spread. r 3 is the intrinsic growth rate of the population, while m 4 is the logistic rate. s 2 shows the effect on the hunted M during the interaction between the intermediate host and susceptible class. c denotes the predation rate in the hostparasite scheme.
N represents the natural host (bat population) of COVID-19 in this dynamic system. r 4 is the intrinsic growth rate and m 5 is the logistic rate of the population. d shows the conversion factor of the natural host. e is the attack rate of the bat population to infect the M, while x 0 < x 1 ð Þ represents the fraction of the potential infectivity of the natural host. h 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).
defined on R þ : Definition 2.2. Podlubny [25] Let f : R þ ! R be a continuous function. The Caputo fractional derivative of order a 2 n À 1; n ð Þis given by Definition 2.3. Podlubny [25] The function with C being the set of complex numbers is called the Mittag-Leffler function of one parameter.
Þþdf t ð ÞN t ð Þ: To analyze the stability of model (6), we perturb the equilibrium point by adding e i t ð Þ > 0; i ¼ 1; 2; 3; 4; 5; that is, Thus, we have Thus, we obtain a linearized system about the equilibrium point of the form where Z ¼ e 1 t ð Þ; e 2 t ð Þ; e 3 t ð Þ; e 4 t ð Þ; e 5 t ð Þ ð Þ . Moreover, J is the Jacobian matrix at the equilibrium: where the co-existing equilibrium point is Λ ¼ " and i i ¼ 1; 2; 3; 4; 5 ð Þ are the eigenvalues and B the eigenvectors of J. Therefore, we get whose solutions are given by Mittag-Leffler functions and Evaluating the Jacobian matrix (9) for the co-existing equilibrium point Λ; we obtain where The characteristic equation of the matrix (17) is given as From Eq. (18), we have two quadratic equations, which are 2 À a 11 þ a 22 ð Þ þ a 11 a 22 1 À a 12 a 21 a 11 a 22 ¼ 0 or where R 01 ¼ a 12 a 21 a 11 a 22 and R 02 ¼ a 45 a 54 a 44 a 55 : R 01 is the basic reproduction number, which represents the transmission potential of S À C 1 class, while R 02 shows the transmission potential of the intermediatenatural host classes M À N : For the following theorems in this section, we consider the case, where both R 01 < 1 and R 02 < 1; which hold for the following statements: Theorem 3.1. Let Λ be the co-existing critical point of system (6) and assume that (i)-(iv) hold such that R 01 < 1 and R 02 < 1: Moreover, let r 1 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 Λ is locally asymptotically stable.
Proof. Let us consider the case for a 11 þ a 22 < 0 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 a 44 þ a 55 < 0 to have eigenvalues with negative real parts. Thus, from and From (v) and Eqs. (28)-(29), we obtain 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 S À C 1 and M À N are R 01 < 1 and R 02 < 1. Moreover, the susceptible class and the C 1 class is stable based on two parameters, which are the awareness of the symptoms and the screening rate.
Theorem 3.2. Let Λ be the co-existing critical point of system (6) and assume that (i)-(iv) hold such that R 01 < 1 and R 02 < 1: Furthermore, let and the ratio between the susceptible and intermediate host is given by and tan À1 À 4 Then all roots of Eq. (18) are complex conjugates with positive real parts, which implies that Λ is locally asymptotically stable.
Proof. Let us consider the case for a 11 þ a 22 > 0 to have eigenvalues with positive real parts. This holds if and From (ii) and Eq. (32) we obtain if where r 2 > b 1 1 À e 1 ð Þ 2m 2 . In considering both (iii) and Eq. (34), we obtain Additionally, we get ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Similarly, let us consider the case for a 44 þ a 55 > 0 to have eigenvalues with positive real parts. From and " N , This completes the proof. Remark 3.2. In Theorem 3.2., we emphasize that class C 1 should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the S 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 Λ be the co-existing critical point of system (6) and assume that (i)-(iv) hold such that R 01 < 1 and R 02 < 1: where tan À1 À 4 then the S À C 1 class represents real or complex conjugates with negative real parts, while the M À N class shows complex conjugates with positive real parts.

and the ratio between the susceptible and intermediate host is given by
then the S À C 1 class represents complex conjugates with positive real parts, while the M À N class shows real or complex conjugates with negative real parts.
In Fig. 1 the blue graph denotes the susceptible class S and the red graph shows C 1 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 %1 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 %1. This changes the endemic spread from epidemic to an uncontrolled pandemic form.
In Fig. 2, we keep the screening parameter as h ¼ 0:01, 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 e 1 = e 2 ¼ 0:4. 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.
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 h and e i (i ¼ 1; 2) 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 C 1 class who does not know they are infected.

Existence and Uniqueness of the Initial Value Fractional-Order Problem
Considering system (6) with the initial conditions S 0 ð Þ > 0; C 1 0 ð Þ > 0; C 2 0 ð Þ > 0; M 0 ð Þ > 0 and N 0 ð Þ > 0, the initial value problem can be written in matrix form as Let us assume that 0 < M 0 ð Þ #; and S 0 ð Þ > 0; when t > r ! 0: In this case, the following definitions can be adopted to the main theorems in this section.
Theorem 4.1. The initial value problem Eq. (39) has a unique solution U 2 C Ã 0; T ½ : Proof. Because of Eq. (39), we have Operating I a on Eq. (40), we obtain Define the operator F : It follows that e ÀWt k FU À FV k¼ e ÀWt I a ðAðU ðtÞ À V ðtÞÞ þ SðtÞBðUðtÞ À V ðtÞÞ þ C 1 ðtÞCðU ðtÞ À V ðtÞÞ þ C 1 ðtÞDðU ðtÞ ÀV ðtÞÞ þ M ðtÞEðU ðtÞ À V ðtÞÞ þ M ðtÞFðUðtÞ À V ðtÞÞÞ Therefore, using the Banach fixed point theorem, we conclude that the operator F given by Eq. (42) has a unique fixed point. Consequently, Eq. (41) has a unique solution U 2 C Ã 0; T ½ : From Eq. (41), we have from which we can deduce that U 0 2 C Ã r 0; T ½ : Thus, we have It follows that Therefore, this IVP is equivalent to Eq. (39), which completes the proof.

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 [30][31][32][33][34].
Let us incorporate an Allee function to the C 1 t ð Þ class at time t such as where we obtain @M t ð Þ and 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 where For a strong Allee effect, let us assume that the Allee function is given by where K 0 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.
(iii) Let r 1 and " M 2 0:5a 4 À1 ; d À 2m 4 r 3 2m 4 r 3 hex ; where tan À1 À 4 then the S À C 1 class represents real or complex conjugates with negative real parts, while the M ÀN class shows complex conjugates with positive real parts.
Proof. Let us first consider the statements in 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 M 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 S À C 1 and M À N is R 01 < 1 and R 02 < 1; respectively. Also, the susceptible class and the C 1 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 C 1 class should be more aware of the symptoms that might become from the susceptible class as well as from the intermediate class, than the S 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 t. 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.