A fractional complex network model for novel corona virus in China

As is well known the novel coronavirus (COVID-19) is a zoonotic virus and our model is concerned with the effect of the zoonotic source of the coronavirus during the outbreak in China. We present a SEIS complex network epidemic model for the novel coronavirus. Our model is presented in fractional form and with varying population. The steady states and the basic reproductive number are calculated. We also present some numerical examples and the sensitivity analysis of the basic reproductive number for the parameters.

outside China. Wuhan also contains a large market for seafood and animals, which is the source of the emergence and spread of  And when looking at how COVID-19 spreads from person to person, we find that the pattern of spread is not known yet, but most of the current information about the method of spread is based on previous information on corona viruses. Also, the spread of COVID-19 from a person infected with the virus to a healthy person needs close communication with the infected person where there will be an effect of cough and sneezing droplets. It turned out from the current cases of infection, whether simple or severe, that symptoms of this disease  appear in the form of fever, shortness of breath and cough. To date, there is no vaccine for this virus, so general prevention instructions such as avoiding direct contact with infected people and using gloves and face masks should be adhered to.
The study focused in this model on the zoonotic nature of the virus because of its continuous effect on the spread of the virus, especially at the beginning of the spread. In addition, the model was placed in a fractional form, with the community being represented by a heterogeneous network, in order for the model to be more realistic.
In the following section we present a heterogeneous network epidemic model for COVID-19 in a fractional form [8][9][10] using the Caputo definition. The SEIS scenario was chosen as the mode of diffusion, as it was considered more suitable than the SEIR because some cases have been confirmed to be re-infected with COVID-19 [11][12][13]. For more information about the basics of fractional calculus and fractional model stability, see [14][15][16][17][18][19][20] and [21][22][23][24], for networks see [25]. In Sect. 2 we described the model. In Sect. 3 we find the steady states and the basic reproductive function. In Sect. 4 we proved the local stability of the steady states. In Sect. 5 we present the sensitivity analysis to get the most effective parameter and some numerical examples. Section 6 is the conclusion.

Fractional SEIS model description
In this model we divided the population into three compartments susceptible, exposed and infected. The susceptible individuals can be exposed because of being in close contact with infected one. Also, the infection could be transmitted to a susceptible individual from a zoonotic source of COVID-19 (an unknown animal embracing the virus). This interaction between susceptible and the zoonotic source happened in a homogeneous pattern during buying and walking in the seafood market. Also, the number of zoonotic sources is considered to be constant in the seafood market (sellers put other animals after the animals that were sold). The exposed individual become infected after the incubation period. The infected individual became susceptible again after the infectious period. The city's population (Wuhan city) is changing as a result of traveling continuously to and from the city. In this model we ignored the births and the deaths.
According to the above system dynamic description, the model is defined as  where k is the degree of the node, 1 ≤ k ≤ n, n is the maximum degree of a node. (t) is the probability to be linked with an infected node and defined as where k = k kP(k). P(k) is the degree distribution of the population. N k (t) is the total population of degree k.
is a Heaviside function representing the zoonotic infection force. This function affects only before seafood market closure (from 1 December 2019 to 31 December 2019). After the seafood market closure on 1 January 2020 this function is equal to 0. Other parameters are described in Table 1. We used the Caputo definition for the fractional order α ∈ (0, 1], which is defined as follows:
We will find the equilibrium points of system (2.1) by putting its equations equal to zero as follows: It is obvious that system (2.1) has a unique free disease equilibrium point, , with respect to z(t) = 0 and an endemic point , The value of the endemic point changes with respect to the existence of z(t). If z(t) = 0, then the endemic point take the form ,

The existence of the endemic point
By substituting with the value of I * * k into the definition of (t) we get the self-consistency equation .
We can put it in the following form: Now, we need to get a solution for g( ) in the interval ∈ (0, 1). By calculating the value of g( ) at both 0 and 1 we get Therefore, we have two cases. Case 1: If z(t) exists, then g(0) > 0. This leads to the function g( ) always having a nontrivial solution in the interval (0, 1).

The basic reproductive number
Only the exposed and infected compartments will be used to find the basic reproductive value [26]. The rate of new infected nodes entering the two compartments E k (t) and I k (t) is represented by the matrix F given by where F 11 , F 12 , F 21 and F 22 are n × n matrices [27]. The following matrix V represents the rate of transferring out of and into the two compartments E k (t) and I k (t): where V 11 , V 12 , V 21 and V 22 are n × n matrices. The basic reproductive number is given by the dominant eigenvalue of FV -1 calculated at the disease-free equilibrium point P 0 and z(t) = 0 (pure population). The elements of F are given by and the elements of matrix V take the form The characteristic equation for the 2n eigenvalues λ of matrix FV -1 is then the basic reproductive number R 0 is defined as Theorem 3.1 Define the basic reproductive number R 0 as follows: 1. If z(t) = 0 and R 0 < 1, then system (2.1) has a unique free disease equilibrium point P 0 . Firstly, we establish the Jacobian matrix of system (2.1) at P 0 with respect to z(t) = 0, which takes the form where each sub-matrix C ij , 1 ≤ i, j ≤ 3 is an n × n matrix and is given by where m ij = ijp(j) k ∀1 ≤ i, j ≤ n. All eigenvalues of the Jacobian matrix (4.1) should satisfy the following condition: After expanding the Jacobian matrix, we get the following characteristic equation: Obviously, we have n negative eigenvalues equal to -B from the first bracket. From the second bracket we have a second degree equation repeated n -1 times in the form which having another two negative Eigenvalues -(γ +B) and -(μ+B). Each one is repeated n -1 times then we have 2n -2 negative Eigenvalues from the second bracket. The third bracket in (4.3) is a second degree equation equal to , Therefore, ρ 0 > 0; if R 0 < 1 then the third bracket has two negative eigenvalues. Hence condition (4.2) is satisfied.
Theorem 4.1 If R 0 < 1 then the free disease steady state P 0 is locally asymptotically stable and unstable if R 0 > 1.
-ε n u n1 · · · -ε n (u nn -(w n + υ n )) + γ The characteristic equation has the form It is clear that Eq. (4.5) has n negative eigenvalues equal to -B. The next 2n eigenvalues could be obtained from the second part of Eq. (4.5), which is defined as a polynomial function of degree 2n as follows: γ + B + μ)) . Now, we will search for the roots of (x) instead of calculating them. In the first case we suppose that which is an equation of degree two with positive coefficients. That means that we have two negative eigenvalues -ξ 1 i , -ξ 2 i depending on w i (w i has an increasing value) and having the values Therefore, we have the last 2n negative eigenvalues. In the second case, we suppose which is a continuous function. We can put the function (x) in a more simple form as follows: we can observe that therefore, we have one root in the interval [-ξ 1 i , -ξ 1 i+1 ]. In general, we have n -1 negative solutions in the interval [-ξ 1 1 , -ξ 1 n ]. Similarly, with ξ 2 i we get n -1 negative solutions in [-ξ 2 1 , -ξ 2 n ]. Searching for the last two roots, we have (-ξ 1 1 ) < 0 and (0) > 0 then we get one more negative solution in the interval [-ξ 1 1 , 0]. Similarly, we can see that (-ξ 2 1 ) < 0. Then we get another negative solution in the interval [-ξ 2 1 , 0]. Finally, the function (x) has 2n negative solutions in the interval [-ξ 2 n , 0]. Hence, condition (4.2) is satisfied and the endemic equilibrium point P 2 is locally asymptotically stable.

Theorem 4.2
The endemic steady state P 2 is always locally asymptotically stable.
Remark 1 When z(t) = 0 and R 0 > 1, then the last proof is valid for P 1 and it will be locally asymptotically stable.

Sensitivity of the parameters
Sensitivity analysis shows us which of the parameters used in our mathematical model is the most effective in spreading the infection [28]. In the definition of R 0 , it is depending on five variables μ, β 1 , γ , B and k where k is the ratio between the second and the first moment of the node degree k as an additional parameter. Using the sensitivity index S R 0 r which mean the sensitivity of the basic reproductive number with respect to r (any chosen parameter) with the definition For example, S R 0 r = 1 means that any increasing (decreasing) of the value of r by v% increases (decreases) the value of R 0 by the same percentage. In the opposite case, S R 0 r = -1 means that any increasing (decreasing) of the value of r by v% decreases (increases) the value of R 0 by the same percentage. After applying the sensitivity analysis, we get the following sensitivity indices: Using the values in Table 2, group 2 we get the following values for the sensitivity indices: It is obvious that the parameter γ is the most sensitive parameter i.e. if this parameter increased by 10%, the value of R 0 will be decreased by 34.14%. Notice that the values of the sensitivity indices can be changed with respect to the parameters value.

Numerical simulation
In this section, we used an Adams-type predictor-corrector method [19,20] for solving system (2.1), showing the results obtained in previous sections. We have a BA random scale free network with p(k) = mk -γ 1 , where m is a constant satisfies k p(k) = 1 and 2 < γ 1 < 3 is the exponent of the power law distribution. Choosing γ 1 = 2.3 and n = 100, we present the following examples.
Example 5. 3 In the case of existence of the zoonotic effect (z(t) = 1000 for t = 100), choosing the values in group 1, Table 2, for model (2.1) parameters we get R 0 = 0.7953 < 1.
In this case, system (2.1) has a unique endemic steady state P 2 which is locally asymptotically stable according to Theorem 4.2. It is shown for S k (t), E k (t) and I k (t) for different values of k with fractional order α = 0.95 in Fig. 8.

Figure 9
The change of R 0 value with respect the value of degree k, the left curve for group 1 parameters and the right curve for group 2 parameters Taking into account the effect of the zoonotic source origin of the disease, as well as the continuous transport movement in Wuhan, the mainland city of the virus. We calculated the basic reproduction number, which significantly depends on traveling and movement rates from and outside the city. In addition we calculated the equilibrium positions for this system, as well as showing the local stability of the disease-free situation if the value of the R 0 < 1. Likewise, the epidemiological situation is locally asymptotically stable, if the value of R 0 > 1. And the danger of this virus (COVID-19) appears in the speed of its spread among individuals and the danger of its transmission to many countries around the world. There is a great fear of the formation of (COVID-19) for another large infection area outside the mainland and containing another strain of the corona family. We cannot deny the effective influence of the zoonotic source through which the virus was transmitted to humans and which in turn has spread among humans. It is possible that the impact of the zoonotic source continues until now and is not limited to the closure of the seafood market in Wuhan, China, which is considered as a possible explanation for the increasing numbers of infection.