In this section, we investigate the existence of equilibria of system (1.4).
In order to find the equilibria of system (1.4), we put
(3.1)
Following the analysis in [12], we find that system (3.1) has always the uninfected equilibrium , where
We define the parameter as
and we also find that, if , system (3.1) has a unique positive equilibrium, . If , system (3.1) has a unique positive equilibrium , where
Next, we shall discuss the stability for the local asymptotic stability of the viral free equilibrium and the infected equilibrium .
To discuss the stability of system (1.4), let us consider the following coordinate transformation:
where denotes any equilibrium of (1.4). So we see that the corresponding linearized system of (1.4) is of the form
(3.2)
The characteristic equation of system (3.2) at is given by
For the local asymptotic stability of the viral free equilibrium , we have the following result.
Theorem 3.1 If , the uninfected state is locally asymptotically stable for .
Proof The associated transcendental characteristic equation at is given by
Obviously, the above equation has the characteristic root
where .
Next, we consider the transcendental polynomial
For , we get
Then we note that
we easily see that
We have
if , the characteristic roots have negative real parts for .
For , we get
Assume that the above equation has roots , for and ; we get
Separating the real and imaginary parts gives
(3.3)
From the second equation of (3.3), we have
that is , .
For , , substituting into the first equation of (3.3), we have
(3.4)
For the parameter values given in Table 1, we take any , the infected equilibrium , and we find that the above equation is unequal for . Therefore, .
According to Lemma 2.1, the uninfected equilibrium is locally asymptotically stable. The proof is completed. □
Remark 3.1 ([2])
The stability region of a system with fractional order is always larger than that of a corresponding ordinary differential system. This means that a unstable equilibrium of an ordinary differential system may be stable in a fractional differential system.
Next, for the sake of convenience, at , we define the following symbols:
Then the characteristic equation of the linear system is
(3.5)
Using the results in [31], we get
and
Denote
Theorem 3.2 Let , , and , then the infected equilibrium is asymptotically stable for any time delay if either
or
Proof According to (3.5).
For , we have
Using the result in [31], the infected steady state is asymptotically stable if the Routh-Hurwitz condition is satisfied, i.e.
or
For , we get
Assume that the above equation has roots , for and ; we get
Separating the real and imaginary parts yields
(3.6)
From the second equation of (3.6), we have
that is, , .
For , , substituting into the first equation of (3.6), we have
For the parameter values given in Table 1, we take any ; then we get the specific value on the infected equilibrium and we can see that the above equation is unequal for .
For , , substituting into the first equation of (3.6), we have
(3.7)
According to the development of Taylor type, we have
We take , and (3.7) becomes
(3.8)
Let
then (3.8) becomes
(3.9)
Notice that
Set
Then the roots of (3.10) can be expressed as
Due to , we have . Hence, neither nor is positive. Thus, (3.10) does not have positive roots. Since , , it follows that (3.9) has no positive roots.
Because of , the roots of (3.7) are positive, that is, .
The proof is completed. □