Since we are interested in virus pathogenesis and not initial processes of infection, we assume that the initial data for the system (1.6) are such that

The objective of this section is to perform a qualitative analysis of system (1.6) and derive the Allee-type dynamics. Clearly, the solutions of system (1.6) with positive initial values are positive and bounded. Let

, and note that (1.6) has one and only one infection-free equilibrium

. Then by using the formula of a basic reproduction number for the compartmental models in van den Driessche and Watmough [

24], we know that the basic reproduction number of virus of (1.6) is

which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process as zero. Although it is zero, we will show that the virus can still persist in host.

We start by studying the equilibria of (1.6). Obviously, the infection-free equilibrium
always exists and is a stable hyperbolic node because the corresponding characteristic equation is
.

In order to find the positive (infection) equilibria, set

then we have the equation

Based on (2.4), we can obtain that

(i) there is no infection equilibria if
;

(ii) there is a unique infection equilibrium
if
;

(iii) there are two infection equilibria
and
if
.

is a *Saddle-Node bifurcation* surface, that is, on one side of the surface
system (1.6) has not any positive equilibria; on the surface
system (1.6) has only one positive equilibrium; on the other side of the surface
system (1.6) has two positive equilibria. The detailed results will follow.

Next, we determine the stability of

and

. The Jacobian matrix at

is

After some calculations, we have

Since
in this case,
is valid. Thus,
and the equilibrium
is a saddle.

The Jacobian matrix at

is

By a similar argument as above, we can obtain that
. Thus, the equilibrium
is a node, or a focus, or a center.

For the sake of simplicity, we denote

We have the following results on the stability of
.

Theorem 2.1.

Suppose that equilibrium
exists; that is,
. Then
is always stable if
. When
, we have

(i)
is stable if
;

(ii)
is unstable if
;

(iii)
is a linear center if
.

Proof.

After some calculations, the matrix trace of

is

and its sign is determined by

which means that
is a monotone decreasing function of variable
.

Note that

implies that

Squaring (2.15) we find that

This means that
. Thus, under the condition of
and the sign of
,
is always valid if
. When
,
if
,
if
, and
if
.

For (1.6), its asymptotic behavior is determined by the stability of
if it does not have a limit cycle. Next, we begin to consider the nonexistence of limit cycle in (1.6).

Note that

is a saddle and

is a node, a focus, or a center. A limit cycle of (1.6) must include

and does not include

. Since the flow of (1.6) moves toward down on the line where

and

and moves towards up on the line where

and

, it is easy to see that any potential limit cycle of (1.6) must lie in the region where

. Take a Dulac function

, and denote the right-hand sides of (1.6) by

and

, respectively. We have

which is negative if
. Hence , we can obtain the following result.

Theorem 2.2.

There is no limit cycle in (1.6) if

Note that

as long as it exists. Thus, inequality (2.19) is always valid if

. When

, using the expression of

in (2.5), we have that inequality (2.19) that is equivalent to

we have (2.19) that is equivalent to

On the other hand, squaring (2.23) we find that

The combination of (2.24) and (2.26) yields (2.20).

is equivalent to

, both

are equivalent to
. Consequently, we have the following.

Corollary 2.3.

There is no limit cycle in (1.6) if either of the following conditions hold:

(i)
and
;

(ii)
and
.

When

, system (1.6) has a unique infection equilibrium

. The Jacobian matrix at

is

The determinant of

is

and the trace of

is

Thus,
is a degenerate singular point. Since its singularity, complex dynamic behaviors may occur, which will be studied in the next section.