Global Stability and Hopf Bifurcation for a Virus Dynamics Model with General Incidence Rate and Delayed CTL Immune Response ()
1. Introduction
Immunity can be broadly categorized into adaptive immunity and innate immunity. Adaptive immunity is mediated by clonally B-cells and T-cells are also called lymphocytes, humoral and cellular immunity is characterized by specificity and memory. Cytotoxic T Lymphocytes (CTLs) play an important role in antiviral defense by attacking infected cells. Many mathematical models have been developed to describe HIV-1 (human immunodeficiency virus type 1) (see for example [1] - [15] ). First of all, we introduced the standard viral infection model with CTL immune response considered by Nowak and Bangham [16] as follows:
(1.1)
where
,
,
and
represent the densities of uninfected target cells, infected cells, virus and CTL cells at time t, respectively. Uninfected cells are produced at rate s, die at rate d, and become infected cells at rate
. Infected cells are produced from uninfected cells at rate
and die at rate
. The parameter p accounts for the strength of the lytic component. Free virions are produced from infected cells at rate
and are removed at rate u. The parameter b is the death rate for CTLs, and
describes the rate of CTL immune response activated by the infected cells. Recently many studies have been done to improve the model (1.1) by introducing delays and changing the incidence rate according to different practical backgrounds. These studies used different delayed models with different forms of incidence rates. In [17], Wang
etal. used the model (1.1) and replaced the incidence function rate by a Beddington-DeAngelis functional response of the form
, which was introduced by Beddington [18] and DeAngelis et al. [19]. In [10], Yuan et al. have
presented a model HIV-1 with an incidence rate of the form
. This incidence rate considered in this paper generalized many forms of commonly used incidence rate, including simple mass action, saturation incidence rate, Beddington-DeAngelis functional response form and the Crowly-Martin functional response form introduced by Crowly-Martin (see [20] ). The global dynamic of a virus dynamics model with Crowly-Martin functional response was discussed in [21]. In this paper, we propose a virus dynamics model with three delays and general incidence rate as follows:
(1.2)
where the parameters have the same meanings as in system (1.1).
represents the time between viral entry into a target cell and the production of new virus particles,
represent the time necessary for the newly produced viruses to become mature and then infection, that is, the maturation time of newly produce viruses and
represents the time that antigenic stimulation needs for generating immunity response [22]. The terms
,
denote the surviving rate of infected cells and viruses during the delay period, respectively. The function f is assumed to be continuously differentiable in the interior of
and satisfies the following hypotheses:
(H1)
(H2)
(H3)
(H4)
Under the hypotheses (H1) - (H4) we show that, for
and
, the dynamics of the model (1.2) is exactly like as for the standard basic model, if
, as well as for
if
and
. But for
, if in particular,
and
, we show that the dynamics of the model changes completely.
remains locally stable if
remains below a certain critical value
. Moreover, in
, there is appearance of a periodic phenomenon, obtained by Hopf bifurcation from
. When
passes
,
becomes unstable. Note that the incidence rate considered in this paper generalizes many forms of commonly used incidence rate, including simple mass action and saturation incident rate and Beddington-DeAngelis functional response form and Crowly-Martin functional response form.
The present paper is organized as follows. In Section 2, we derived the basic reproduction ratios for viral infection and CTL immune response
and
, respectively, and the existence of three equilibria is established. By means of suitable Lyapunov functionals and LaSalle’s invariant principle, we proved that, if
, the infection-free equilibrium
is globally asymptotically stable for all
(Section 3). In Section 4 we study the global stability of
and
and we prove that if
, the CTL-inactivated infection equilibrium
is globally asymptotically stable for all
, and if
and
, the CTL-activated infection equilibrium
is globally asymptotically stable for all
. In Section 5, we analyze the local stability of
in the case
,
and the existence of Hopf bifurcation from
. In Section 6, we present some numerical simulations to illustrate our results. In Section 7, some examples are given.
The initial conditions of (1.2) are given as:
(1.3)
where
, here
, with
denotes the Banach space of continuous functions mapping the interval
into
.
2. Preliminary Results
In this section we established the positivity and the boundedness of solutions of (1.2) and we define the basic reproduction numbers
and
and the existence of three possible equilibrium points is studied. The following theorem establishes the non-negativity and boundedness of solutions of (1.2).
Theorem 2.1. Under hypothesis (H1) and with the initial conditions (1.3), all solutions
of system (1.2) are non-negative and bounded on
.
Proof. Let put system (1.2) in a vector form
by setting
, and for all
.
where
and
. It is easy to check that
,
. Due to lemma 2 in [23], any solution of (1.2) with
, say
, is such that
for all
.
Next we show that the solutions are also bounded.
From (1.2)1 we have
. This implies
, so
is bounded.
Let:
where
. This implies that
is bounded and so is
and
. Thus, there exists a
such that
. It follows from (1.2)3 that:
and consequently
is bounded. Finally, all the solutions of system (1.2) are bounded. This completes the proof.
Global behaviour of system (1.2) may depends on the basic reproduction numbers
and
given by:
(2.1)
where,
, and:
(2.2)
with
. Here,
and
are the basic reproduction ratios
for viral infection and CTL immune response of system (1.2), respectively. Based on the hypotheses (H2) and (H3) it is clear that
.
System (1.2) always has the infection-free equilibrium
. For other possible equilibrium , we have the following theorem:
Theorem 2.2. Suppose that the conditions (H1) - (H3) are satisfied.
1) If
, then system (1.2) has a CTL-inactivated infection equilibrium of the form
with
.
2) If
, then system (1.2) has a CTL-activated infection equilibrium of the form
with
.
Proof. The steady states of system (1.2) satisfy the following equations:
(2.3)
From the last equation of (2.3), we have:
(2.4)
Equations (2.4) has two possible solutions,
or
.
If
, (2.3)3 yields
.
By substituting this into (2.3)2, we obtain:
(2.5)
which gives
or
.
If
, we obtain the infection-free equilibrium
.
If
, (2.3)1 and (2.3)2 yields:
(2.6)
with Equation (2.3)3, yields:
(2.7)
Since,
and
this implies that
.
Now, from (H1), (H2) and (H3), the following functional:
(2.8)
satisfy:
for
and:
.
Hence, we obtain the CTL-inactivated infection equilibrium:
with
is the unique zero in
of K and y and v are given by (2.6), (2.7) respectively.
If
, from (2.4) and (2.3)3 we obtain:
,
and:
,
and from the first and second equation of (2.3) we obtain:
(2.9)
which implies that
.
Now, from (2.3)1 the functional:
satisfy:
for
,
Hence, we obtain the CTL-activated infection equilibrium of the form
with
is the unique zero of L in
and z is given by (2.9). This completes the proof.
Remark 1. From (2.8) we have
if
. So, as K is increasing in the interval
; we deduces that
and consequently
.
3. Global Stability of the Infection-Free Equilibrium
In this section, we study the global stability of the infection-free equilibrium
of System (1.2).
Theorem 3.1. If
, the infection-free equilibrium
of System (1.2) is globally asymptotically stable for all
.
Proof. Define a Lyapunov function
as follows:
Calculating the time derivative of
along the positive solutions of System (1.2), we obtain:
From (H1), we have
. Since
, we have
. Hence, the infection-free equilibrium
is stable under the condition
. Let
be the largest invariant set in the set:
Note that for each
,
if and only if
,
and
. Therefore we have two cases:
•
, then
;
•
. From
and the first equation of (1.2), we have
. Since
from (H1) and (H2) we get
. Hence
.
By the above discussion, we deduce that
. It follow from LaSalle invariance principle [24] that the infection-free equilibrium
is globally asymptotically stable.
4. Global Stability of the Infected Equilibria
In this section, we study the global stability of the CTL-inactivated infection equilibrium
and the CTL-activated infection equilibrium
of system (1.2) by the Lyapunov direct method.
We set:
It is clear that for any
,
and
has the global minimum
, with
.
Theorem 4.1. Suppose that the conditions (H1) - (H4) are satisfied. Then the equilibrium
is globally asymptotically stable if
for all
, and the equilibrium
is globally asymptotically stable if
for all
and
.
Proof. Let
an infected equilibrium point of (1.2), i.e.,
or
. Define a Lyapunov functional:
where:
The function,
, verifies:
From (H2), we have
for
,
for
and
. So
. Consequently
is nonnegative defined with respect to the endemic equilibrium
, which is a global minimum.
We now prove that the time derivative of
is non-positive. Calculating the time derivative of
along the positive solutions of (1.2), we obtain:
At
, by using
,
and
, we have:
Calculating the time derivative of
, we obtain:
Combining (4.1) and (4.2), we obtain:
(4.1)
From (H2), we have:
and from (H4) we have:
and as g is positive, we have:
Now, if we approach
by
; in particular
and
we obtain:
From Remark 1, we have
for all
. In this case, it is easy to verify that from (4.3), the largest invariant set in
is the singleton
. Using LaSalle invariance principle [24], if
then the equilibrium
is globally asymptotically stable.
And, if we approach
by
and we suppose
, in particular
we obtain
Thus, the equilibrium
is stable. In this case, note that
if and only if
,
, and
and using the first and second equation of (1.2), we obtain
. Therefore, it follows from LaSalle's invariance principle [24] that the CTL-activated infection equilibrium
is globally asymptotically stable. This completes the proof.
We have so obtained the results of [1] [3] [4] [8] [9] [17] on the passage of the global stability from one equilibrium to another, depending on the parameters
and
. In what follows, we will show that stability of
depends mainly on the delay
.
5. The CTL-Activated Equilibrium and Hopf Bifurcation
In this section we will take
and
. The characteristic equation of system (1.2) at the CTL-activated equilibrium
is of the form:
(5.1)
where:
(5.2)
We have already seen in Theorem 4.1 for
and
that the equilibrium
is globally asymptotically stable in the case
and in particular is locally asymptotically stable in this case.
Now, let see to what value of
, this stability persists.
If
is a solution of (5.1), separating real and imaginary parts, it follows that:
(5.3)
Squaring and adding the two equations of (5.3), we obtain that:
where:
(5.4)
Letting
, Equation (5.4) can be written as:
(5.5)
Denote:
and define:
where
is one of cube roots of the complex number
and
.
According to [25] we have the following Lemma.
Lemma 1. [25]
For the polynomial Equation (5.6), the following states are true.
1) If
and one of the following conditions holds,
a)
and
;
b)
and
;
c)
and
.
Then Equation (5.6) has no positive root.
2) If
or
and one of the following conditions holds,
a)
,
and
;
b)
,
and
;
c)
,
and
.
Then Equation (5.6) has at least one positive root.
Noting that:
So Equation (5.6) has at most three positive roots. Suppose that Equation (5.6) has
positive real roots
, respectively. Then Equation (5.4) has
positive roots
. From (5.3), we have:
Let:
(5.6)
where
and
. Then
is a pair of purely imaginary roots of Equation (5.1) with
.
Define:
(5.7)
Let
be a root of Equation (5.1) satisfying
.
Differentiating the two sides of Equation (5.1) with respect to
, we obtained
and we have:
From (5.4), we get:
It therefore follows that:
Since
, we conclude that
and
have the same sign.
After this discussion, we have the following theorem.
Theorem 5.1. Let
and
,
be defined by (5.10) and (5.11), respectively. If
we have the following results:
1) If
and one of the following conditions holds,
a)
and
;
b)
and
;
c)
and
.
Then the CTL-activated infection equilibrium
of system (1.2) is locally asymptotically stable for all
.
2) If
or
and one of the following conditions holds:
a)
,
and
;
c)
,
and
;
c)
,
and
.
Then the CTL-activated infection equilibrium
of system (1.2) is locally asymptotically stable for
.
3) System (1.2) undergoes a Hopf bifurcation at CTL-activated infection equilibrium when
if the condition as stated in (2) are satisfied and:
.
6. Numerical Simulations
In this section, we give some numerical simulations supporting the theoretical analysis given in Sections 3, 4 and 5. We assume
.
Parameters in Figure 1 are
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(see [26] [27] [28] [29] [30] ).
By making the calculations we have
, and that system (1.2) has a infection-free equilibrium
. By Theorem 3.1, we get that the infection-free equilibrium
is globally asymptotically stable. Numerical simulation illustrates this fact.
Parameters in Figure 2 are
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. By making the calculations we have
and
, in this case, the system (1.2) has a CTL-inactivated infection equilibrium
who is globally asymptotically stable according to the Theorem 4.1, Therefore Numerical simulation prove this result.
Parameters in Figure 3 are
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. By making the calculations we have
, and that system (1.2) has a CTL-activated infection equilibrium
. By Theorem 4.1, we obtain that the CTL-activated infection equilibrium
of system (1.2) is globally asymptotically stable. Numerical simulation illustrates our result.
Figure 1. Global stability on the infection-free equilibrium
of system (1.2).
Figure 2. Global stability of the CTL-inactivated infection equilibrium
of system (1.2).
Figure 3. Global stability of the CTL-activated infection equilibrium
of system (1.2) in the case
.
Parameters in Figure 4 are
,
,
,
,
,
,
,
,
,
,
,
. By making the calculations we have
. By Theorem 5.1, we see that the CTL-activated infection equilibrium
is locally asymptotically stable if
and unstable if
. Further, system 1.2 undergoes a Hopf bifurcation at the CTL-activated infection equilibrium when
. Numerical simulation illustrates our result.
7. Examples
In this section, we give some particular examples.
Example 1. In (1.2), if
and
, we obtain the model studied by Zhu et al. [1].
Example 2. Let
and
, in this case we obtain the model presented by Li et al. [4]. So the work presented in [4] is a particular case of (1.2) because the function
satisfies the hypotheses (H1) - (H4).
Figure 4. Stability and Hopf bifurcation of the CTL-activated infection equilibrium
of system (1.2) in the case,
and
.
Example 3. In (1.2), if
and
, we obtain a virus dynamics model with Beddington-Deangelis incidence rate and CTL immune response. This model is presented in [17]. The function
satisfies the hypotheses (H1) - (H4), so the model presented in [17] is a particular case of (1.2).
Example 4. Let
and
. The
hypotheses (H1) - (H4) are satisfied. In this case we obtain a virus dynamics model with Crowly-Martin functional response. The global properties of this model were studied in [21].
Example 5. A last example, in (1.2), if
and
, we obtain the results presented in [13].
8. Conclusion
In this paper, we have considered a virus dynamics model with a general incidence rate and three delays
and
. This general incidence represents a variety of possible incidence functions that could be used in virus dynamics models. We establish that the global dynamics are determined by two threshold parameters the basic reproduction ratios for viral infection and CTL immune response
and
, respectively. We have proved that the infection-free equilibrium
is globally asymptotically stable if the basic reproduction ratios viral infection
for all
,
and
. The hypotheses on the general incidence function are used to assure the existence of the CTL-inactivated infection equilibrium
and the CTL-activated infection equilibrium
. We prove that if
, the CTL-inactivated infection equilibrium
is globally asymptotically stable for all
.
, assure the existence of
and when
,
is globally asymptotically stable for all
, and when
and
,
is only locally asymptotically stable for
less than a critical value
. Moreover, in
, there is the appearance of a periodic phenomenon from
, obtained by Hopf bifurcation and when
passes
,
becomes unstable. In the end, we have presented some numerical simulations.
Acknowledgements
We would like to thank the anonymous referees for their very helpful suggestions and comments.