3.2. Stability Result of Tumour-Free Equilibrium Point,
To examine the local stability of tumour-free equilibrium
(case for
), the variational matrix or the Jacobian becomes
Clearly
and
are eigenvalues, the remaining eigenvalues are given as the solutions to the characteristic equation
where
is always positive. By Routh Hurwitz criteria, sufficient and necessary conditions for
to become negative real part is
such that
Different values of
,
,
a and
d will affect the stability of the fixed point of the system.
a represents the total cell lost at the interphase where
d represents the total cell lost at the mitosis. The cell lost is the summation of natural death (apoptosis) of the tumour cell and the tumour cells that are killed by the immune system in each phase of the cell cycle. Based on the condition in (
5), the tumour would be defeated when there is no immune system interaction since the death rates dominate the growth rate. The literature shows that the rate of natural death (apoptosis) of the tumour cell is very low in the malignant tumour. Hence, we consider the death rate of tumour cells killed by the immune system is higher than the natural death rate of the tumour cells,
and
. The value of
a and
d should be higher than proliferation rate of the tumour cell,
and
, so that the tumour growth can be diminished. In the absence of immune response, these two parameters should be in control in order to maintain the growth of the tumour. However, as far as the immune responses are concerned, it greatly helps to suppress tumour growth by inhibiting their outgrowth. Villasana [
24] and Liu [
25] have focused on involving one immune response with results showing that the immune cell can suppress the tumour growth. From the results, we extend the model by adding another immune response since there are two types of immune system in the human body, i.e., innate and adaptive immune system. NK cell is a part of the innate system and the first line of defence against viruses and tumours. Once NK cells detect a foreign cell (tumour cell), they will directly attack the latter and give an alert signal to T cells (CD
T cell) and generate tumour-specific CD
T cell, which is represented by CD
T cell in the presented model. Tumour-specific CD
T cell is only present in the presence of a tumour cell and helps CD
T cell to kill the tumour cell. By including both immune responses from both immune systems, the model is more realistic.
If , a low death rate of the NK cell, a high NK cell production rate, k or a high immunotherapy effectiveness is necessary for tumour eradication.
It is indeed practical to have a stability conditions in term of immune parameter, k as that can be controlled from the outside, for example with chemotherapeutic treatments.
Now the effect of the delay is added in the system. As the delay is varied the conditions of the tumour growth changes. When the delay is positive, the the characteristic equation for the linearised equation about the
is given by
Clearly
and
are negative real roots. Then, the stability of
is determined by the distribution of the roots of equation
Now substituting
where
. We obtained the system transcendental equations to determine
and
.
We break the polynomial up into its real and imaginary parts, and write the exponential in terms of trigonometric functions to obtain
where
In order for (8) to hold, both the real and imaginary part must be 0, so we obtain the pair of equations
which can be rewritten as
Squaring each equation and summing the results yields
Solving the equation we obtain the polynomial
By letting
, the equation becomes a quadratic with roots given by
where
Equation (
14) has a positive real root in two circumstances. Since the lead coefficient is positive, if
then there is a single positive real root. If
, the root of (
14) are
and there is a simple positive root if and only if
and
. Thus, we can conclude.
Theorem 1. A steady state with characteristic Equation (6) is stable in the absence of delay, and becomes unstable with increasing delay if and only if - 1.
and , and
- 2.
either , or , and .
Noted that we are only interested in
, and thus if all of the real roots of (
14) are negative, we will have shown that there can be no simultaneous solution
of (11). Conversely, if there is a positive real root
to (
14), there is a delay
corresponding to
, which solves the equation in (11).
Now, suppose that (
14) has a positive real root. Then, there exists a positive root,
, satisfying (11) and hence the characteristic equation has a pair of imaginary roots
. Eliminating
and we obtain the expression for the time delay as
where for
the equilibrium point
is stable. When
, the characteristic equation has a imaginary root where bifurcation will occurs. As
, there is at least one root with positive real part.
The tumour conditions are represented by the region in
Figure 2. Assume that there is no natural death of the tumour cell,
. The resulting curves are obtained when the parameters
,
,
,
and
. The curves show the regions of growth, decay and stability switching. In the absence of delay,
, tumour growth is as shown in
Figure 3. However, when
increases, the region of tumour decay starts to change and becomes smaller. In the case
, tumour growth remains at the same region (R-I), while the decay region is subdivided into four regions including the regions of stability switching. For
, tumour regions are given by R-II and R-IV and as
increases the stability switching occurs at R-III and R-V as shown in
Figure 2.
Stability switching is very important in cancer chemotherapy. Cycle-specific drugs perform the primary action during specific phase of the cycle, i.e., interphase and M phase. They trap the tumour cell at some points and prevent the cell from continuing their cycle. Then, the cell will die due to natural causes and not undergo mitosis process. As a result, tumour growth stops. However, the trapping cells may have an adverse effect since some tumours can inhibit the immune system and become active. The active cell will enter the mitosis phase and continue to grow.
3.3. Stability Result of Tumour Equilibrium Point with Immune Responses
Let
, then the linearised system of (
1) at
generates a characteristic equation
where
The steady state is stable if the real part of roots of the characteristic polynomial (
17) are negative. By the Routh–Hurwitz theorem, it can be shown that the necessary and sufficient conditions for the system are as follows:
Theorem 2. Consider system as in (1). For the system without delay , the steady state is locally asymptotically stable if - a.
,
- b.
- c.
, and
- d.
.
Proof. Consider system (
1) with
holds. Assume the steady state of the system is locally asymptotically stable. Then the system (
1) is said to be stable if and only if the roots of its characteristic equation have negative real parts. The characteristic equation of system (
1) with
is Equation (
17). All the roots of (
17) have negative real part if (
17) satisfies conditions a, b, c and d. □
In case of positive delay (
), the system in (
1) is linearised around the fixed point
and the characteristic equation is
Now substituting
where
is positive in the characteristic equation. We obtained the system transcendental equations to determine
and
.
We break the polynomial up into its real and imaginary parts, and write the exponential in terms of trigonometric functions to obtain
where
In order for (
20) to hold, both the real and imaginary part must be 0, so we obtain the pair of equations
which can be rewritten as
Squaring each equation and summing the results yields
Solving the equation we obtain the polynomial
where
By letting
, the equation becomes a polynomial with roots given by
Noted that we are only interested in , and thus if all of the real roots of (26) are negative, we will have shown that there can be no simultaneous solution of (22). Conversely, if there is a positive real roots to (26), there is a delay corresponding to which solve equation in (22).
Now, suppose that (26) has a positive real root. Then there exist a positive
, satisfying (22) and hence the characteristic equation has a pair of imaginary roots
. Eliminating
, we obtain the expression for the time delay as
where for
the steady state
is stable. When
, the characteristic Equation (
18) has a pair of imaginary roots,
, where bifurcation will occurs. As
, there is at least one root with positive real part which means that at least one stability switching occurred. If there is no positive root, then the system does not have bifurcation and cannot lead to a stability switching.