*2.2. Analysis Stability of Equilibrium Points*

According to the concept of the Hartman–Grobman theorem, the hyperbolic equilibrium point in the neighborhood of a nonlinear dynamical system is topologically equivalent to its linearization. Therefore, the Jacobian of the nonlinear dynamic of (1) is computed as follows:

$$J[N\_\prime I] = \begin{bmatrix} F\_N[N\_\prime I] & F\_I[N\_\prime I] \\ G\_N[N\_\prime I] & G\_I[N\_\prime I] \end{bmatrix} \tag{2}$$

where *<sup>F</sup>*[*<sup>N</sup>*, *I*] = *dNdt* and *<sup>G</sup>*[*<sup>N</sup>*, *I*] = *dI dt* . The stability cases according to the above equilibrium points are as follows:

1. Stability of primary response stage- *p*1: Under the hypothesis of the IHDM, the immune system is able to protect the human body from developing diseases. This means that it responds directly in cases of emergency such as the appearance of abnormal cells in the tissue. The Jacobian (2) at equilibrium point *p*1 is computed as:

$$J[N,I]\_{p\_1} = \begin{pmatrix} r - \eta \frac{\sigma}{\delta} & -0\\ \frac{m\rho\sigma}{m^2\delta} - \mu \frac{\sigma}{\delta} & -\delta \end{pmatrix} . \tag{3}$$

**Proposition 1.** *Since the immune system is strong, the equilibrium point p*1 *is a stable node.*

**Proof.** From the Jacobian (3), the eigenvalues are given by

$$
\lambda\_1 = -\delta < 0,
$$

$$
\lambda\_2 = r - \eta \frac{\sigma}{\delta} = r - \eta I(0) < 0 \Leftrightarrow I(0) > \frac{r}{\eta},
$$

where 0 < *δ* < 1, then,

$$
\delta^{-1} \ge 1 \Rightarrow \lambda\_2 < \lambda\_1 < 0.
$$

Hence, the equilibrium point *p*1 is a stable node, see Figure 2.

2. Stability of interaction stage- *p*2: This stage describes the ability of the immune cells to inhibit and eliminate the abnormal cells to prevent them from progressing to cancer over many years. We consider the model as having a significant interaction if abnormal cells are dying or being inhibited by the immune cells. This means that parameter *β* → 0 at *t* → ∞. To examine the stability of equilibrium point *p*2 = ( −*<sup>r</sup>*(*<sup>δ</sup>*+*mμ*−*ρ*)+*ησ* √Δ 2*rμ* , *r η* ), we compute the Jacobian (2) at this point as

*J*[*<sup>N</sup>*, *<sup>I</sup>*]*<sup>p</sup>*2 = ⎛ ⎝ −*β μ x* −*η* 2*rμ x rμ η* (−<sup>1</sup> + <sup>4</sup>*mμρr*<sup>2</sup> *y*2 ) −*ησ r* ⎞ ⎠ , (4)

where

$$\begin{array}{rcl} x & = & -r(\delta + m\mu - \rho) + \eta\sigma + \sqrt{\Delta}, \\ y & = & -r\delta + rm\mu + r\rho + \eta\sigma + \sqrt{\Delta}. \end{array}$$

**Proposition 2.** *There is an unstable saddle equilibrium point during the interaction stage.*

**Proof.** From the Jacobian (4), the characterized equation is given by

$$(\frac{-\beta}{\mu}\mathbf{x} - \lambda)(\frac{-\eta\sigma}{r} - \lambda) - \frac{r\mu}{\eta}(-1 + \frac{4m\mu\rho r^2}{y^2})(\frac{\eta}{2r\mu}\mathbf{x}) = 0\tag{5}$$

$$(\frac{-\beta}{\mu}x - \lambda)(\frac{-\eta\sigma}{r} - \lambda) - (-1 + \frac{4m\mu\rho r^2}{y^2})(\frac{1}{2}x) = 0. \tag{6}$$

We assume that *C*1 = *x*2 (−<sup>1</sup> + <sup>4</sup>*mμρr*<sup>2</sup> *y*2 ) > 0. Then, we rewrite (6) as follows:

$$
\lambda^2 + (\frac{\beta \mathbf{x}}{\mu} + \frac{\eta \sigma}{r})\lambda + (\frac{-\eta \sigma}{r})(\frac{\beta \mathbf{x}}{\mu}) - \mathbb{C}\_1 = 0. \tag{7}
$$

Since, under the hypotheses of the IHDM, *β* → 0 at *t* → <sup>∞</sup>, Equation (7) is computed as

$$
\lambda^2 + \frac{\eta \sigma}{r} \lambda - \mathbb{C}\_1 = 0. \tag{8}
$$

Now, we apply the Routh–Hurwitz theorem for (8), giving

$$
\begin{vmatrix}
\lambda^2 & 1 & -\mathbf{C}\_1 \\
\lambda^1 & \frac{\eta \sigma}{r} & 0 \\
\lambda^0 & -\mathbf{C}\_1 & 0
\end{vmatrix}
$$

Since the first column has one sign change, the equilibrium point *p*2 is an unstable saddle.

3. Stability of recovery stage- *p*3: According to the physiological process, the number of immune cells which are involved in the interaction starts to reduce automatically after inhibiting and eliminating the abnormal cells. Furthermore, the normal cells divide and grow, taking the place of the removed abnormal cells. To examine the stability of this point, we compute the Jacobian at *p*3 = (*β*−1, *β*(<sup>1</sup>+*mβ*)*σ* (<sup>1</sup>+*mβ*)(*βδ*+*μ*)−*βρ* ) as follows:

$$J[N,I]\_{\mathbb{P}3} = \begin{pmatrix} \frac{-rz + \beta \eta (1 + m\beta)\tau}{z} & \frac{-\eta}{\beta} \\ \frac{\beta(-(1 + m\beta)^2 \mu + m\beta^2 \rho)\tau}{z(1 + m\beta)} & -\frac{z}{\beta + m\beta^2} \end{pmatrix} \tag{9}$$

where *z* = *β*(*δ* + *mμ* − *ρ*) + *mδβ*<sup>2</sup> + *βη*(1 + *mβ*)*σ* > 0.

**Proposition 3.** *In the recovery stage, the system might to be stable.*

**Proof.** The characterized equation of (9) is given by

$$(\frac{-rz+\beta\eta(1+m\beta)\sigma}{z}-\lambda)(-\frac{z}{\beta+m\beta^2}-\lambda)-\mathcal{C}\_2=0,\tag{10}$$

where

$$\mathcal{C}\_2 = \left(\frac{-\eta}{\beta}\right) \left(\frac{\mathbb{B}(-(1+m\beta)^2\mu + m\beta^2\rho)\sigma}{z(1+m\beta)}\right) > 0.1$$

To simplify, we let

$$A = \frac{-rz + \beta \eta (1 + m\beta)\sigma}{z} - \frac{z}{\beta + m\beta^2} < 0$$

and

$$B = \left(\frac{-rz + \beta \eta (1 + m\beta) \sigma}{z}\right) \left(-\frac{z}{\beta + m\beta^2}\right) > 0.1$$

Then, Equation (10) is rewritten as

$$
\lambda^2 - A\lambda + D = 0,\tag{11}
$$

where *D* = *B* − *C*2 > 0. We apply the Routh–Hurwitz theorem for (11) to determine the sign of roots:

$$
\begin{vmatrix}
\lambda^2 & 1 & D \\
\lambda^1 & -A & 0 \\
\lambda^0 & D & 0
\end{vmatrix}
$$

Since *A* < 0, the sign of elements in first column is positive and the equilibrium point *p*3 is called a stable node point for the IHDM.

With reference to propositions (1,2,3), we conclude this section with the following remark:
