**4. Numerical Simulation**

Both the IHDM and the IUNHDM were simulated using Mathematica software with the built-in functionality NDSolve. The proposed system of ordinary differential equations can be solved with any standard numerical method. For this, we used an explicit Runge–Kutta with a difference order of four and a step size of 1/10,000. The reliability and accuracy of the proposed numerical method can be seen from the residual error which is shown in Figures 6–9.

**Figure 6.** The residual error at steps for the proposed numerical method for the IHDM.

**Figure 7.** The residual error at time t for the proposed numerical method for the IHDM.

**Figure 8.** The residual error at steps for the proposed numerical method for the IUNHDM.

**Figure 9.** The residual error at time t for the proposed numerical method for the IUNHDM.

The simulation of the IHDM and the IUNHDM indicated that the immune system response is affected by specific parameters, namely, *ρ*, *m*, *η*, and *μ*. These parameters are put as *m* = 0.4787, *ρ* = 0.2206, *η* = 0.8791, and *μ* = 0.6986 for the IHDM and *m* = 0.3389, *ρ* = 0.2710, *η* = 0.1379, and *μ* = 0.8130 for the IUNHMD. In addition, the immune system can respond to an emergency case if and only if the threshold rate achieves its peak, which is given by 0.478; it is reduced to 0.3389 in the IUNHDM. This reveals a weakness in the immune system. Furthermore, the impact of the interaction between the immune system and abnormal cells depends on the rate of two parameters: *η* and *μ*. By comparing both models, we can deduce that the immune system succeeded in performing its function if and only if

> the rate of parameter *η* > the rate of parameter *μ*.

The behavior of the cells in the IHDM and the IUNHDM are shown in Figures 10 and 11, respectively. Furthermore, The numerical simulation revealed that in the IHDM, the immune system recognized abnormal cells as foreign bodies and started to inhibit or eliminate them within the first five days. In addition, the immune system put a copy of the abnormal cells in its memory, which helps it to eliminate them if they appear again or begin to progress. This is clearly seen as the population of immune cells returns to its initial value after the 10th day of interaction. Furthermore, the horizontal growth of normal cells indicates that normal cells divide and grow by following the signals of control cellular growth and death [5]. On the other hand, the response of the immune system and attitude of their cells were delayed in the IUNHDM. Hence, the immune cells failed to become involved in the interaction and reduced to zero, while the normal cells had a mutation which forced them to grow vertically, something which can lead to carcinoma [5,40].

**Figure 10.** The behavior of the IHDM where *r* = 0.431201, *β* = 2.99 × <sup>10</sup>−6, *σ* = 0.7, *δ* = 0.57, *m* = 0.4787, *ρ* = 0.2206, *η* = 0.8791, and *μ* = 0.6986.

**Figure 11.** The behavior of the IUNHDM where *r* = 0.431201, *β* = 2.99 × <sup>10</sup>−6, *σ* = 0.7, *δ* = 0.57, *m* = 0.3389, *ρ* = 0.2710, *η* = 0.1379, and *μ* = 0.8130.
