*3.2. Stability of Equilibrium Points*

1. Stability of primary response stage: Since this point is identical to the primary response stage for the IHDM, we use (3) to examine the stability of this stage for the IUNHDM.

**Proposition 4.** *The point which represents the primary response stage is the unstable node for the IUNHDM.*

**Proof.** On the basis of the weakness of the immune system, we can hypothesize that *η I*<*r*, where *I* = *σδ* . That means that the parameter of *η* failed to achieve its highest peak rate. Therefore, the eigenvalues of matrix (3) have a different sign. Accordingly, the equilibrium point *u*1 is an unstable node.

2. Stability of coexistence stage: This stage is considered to be a trigger for cancer because the immune cells die without inhibiting or eliminating the abnormal cells. This era of rapid development affecting our lifestyle, especially our dietary habits, is one of the main causes of abnormal cells progressing early into tumor cells. Since this is mathematically similar to the recovery point in the IHDM, the stability case is given by the following proposition:

**Proposition 5.** *The point which represents the coexistence stage is stable.*

**Remark 4.** *The phase portrait of the IUNHDM and its solutions around the equilibrium points is shown in Figures 4 and 5.*

**Figure 4.** The phase portrait of the immune-unhealthy diet model (IUNHDM) and its solutions around the response equilibrium point.

**Figure 5.** The phase portrait of the IUNHDM and its solutions around the coexistence equilibrium point.
