**2. The Immune—Healthy Diet Model (IHDM)**

The immune system is very complex. One of its main functions is to recognize pathogens and to protect the body from developing diseases. However, the response of the immune system is affected by dietary habits, physical activity, stress, and sleep habits. A balanced diet and adequate intake of vitamins act to boost the immune system [29].

Our model is composed of ordinary differential equations, and it supposes that the individual follows a healthy diet as per the food pyramid, which is shown in Figure 1. This food pyramid follows recommendations from a report from the World Cancer Research Fund (WCRF) and the American Institute for Cancer Research (AICR), which recommends drinking water, eating a diet rich in wholegrains, vegetables, fruits, and beans, and having a lower intake of red and processed meats, as well as sugars and sweets [30,31]. The IHDM is formulated into two main populations: normal cells and immune cells activated as a result of the inability of the abnormal cells to eliminate themselves automatically. According to a cell's life cycle, we formulated the first equation to describe the behavior of normal cells during the process of their dynamic division and growth and to consider that some cells might divide abnormally. In addition, the natural death of the normal cells where the elimination of these cells occurs by apoptosis was not considered. The dynamic process of the interaction between the immune cells with the abnormal cells is presented in the second equation. This considers abnormal cells in the primary stage, which is where most cancers develop as a consequence of multiple abnormalities accumulating over many years. At this stage, the immune system eliminates them from growing before they turn into tumor cells (by attacking and repairing processes).

The IHDM is given as follows:

$$\begin{array}{rcl} \frac{dN}{dt} &=& rN[1 - \beta N] - \eta N I\_{\prime} \\ \frac{dI}{dt} &=& \sigma - \delta I - \frac{\rho N I}{m + N} - \mu N I\_{\prime} \end{array} \tag{1}$$

with initial values *N*(0) = 1 and *I*(0) = 1.22, where the dependent variables *N* and *I* represent the population of normal cells and immune cells, respectively. The parameters *r*, *β*, *η*, *σ*, *δ*, *ρ*, *m*, and *μ* are real and positive. The rate of growth of normal cells is represented by the parameter *r* and the rate of appearance of abnormal cells during the cell life cycle of the normal cell is given by *β*. Furthermore, the fixed source of immune cells is represented by *σ* and their rate of natural death is represented by *δ*. The ability of immune cells to eliminate abnormal cells determined by the Michaelis–Menten term *ρIN m*+*N* [15]. The coexistence of abnormal cells stimulates the immune system to respond [20]. The rate of this response is represented by *ρ* and the parameter *m* represents the threshold rate of the immune system. The parameters *η* and *μ* display the interaction between the abnormal cells and immune cells. The ability of the immune cells to eliminate abnormal cells or inhibit them is given by parameter *η*, whereas the parameter *μ* illustrates the decreasing number of immune cells as a result of their interaction with abnormal cells. In the IHDM, the rate of the parameter *η* > *μ* represents the

case where the immune system is strong and succeeding in performing its function and the person is following a healthy diet [29].

**Figure 1.** The dietary managemen<sup>t</sup> food pyramid according to the World Cancer Research Fund (WCRF) and American Institute for Cancer Research (AICR) where the amounts of food are estimated based on nutritional and practical considerations.

*2.1. Equilibrium Points*

> In this section, we use nullclines from system (1) to compute the equilibrium points as follows:

1. *dNdt* = 0 ⇔ *N* = 0, *r*[<sup>1</sup> − *βN*] − *ηI* 2. *dI dt* = 0 ⇔ −−*ρN I* −

$$
\sigma - \delta I - \frac{\rho N I}{m + N} - \mu N I = 0.
$$

= 0.

We classify the equilibrium points according to their biological terms as the following:


$$p\_2 = (\frac{-r(\delta + m\mu - \rho) + \eta \varphi + \sqrt{\Delta}}{2r\mu}, \frac{r}{\eta})\_{\prime\prime}$$

where Δ = *r*(*<sup>δ</sup>* + *mμ* − *ρ*)<sup>2</sup> + <sup>4</sup>*mrμ<sup>δ</sup>*(−*<sup>r</sup>* + *η σδ* ) > 0.

3 Recovery stage: Immune cells that are involved in the reaction tend to zero and all abnormal cells are substituted with normal cells. This point is represented by

$$p\_3 = (\beta^{-1}, \frac{\beta(1+m\beta)\sigma}{(1+m\beta)(\beta\delta+\mu)-\beta\rho}), \text{ where } 0 < \beta < 0.1.$$
