# A Dynamic Simulation of the Immune System Response to Inhibit and Eliminate Abnormal Cells

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## Abstract

**:**

## 1. Introduction

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

#### 2.1. Equilibrium Points

- $\frac{dN}{dt}=0\iff $$$N=0,$$$$r[1-\beta N]-\eta I=0.$$
- $\frac{dI}{dt}=0\iff $$$\sigma -\delta I-\frac{\rho NI}{m+N}-\mu NI=0.$$

- 1
- Primary response stage: The immune system recognizes the appearance of abnormal cells in the tissue, immune cells start to grow, and $N=0$; this point is given by ${p}_{1}=(0,\frac{\sigma}{\delta}).$
- 2
- Interaction stage: The immune cells eliminate or inhibit the abnormal cells, this means $\beta \to 0$ at the end of the interaction. This point is given by$${p}_{2}=(\frac{-r(\delta +m\mu -\rho )+\eta \sigma +\sqrt{\Delta}}{2r\mu},\frac{r}{\eta}),$$
- 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$${p}_{3}=({\beta}^{-1},\frac{\beta (1+m\beta )\sigma}{(1+m\beta )(\beta \delta +\mu )-\beta \rho}),\mathrm{where}0<\beta <0.1.$$

#### 2.2. Analysis Stability of Equilibrium Points

- 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:$$J{[N,I]}_{{p}_{1}}=\left(\begin{array}{cc}r-\eta \frac{\sigma}{\delta}\hfill & \phantom{\rule{25.6073pt}{0ex}}-0\hfill \\ \frac{m\rho \sigma}{{m}^{2}\delta}-\mu \frac{\sigma}{\delta}\hfill & \phantom{\rule{25.6073pt}{0ex}}-\delta \hfill \end{array}\right).$$

**Proposition**

**1.**

**Proof.**

- 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 $\beta \to 0$ at $t\to \infty $. To examine the stability of equilibrium point ${p}_{2}=(\frac{-r(\delta +m\mu -\rho )+\eta \sigma \sqrt{\Delta}}{2r\mu},\frac{r}{\eta}),$ we compute the Jacobian (2) at this point as$$J{[N,I]}_{{p}_{2}}=\left(\begin{array}{cc}\frac{-\beta}{\mu}x\hfill & \frac{-\eta}{2r\mu}x\hfill \\ \frac{r\mu}{\eta}(-1+\frac{4m\mu \rho {r}^{2}}{{y}^{2}})\hfill & \frac{-\eta \sigma}{r}\hfill \end{array}\right),$$$$\begin{array}{ccc}x\hfill & =& -r(\delta +m\mu -\rho )+\eta \sigma +\sqrt{\Delta},\hfill \\ y\hfill & =& -r\delta +rm\mu +r\rho +\eta \sigma +\sqrt{\Delta}.\hfill \end{array}$$

**Proposition**

**2.**

**Proof.**

- 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 ${p}_{3}=({\beta}^{-1},\frac{\beta (1+m\beta )\sigma}{(1+m\beta )(\beta \delta +\mu )-\beta \rho})$ as follows:$$J{[N,I]}_{{p}_{3}}=\left(\begin{array}{cc}\frac{-rz+\beta \eta (1+m\beta )\sigma}{z}\hfill & \frac{-\eta}{\beta}\hfill \\ \frac{\beta (-{(1+m\beta )}^{2}\mu +m{\beta}^{2}\rho )\sigma}{z(1+m\beta )}\hfill & -\frac{z}{\beta +m{\beta}^{2}}\hfill \end{array}\right),$$

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

- The system has three equilibrium points;
- The system has two equilibrium points which are stable nodes, which shows that the immune system plays a pivotal role in protecting the human body from diseases.
- The system has only one equilibrium point which is an unstable saddle, which shows the interaction between the immune system and abnormal cells.

## 3. Immune-Unhealthy Diet Model (IUNHDM)

#### 3.1. Equilibrium Points:

- Primary response stage: In this stage, the immune system recognizes abnormal cells as foreign; this point is represented by ${u}_{1}=(0,\frac{\sigma}{\delta}).$ This is a similar equilibrium point to that of the IHDM.
- Coexistence stage: In this stage, the immune cells treat abnormal cells as normal cells; this point is represented by$${u}_{2}=({\beta}^{-1},\frac{\beta (1+m\beta )\sigma}{(1+m\beta )(\beta \delta +\mu )-\beta \rho}),\mathrm{where}0<\beta <0.1.$$

**Remark**

**2.**

**Remark**

**3.**

- There was a response from the immune system but the immune cells did not become involved in the interaction because the immune system was weak, or;
- The immune cells became involved in the interaction but failed to inhibit or eliminate the abnormal cells. Hence, this type of interaction damages the immune cells. In other words, the population of $I\to zero$ before inhibiting or eliminating the abnormal cells.

#### 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.**

**Proof.**

- 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.**

## 4. Numerical Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The dietary management 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.

**Figure 2.**The phase portrait of the immune–healthy diet model (IHDM) and its solutions around the response and interaction equilibrium points.

**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.

**Figure 10.**The behavior of the IHDM where $r=0.431201,\beta =2.99\times {10}^{-6}$, $\sigma =0.7,$$\delta $$\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$$0.57$, $m=0.4787$, $\rho =0.2206,\eta =0.8791$, and $\mu =0.6986.$

**Figure 11.**The behavior of the IUNHDM where $r=0.431201,\beta =2.99\times {10}^{-6},\phantom{\rule{3.33333pt}{0ex}}$$\sigma =0.7,$$\delta $$\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$$0.57$, $m=0.3389,$$\rho =0.2710,\eta =0.1379$, and $\mu =0.8130.$

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**MDPI and ACS Style**

Alharbi, S.A.; Rambely, A.S.
A Dynamic Simulation of the Immune System Response to Inhibit and Eliminate Abnormal Cells. *Symmetry* **2019**, *11*, 572.
https://doi.org/10.3390/sym11040572

**AMA Style**

Alharbi SA, Rambely AS.
A Dynamic Simulation of the Immune System Response to Inhibit and Eliminate Abnormal Cells. *Symmetry*. 2019; 11(4):572.
https://doi.org/10.3390/sym11040572

**Chicago/Turabian Style**

Alharbi, S. A., and A. S. Rambely.
2019. "A Dynamic Simulation of the Immune System Response to Inhibit and Eliminate Abnormal Cells" *Symmetry* 11, no. 4: 572.
https://doi.org/10.3390/sym11040572