# Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. 2I2SQR Rumor Model without Time-Delay

#### 2.1. Model Formulation

**Remark 1.**

#### 2.2. Existence of Equilibria

- (i).
- If ${\Re}_{01}>1$, Equation (4) has a positive solution.
- (ii).
- If ${\Re}_{01}=1$, Equation (4) is transformed into$$\begin{array}{c}\hfill {a}_{2}{({S}_{1}^{*})}^{2}+{a}_{1}{S}_{1}^{*}=0.\end{array}$$Obviously, ${S}_{1}^{*}=0$ or ${S}_{1}^{*}=-\frac{{a}_{1}}{{a}_{2}}>0$ if ${a}_{1}<0$.
- (iii).
- If ${\Re}_{01}<1$ and ${a}_{1}<0$, the following results can be easily verified.(1). If ${\Re}_{01}>{\widehat{\Re}}_{01}$, Equation (4) has two positive roots.(2). If ${\Re}_{01}={\widehat{\Re}}_{01}$, Equation (4) has two equal positive roots $-\frac{{a}_{1}}{2{a}_{2}}$.(3). If ${\Re}_{01}<{\widehat{\Re}}_{01}$, Equation (4) has no positive root.

- (i).
- If ${\Re}_{02}>1$, Equation (8) has a positive solution.
- (ii).
- If ${\Re}_{02}=1$, Equation (8) has a solution ${S}_{2}^{*}=0$ or ${S}_{2}^{*}=-\frac{{\tilde{b}}_{2}}{{b}_{3}}>0$ if and only if ${\tilde{b}}_{2}>0$.
- (iii).
- If ${\Re}_{02}<1$ and ${\tilde{b}}_{2}>0$, one of the following three cases holds(1). If ${\Re}_{02}>{\widehat{\Re}}_{02}$, Equation (8) has two positive roots.(2). If ${\Re}_{02}={\widehat{\Re}}_{02}$, Equation (8) has two identical positive roots ${S}_{2}^{*}=-\frac{{\tilde{b}}_{2}}{2{b}_{3}}$.(3). If ${\Re}_{02}<{\widehat{\Re}}_{02}$, Equation (8) does not have a positive root.Case (2). When ${S}_{1}^{*}>0$, it has ${b}_{0}>0$. We denote$$\begin{array}{c}\hfill G({S}_{2}^{*})={b}_{3}{({S}_{2}^{*})}^{3}+{b}_{2}{({S}_{2}^{*})}^{2}+{b}_{1}{S}_{2}^{*}+{b}_{0}=0.\end{array}$$

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

- (i).
- $D<0$. Using the fact that $G(0)={b}_{0}>0$ and ${lim}_{{S}_{2}^{*}\to \infty}G({S}_{2}^{*})=-\infty $, it follows that the real root is positive.
- (ii).
- (iii).
- $D>0$. The analysis method is similar to (ii), and Equation (5) has one positive root.

**Theorem 1.**

#### 2.3. Stability and Hopf Bifurcation of the Equilibria

**Theorem 2.**

**Proof.**

**Lemma**

**3**

**.**For any $a>0$ and $b>0$, if $\frac{\mathrm{d}x(t)}{\mathrm{d}t}\ge b-ax(t)$ for $t\ge 0$ and $x(0)>0$, it has ${lim}_{t\to +\infty}infx(t)\ge \frac{b}{a}$; if $\frac{\mathrm{d}x(t)}{\mathrm{d}t}\le b-ax(t)$ for $t\ge 0$ and $x(0)>0$, it has ${lim}_{t\to +\infty}supx(t)\le \frac{b}{a}$.

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Remark 2.**

**Theorem 5.**

**Proof.**

**Remark 3.**

## 3. Optimal Control Model

**Theorem 6.**

**Proof.**

**Remark 4.**

## 4. 2I2SQR Rumor Model with Time-Delays

#### 4.1. Model Formulation

#### 4.2. Stability and Hopf Bifurcation of Equilibria

**Theorem 7.**

**Theorem 8.**

**Lemma 4.**

**Proof.**

**Theorem 9.**

**Remark 5.**

**Remark 6.**

## 5. Numerical Simulations

#### 5.1. The Stability of Rumor-Free Equilibrium

**Example 1.**

#### 5.2. The Stability of Rumor Equilibrium

**Example 2.**

#### 5.3. The Hopf Bifurcation of Rumor Equilibrium

**Example 3.**

#### 5.4. Feasibility of Optimal Control

**Example 4.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

OSNs | online social networks |

DK | Daley and Kendall |

MT | Maki and Thompson |

2I2SQR model | the model with 2 ignorants, 2 spreaders, quarantined and recovered users |

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**Figure 2.**(

**a**) The backward bifurcation graph of ${S}_{1}^{*}$ and the basic reproduction number ${\Re}_{01}$. (

**b**) The backward bifurcation graph of ${S}_{2}^{*}$ and the basic reproduction number ${\Re}_{02}$.

**Figure 3.**The dynamics of system (1) with ${\Re}_{0}<1$. (

**a**) The trajectories of ${S}_{1}(t)$ and ${S}_{2}(t)$. (

**b**) The trajectories of ${I}_{1}(t)$, ${I}_{2}(t)$, $Q(t)$ and $R(t)$.

**Figure 4.**(

**a**) The trajectories of ${S}_{1}(t)$ with different time delays. (

**b**) The trajectories of ${S}_{2}(t)$ with different time delays.

**Figure 5.**(

**a**) The evolution between ${\tau}_{10}$ and $\frac{{r}_{1}}{{\sigma}_{1}}$. (

**b**) The evolution between ${\tau}_{20}$ and $\frac{{r}_{2}}{{\sigma}_{2}}$.

**Figure 6.**(

**a**) The trajectories of ${S}_{1}(t)$ with different time delays. (

**b**) The trajectories of ${S}_{2}(t)$ with different time delays.

**Figure 7.**The dynamics of system (1) with ${\Re}_{01}=2.14>1$ and ${\Re}_{02}=0.97<1$. (

**a**) The trajectories of ${S}_{1}(t)$ and ${S}_{2}(t)$. (

**b**) The trajectories of ${I}_{1}(t)$, ${I}_{2}(t)$, $Q(t)$ and $R(t)$.

**Figure 8.**The dynamics of system (1) with ${\Re}_{01}=2.14>1$ and ${\Re}_{02}=1.9>1$. (

**a**) The trajectories of ${S}_{1}(t)$ and ${S}_{2}(t)$. (

**b**) The trajectories of ${I}_{1}(t)$, ${I}_{2}(t)$, $Q(t)$ and $R(t)$.

**Figure 9.**The dynamics of system (1) with ${\Re}_{01}=1.48>1$ and ${\Re}_{02}=1$. (

**a**) The trajectories of ${S}_{1}(t)$ and ${S}_{2}(t)$. (

**b**) The trajectories of ${I}_{1}(t)$, ${I}_{2}(t)$, $Q(t)$ and $R(t)$.

**Figure 10.**The phase diagram of ${S}^{1}(t)+{S}^{2}(t)$ and ${I}^{1}(t)+{I}^{2}(t)$. (

**a**) ${\Re}_{01}=2.14>1$ and ${\Re}_{02}=0.97<1$. (

**b**) ${\Re}_{01}=2.14>1$ and ${\Re}_{02}=1.9>1$. (

**c**) ${\Re}_{01}=1.48>1$ and ${\Re}_{02}=1$.

**Figure 11.**The dynamics of ${S}_{1}(t)$ and ${S}_{2}(t)$. (

**a**) ${\Re}_{01}=0.95<1$ and ${\Re}_{02}=1.5$. (

**b**) ${\Re}_{01}=0.28<1$ and ${\Re}_{02}=1$.

**Figure 12.**(

**a**) The trajectories of ${S}_{1}(t)$ with $\tau =11$ and $\tau =11.25$. (

**b**) The trajectories of ${S}_{2}(t)$ with $\tau =11$ and $\tau =11.25$.

**Figure 13.**(

**a**) The trajectory of ${S}_{1}(t)$ with $\tau =11.264$. (

**b**) The trajectory of ${S}_{2}(t)$ with $\tau =11.264$.

**Figure 14.**(

**a**) The phase diagram of ${S}^{1}(t)+{S}^{2}(t)$ with $\tau =11$. (

**b**) The phase diagram of ${S}^{1}(t)+{S}^{2}(t)$ with $\tau =11.26$.

**Figure 15.**(

**a**) The trajectory of ${S}_{1}(t)$ with ${\tau}_{1}=11$. (

**b**) The trajectory of ${S}_{2}(t)$ with ${\tau}_{2}=9$.

**Figure 16.**(

**a**) he trajectory of ${S}_{1}(t)$ with ${\tau}_{1}=11.5$. (

**b**) The trajectory of ${S}_{2}(t)$ with ${\tau}_{2}=9.5$.

**Figure 17.**(

**a**) The trajectory of ${S}_{1}(t)$ with ${\tau}_{1}=11.6$. (

**b**) The trajectory of ${S}_{2}(t)$ with ${\tau}_{2}=9.8$.

**Figure 20.**(

**a**) The trajectories of optimal control ${r}_{i}(t)$ $i=1,2$. (

**b**) The objective function. $J(t)$ with $T=2,5,8,10$. (

**c**) The consumption of $J(t)$ under different ${r}_{i}(t)$, $i=1,2$.

Parameters | Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | Set 6 | Set 7 |
---|---|---|---|---|---|---|---|

${B}_{1}$ | 0.1 | 0.11 | 0.11 | 0.11 | 0.09 | 0.03 | 0.1 |

${B}_{2}$ | 0.1 | 0.1 | 0.1 | 0.1 | 0.13 | 0.18 | 0.1 |

${\alpha}_{1}$ | 0.27 | 0.25 | 0.25 | 0.23 | 0.2 | 0.32 | 0.23 |

${\alpha}_{2}$ | 0.26 | 0.14 | 0.25 | 0.15 | 0.25 | 0.25 | 0.15 |

${\mu}_{1}$ | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

${\mu}_{2}$ | 0.3 | 0.01 | 0.01 | 0.01 | 0.01 | 0.3 | 0.01 |

${\beta}_{1}$ | 0.13 | 0.13 | 0.13 | 0.13 | 0.24 | 0.08 | 0.13 |

${\beta}_{2}$ | 0.12 | 0.15 | 0.15 | 0.14 | 0.15 | 0.08 | 0.14 |

c | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

${\sigma}_{1}$ | 0.003 | 0.52 | 0.52 | 0.41 | 0.5 | 0.58 | 0.5 |

${\sigma}_{2}$ | 0.0028 | 0.51 | 0.51 | 0.4 | 0.52 | 0.5 | 0.5 |

${r}_{1}$ | 0.001 | 0.02 | 0.02 | 0.05 | 0.1 | 0.1 | 0.06 |

${r}_{2}$ | 0.001 | 0.05 | 0.03 | 0.05 | 0.08 | 0.0005 | 0.06 |

d | 0.2 | 0.21 | 0.21 | 0.21 | 0.21 | 0.21 | 0.2 |

$\rho $ | 0.08 | 0.03 | 0.03 | 0.08 | 0.05 | 0.08 | 0.08 |

$\langle k\rangle $ | 3.5 | 7 | 7 | 7 | 7 | 3.5 | 7 |

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**MDPI and ACS Style**

Yu, S.; Yu, Z.; Jiang, H.
Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism. *Mathematics* **2022**, *10*, 4556.
https://doi.org/10.3390/math10234556

**AMA Style**

Yu S, Yu Z, Jiang H.
Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism. *Mathematics*. 2022; 10(23):4556.
https://doi.org/10.3390/math10234556

**Chicago/Turabian Style**

Yu, Shuzhen, Zhiyong Yu, and Haijun Jiang.
2022. "Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism" *Mathematics* 10, no. 23: 4556.
https://doi.org/10.3390/math10234556