# Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Model

#### 2.1. Basic Properties of the Model

**Proof.**

#### 2.2. The Basic Reproduction Number

## 3. Dynamics of the System

#### 3.1. Existence of Equilibria

#### 3.2. Stability Analysis of ${E}_{0}$

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

#### 3.3. Stability of ${E}^{*}$ and Hopf Bifurcation

**Theorem**

**3.**

- i.
- $\varphi \left({\theta}^{*}\right)=0$ and $\frac{d\varphi}{d\theta}{|}_{\theta ={\theta}^{*}}\ne 0$, where$$\varphi \left(\theta \right)=({a}_{3}-{a}_{1}{a}_{2})({a}_{5}{a}_{2}-{a}_{3}{a}_{4})-{({a}_{5}-{a}_{1}{a}_{4})}^{2},$$$$\phi =\frac{{a}_{5}-{a}_{1}{a}_{4}}{{a}_{3}-{a}_{1}{a}_{2}}>0,\phantom{\rule{1.em}{0ex}}{a}_{3}-{a}_{1}\phi \ne 0,$$
- ii.
- ${a}_{5}={a}_{1}{a}_{4}$, ${a}_{3}={a}_{1}{a}_{2}$, ${a}_{4}<0$, ${a}_{1}{a}_{3}\ne 0$,$$\left[{a}_{1}^{\prime}{\phi}^{2}+({a}_{1}{a}_{2}^{\prime}-{a}_{3}^{\prime})\phi -({a}_{1}{a}_{4}^{\prime}-{a}_{5}^{\prime})\right]{|}_{\theta ={\theta}^{*}}\ne 0.$$$$\phi =\frac{1}{2}\left({a}_{2}+\sqrt{{a}_{2}^{2}-4{a}_{4}}\right)>0.$$

**Lemma**

**1.**

## 4. The Optimal Control Problem

**Theorem**

**4.**

**Proof.**

## 5. Numerical Simulation

#### Numerical Solution of the Optimal Control Problem

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Forward bifurcation is shown. $\lambda $ is varied in the interval $(0,\phantom{\rule{3.33333pt}{0ex}}0.0005)$ and the rest of the parameter values are taken from Table 1.

**Figure 3.**Stability of ${E}_{0}$ in (

**a**) $\alpha -\omega $, (

**b**) $\lambda -\omega $ parameter planes. Other parameter values are taken from Table 1.

**Figure 5.**(

**a**–

**e**): Hopf bifurcation taking $\lambda $ as main parameter. Values of the parameters are same as Figure 4.

**Figure 6.**(

**a**–

**e**): Hopf bifurcation taking global awareness rate $\omega $ as the bifurcating parameter. Here, $\lambda =0.001$ and the rest of the parameters’ values are same as Figure 5.

**Figure 7.**(

**a**–

**e**): Solution trajectories for two different values of the local awareness rate $\eta $.

Parameter | Definition | Reference | Value |
---|---|---|---|

(day${}^{-1}$) | |||

b | Constant recruitment rate | [12] | 12 |

$\lambda $ | Disease transmission rate | [22,31] | 0.0005 |

$\alpha $ | Contact rate between unaware | [12] | 0.002 |

susceptible with media | |||

$\omega $ | Rate of media campaigns by global sources | [21,22] | 0.03 |

d | Susceptible class natural death rate | [12,32] | 0.01 |

$\delta $ | Additional death rate due to infection | [32] | 0.007 |

$\beta $ | Rate at which aware human becomes unaware | [22] | 0.0025 |

r | Rate of recovery of infected human | [12] | 0.01 |

$\gamma $ | Rate at which recovered class becomes | [12] | 0.0015 |

susceptible after immunity loss | |||

p | Portion of recovered class becoming | [12] | 0.3 |

susceptible unaware class | |||

$\eta $ | Rate of awareness programs by local sources | [12] | 0.25 |

$\theta $ | Depletion rate of awareness program | [12,22] | 0.015 |

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**MDPI and ACS Style**

Al Basir, F.; Rajak, B.; Rahman, B.; Hattaf, K.
Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment. *Axioms* **2023**, *12*, 608.
https://doi.org/10.3390/axioms12060608

**AMA Style**

Al Basir F, Rajak B, Rahman B, Hattaf K.
Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment. *Axioms*. 2023; 12(6):608.
https://doi.org/10.3390/axioms12060608

**Chicago/Turabian Style**

Al Basir, Fahad, Biru Rajak, Bootan Rahman, and Khalid Hattaf.
2023. "Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment" *Axioms* 12, no. 6: 608.
https://doi.org/10.3390/axioms12060608