# Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model Derivation

## 3. Basic Properties of the Model

#### 3.1. Non-Negativity and Boundedness of the Solutions

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 3.2. The Basic Reproduction Number

## 4. Existence of Equilibrium Points

#### 4.1. The Disease-Free Equilibrium (DFE)

#### 4.2. The Endemic Equilibrium Point (EEP)

**Remark**

**1.**

## 5. Jacobian Matrix and Stability Analysis of Equilibrium Points

#### 5.1. Local Stability Analysis of Disease-Free Equilibrium (DFE)

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

#### 5.2. Global Stability of DFE

- (i)
- For $\frac{dX}{dt}=F(X,0)$ ${X}^{0}$, is globally asymptotically stable,
- (ii)
- $\frac{dZ}{dt}={D}_{Z}G(X,0)Z-\widehat{G}(X,Z),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\widehat{G}(X,Z)\ge 0$,

**Theorem**

**2.**

**Proof.**

#### 5.3. Local Stability of EEP

## 6. The Optimal Control Problem

#### 6.1. Existence of the Optimal Control Triple

**Theorem**

**3.**

- (i)
- (ii)
- The control set $\mathcal{U}$ is convex and closed;
- (iii)
- Each right-hand side of the state system (16) is: (a) continuous, (b) bounded above by a sum of the bounded control and the state variables, and (c) can be written as a linear function of u with time and the state-dependent coefficients;
- (iv)
- The integrand function of the objective functional is convex on $\mathcal{U}$;
- (v)
- There exist positive numbers ${\ell}_{1},{\ell}_{2},{\ell}_{3},{\ell}_{4}$ and a constant $\ell >1$ such that$${A}_{1}{{C}_{1}}^{2}+{A}_{2}\phantom{\rule{0.166667em}{0ex}}{{C}_{2}}^{2}+{A}_{3}{{C}_{3}}^{2}+{P}_{1}\phantom{\rule{0.166667em}{0ex}}{H}_{i}-{P}_{2}{{H}_{a}}^{2}\ge -{\ell}_{1}+{\ell}_{2}|{C}_{1}{|}^{\ell}+{\ell}_{3}|{C}_{2}{|}^{\ell}+{\ell}_{4}{\left|{C}_{3}\right|}^{\ell}.$$

#### 6.2. Characterization of the Optimal Control

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

## 7. Numerical Simulations

#### 7.1. Numerical Solution of the Optimal Control Problem

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Schematic diagram of the model (1): interactions between model populations are shown.

**Figure 2.**Forward transcritical bifurcation: equilibrium values of (

**a**) infected human and (

**b**) infective vectors are plotted with respect to the basic reproduction number ${\mathcal{R}}_{0}$. The parameter $\beta $ is varied, and the rest of the parameters’ values are taken from Table 1.

**Figure 3.**Numerical solution of system (1) with and without the impact of awareness.

**Figure 4.**Phase portrait is plotted in ${H}_{u}-{H}_{a}-{H}_{i}$ phase space. Parameter values are the same as in Figure 3.

**Figure 6.**Effect of global awareness is shown varying the parameter $\omega $. Other parameters values are as shown in Figure 3.

Variables/ | Descriptions | Values |
---|---|---|

Parameters | ||

${H}_{u}\left(t\right)$ | Number of unaware humans at time t | — |

${H}_{a}\left(t\right)$ | Number of aware humans at time t | — |

${H}_{i}\left(t\right)$ | Number of infected humans at time t | — |

${V}_{s}\left(t\right)$ | Number of susceptible mosquitoes at time t | — |

${V}_{i}\left(t\right)$ | Number of infective mosquitoes at time t | — |

$M\left(t\right)$ | Level of awareness due to media campaign at time t | — |

${\lambda}_{1}$ | Disease transmission from | 0.02 |

infected mosquitoes to unaware humans | ||

$\alpha $ | Rate of awareness by media campaign | 0.001 |

${\lambda}_{2}$ | Disease transmission from | 0.002 |

infected mosquitoes to aware humans | ||

$\beta $ | Infection rate of vector | 0.25 |

infected humans to susceptible mosquitoes | ||

${\Pi}_{h}$ | Recruitment rate of susceptible humans | 400 |

${\Pi}_{v}$ | Recruitment rate of susceptible mosquitoes | 10,000 |

$\mu $ | Natural death rate of mosquitoes | 0.12 |

r | Recovery rate of infected humans due to medication | 0.001 |

${d}_{h}$ | Natural death rate of humans | 0.002 |

$\delta $ | Disease-induced death rate for human population | 0.01 |

$\gamma $ | Efficacy of insecticide | 0.003 |

$\theta $ | Fading of memory | 0.01 |

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**MDPI and ACS Style**

Al Basir, F.; Abraha, T.
Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions. *Mathematics* **2023**, *11*, 1687.
https://doi.org/10.3390/math11071687

**AMA Style**

Al Basir F, Abraha T.
Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions. *Mathematics*. 2023; 11(7):1687.
https://doi.org/10.3390/math11071687

**Chicago/Turabian Style**

Al Basir, Fahad, and Teklebirhan Abraha.
2023. "Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions" *Mathematics* 11, no. 7: 1687.
https://doi.org/10.3390/math11071687