## Abstract

Mathematical modelling techniques have been used extensively during the HIV epidemic. Drug injection causes increased HIV spread in most countries globally. The media is crucial in spreading health awareness by changing mixing behaviour. The published studies show some of the ways that differential equation models can be employed to explain how media awareness programs influence the spread and containment of disease (Greenhalgh et al. 2015). Here we build a differential equation model which shows how disease awareness programs can alter the HIV prevalence in a group of people who inject drugs (PWIDs). This builds on previous work by Greenhalgh and Hay (1997) and Liang et al. (2016). We have constructed a mathematical model to describe the improved model that reduces the spread of the diseases through the effect of awareness of disease on sharing needles and syringes amongst the PWID population. The model supposes that PWIDs clean their needles before use rather than after.

We carry out a steady state analysis and examine local stability. Our discussion has been focused on two ways of studying the influence of awareness of infection levels in epidemic modelling. The key biological parameter of our model is the basic reproductive number R0. R0 is a crucial number which determines the behaviour of the infection. We find that if R0 is less than one then the disease-free steady state is the unique steady state and moreover whatever the initial fraction of infected individuals then the disease will die out as time becomes large. If R0 exceeds one there is the disease-free steady state and a unique steady state with disease present. We also showed that the disease-free steady state is locally asymptotically stable if R0 is less than one, neutrally stable if R0 is equal to one and unstable if R0 exceeds one. In the last case when R0 is greater than one the endemic steady state was locally asymptotically stable. Our analytical results are confirmed by using simulation with realistic parameter values. In non-technical terms the number R0 is a critical value describing how the disease will spread. If R0 is less than or equal to one then the disease will always die out but if R0 exceeds one and disease is present the disease will sustain itself and moreover the numbers of PWIDs with

disease will tend to a unique non-zero value.

We carry out a steady state analysis and examine local stability. Our discussion has been focused on two ways of studying the influence of awareness of infection levels in epidemic modelling. The key biological parameter of our model is the basic reproductive number R0. R0 is a crucial number which determines the behaviour of the infection. We find that if R0 is less than one then the disease-free steady state is the unique steady state and moreover whatever the initial fraction of infected individuals then the disease will die out as time becomes large. If R0 exceeds one there is the disease-free steady state and a unique steady state with disease present. We also showed that the disease-free steady state is locally asymptotically stable if R0 is less than one, neutrally stable if R0 is equal to one and unstable if R0 exceeds one. In the last case when R0 is greater than one the endemic steady state was locally asymptotically stable. Our analytical results are confirmed by using simulation with realistic parameter values. In non-technical terms the number R0 is a critical value describing how the disease will spread. If R0 is less than or equal to one then the disease will always die out but if R0 exceeds one and disease is present the disease will sustain itself and moreover the numbers of PWIDs with

disease will tend to a unique non-zero value.

Original language | English |
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Article number | e12593 |

Number of pages | 28 |

Journal | Engineering Reports |

Volume | 5 |

Issue number | 5 |

Early online date | 8 Dec 2022 |

DOIs | |

Publication status | Published - 31 May 2023 |

## Keywords

- Human Immunodeficiency Virus (HIV)
- mathematical modelling
- reproduction number
- local and global stability