Projects per year

### Abstract

In this paper, we introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay [10]. This was based on the original model constructed by Kaplan [17] which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE) for the fraction of PWIDs who are infected with HIV at time t. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0,1) provided that some infected PWIDs are initially present, and next construct the conditions required for extinction and persistence. Furthermore, we also show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence. Hence in situations where stochastic noise is important predictions of control measures such as needle cleaning or reduction of needle sharing rates needed to eliminate disease may be overly conservative.

Original language | English |
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Article number | 6757928 |

Number of pages | 14 |

Journal | Computational and Mathematical Methods in Medicine |

Volume | 2016 |

DOIs | |

Publication status | Published - 10 Mar 2016 |

### Keywords

- HIV
- AIDS
- parameter perturbation
- extinction
- persistence
- stationary distribution
- stochastic differential equations
- Brownian motion
- environmental stochasticity

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## Projects

- 3 Finished

## Numerical Analysis of Stochastic Differential Equations: New Challenges

1/10/15 → 30/09/17

Project: Research Fellowship

## Epsrc Doctoral Training Grant | Liang, Yanfeng

Greenhalgh, D., Mao, X. & Liang, Y.

EPSRC (Engineering and Physical Sciences Research Council)

1/10/12 → 3/10/16

Project: Research Studentship - Internally Allocated

## Epsrc Doctoral Training Grant / RA8099

EPSRC (Engineering and Physical Sciences Research Council)

1/10/12 → 30/09/16

Project: Research Studentship - Internally Allocated

## Datasets

## A Stochastic Differential Equation Model for the Spread of HIV Amongst People Who Inject Drugs

Liang, Y. (Creator), Greenhalgh, D. (Creator) & Mao, X. (Creator), University of Strathclyde, 2015

DOI: 10.15129/bf5ba8b5-d484-43c7-8006-14fb76819be2

Dataset