### Abstract

Original language | English |
---|---|

Article number | 6757928 |

Number of pages | 14 |

Journal | Computational and Mathematical Methods in Medicine |

Volume | 2016 |

DOIs | |

Publication status | Published - 10 Mar 2016 |

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### Keywords

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

### Cite this

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**A stochastic differential equation model for the spread of HIV amongst people who inject drugs.** / Liang, Yanfeng; Greenhalgh, David; Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A stochastic differential equation model for the spread of HIV amongst people who inject drugs

AU - Liang, Yanfeng

AU - Greenhalgh, David

AU - Mao, Xuerong

PY - 2016/3/10

Y1 - 2016/3/10

N2 - 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.

AB - 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.

KW - HIV

KW - AIDS

KW - parameter perturbation

KW - extinction

KW - persistence

KW - stationary distribution

KW - stochastic differential equations

KW - Brownian motion

KW - environmental stochasticity

UR - http://www.hindawi.com/journals/cmmm/aip/650438/

U2 - 10.1155/2016/6757928

DO - 10.1155/2016/6757928

M3 - Article

VL - 2016

JO - Computational and Mathematical Methods in Medicine

JF - Computational and Mathematical Methods in Medicine

SN - 1748-670X

M1 - 6757928

ER -