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

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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.
LanguageEnglish
Article number6757928
Number of pages14
JournalComputational and Mathematical Methods in Medicine
Volume2016
DOIs
Publication statusPublished - 10 Mar 2016

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Stochastic Equations
Drugs
Differential equations
HIV
Differential equation
Pharmaceutical Preparations
Needles
Persistence
Stochasticity
Noise
Model
Cleaning
Needle Sharing
Parameter Perturbation
Deterministic Model
Stationary Distribution
Unique Solution
Extinction
Sharing
Acquired Immunodeficiency Syndrome

Keywords

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

Cite this

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title = "A stochastic differential equation model for the spread of HIV amongst people who inject drugs",
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.",
keywords = "HIV, AIDS, parameter perturbation, extinction, persistence, stationary distribution, stochastic differential equations, Brownian motion, environmental stochasticity",
author = "Yanfeng Liang and David Greenhalgh and Xuerong Mao",
year = "2016",
month = "3",
day = "10",
doi = "10.1155/2016/6757928",
language = "English",
volume = "2016",
journal = "Computational and Mathematical Methods in Medicine",
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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

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