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

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

Combination interventions for Hepatitis C and Cirrhosis reduction among people who inject drugs: An agent-based, networked population simulation experiment

Hepatitis C virus (HCV) infection is endemic in people who inject drugs (PWID), with prevalence estimates above 60 percent for PWID in the United States. Previous modeling studies suggest that direct acting antiviral (DAA) treatment can lower overall prevalence in this population, but treatment is often delayed until the onset of advanced liver disease (fibrosis stage 3 or later) due to cost. Lower cost interventions featuring syringe access (SA) and medically assisted treatment (MAT) for addiction are known to be less costly, but have shown mixed results in lowering HCV rates below current levels. Little is known about the potential synergistic effects of combining DAA and MAT treatment, and large-scale tests of combined interventions are rare. While simulation experiments can reveal likely long-term effects, most prior simulations have been performed on closed populations of model agents--a scenario quite different from the open, mobile populations known to most health agencies. This paper uses data from the Centers for Disease Control's National HIV Behavioral Surveillance project, IDU round 3, collected in New York City in 2012 by the New York City Department of Health and Mental Hygiene to parameterize simulations of open populations. Our results show that, in an open population, SA/MAT by itself has only small effects on HCV prevalence, while DAA treatment by itself can significantly lower both HCV and HCV-related advanced liver disease prevalence. More importantly, the simulation experiments suggest that cost effective synergistic combinations of the two strategies can dramatically reduce HCV incidence. We conclude that adopting SA/MAT implementations alongside DAA interventions can play a critical role in reducing the long-term consequences of ongoing infection

Int J Drug Policy

Network modelling is a valuable tool for simulating hepatitis C virus (HCV) and HIV transmission among people who inject drugs (PWID) and assessing the potential impact of treatment and harm-reduction interventions. In this paper, we review literature on network simulation models, highlighting key structural considerations and questions that network models are well suited to address. We describe five approaches (Erd\uf6s-R\ue9nyi, Stochastic Block, Watts-Strogatz, Barab\ue1si-Albert, and Exponential Random Graph Model) used to model partnership formation with emphasis on the strengths of each approach in simulating different features of real-world PWID networks. We also review two important structural considerations when designing or interpreting results from a network simulation study: (1) dynamic vs. static network and (2) injection only vs. both injection and sexual networks. Dynamic network simulations allow partnerships to evolve and disintegrate over time, capturing corresponding shifts in individual and population-level risk behaviour; however, their high level of complexity and reliance on difficult-to-observe data has driven others to develop static network models. Incorporating both sexual and injection partnerships increases model complexity and data demands, but more accurately represents HIV transmission between PWID and their sexual partners who may not also use drugs. Network models add the greatest value when used to investigate how leveraging network structure can maximize the effectiveness of health interventions and optimize investments. For example, network models have shown that features of a given network and epidemic influence whether the greatest community benefit would be achieved by allocating hepatitis C or HIV treatment randomly, versus to those with the most partners. They have also demonstrated the potential for syringe services and "buddy sharing" programs to reduce disease transmission.U38 PS004644/PS/NCHHSTP CDC HHSUnited States/P30 DA040500/DA/NIDA NIH HHSUnited States/P30 AI042853/AI/NIAID NIH HHSUnited States/CC/CDC HHSUnited States/R01 DA046527/DA/NIDA NIH HHSUnited States/2022-01-05T00:00:00Z31740175PMC872979210782vault:4066

Incorporation of awareness programs into a model of the spread of HIV/AIDS among people who inject drugs

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

Mathematical models for the transmission dynamics of HIV and its progression to AIDS in Ireland

Despite advances m understanding the basic biology of HIV the aetiological agent of AIDS, medica1, public health and health education planning is plagued by uncertainties Mathematical models of the dynamics of HIV transmission and its progression to AIDS can clarify what data must be collected in order to predict future prevalence, make predictions about the likely effect of future intervention pobcies and provide predictions for several decades ahead. The motivation of this research is to provide reliable estimates of the incidence of HIV infection and AIDS in the Irish population. In Chapters 1 and 2 we discuss the background to the disease in Ireland and the role of mathematical modelling in the spread AIDS. From this we show where key epidemiological data is lacking and how models to date have concentrated on the spread of the disease within the homosexual population. In Chapter 3 we describe the adjustment of the number of AIDS cases to allow for reporting delays Subsequently we consider the solution of the integral equation models generated by the back-projection method for the adjusted AIDS cases. In Chapter 4 we improve upon the estimates of the incidence of HIV infection found in Chapter 3 by evaluating the integral arising in back-projection, in terms of a gamma function plus a remainder in the form of a series in t. We also provide error bounds for the remainder. This new solution allows us to predict new and more reliable estimates of the level of HIV infection m Ireland. In Chapter 5 we provide estimates of the minimum number of deaths from AIDS, based on the number of AIDS cases known to the Department of Health and the distribution of the length of survival times after the onset of AIDS. The results of a HIV transmission survey are presented in Chapter 6 These provide detailed information on the habits and behaviour of those at risk of HIV infection and allow us to derive preliminary model parameters. Finally in Chapter 7 we develop and implement a nonlinear deterministic differential equation model for the spread of HIV and its progression to AIDS m the Irish IVDU and homosexual populations. We examine the effects of likely intervention policies on the extent and spread of the disease and we make recommendations based on our thesis findings

Use of generating functions in HIV/AIDS transmission models

Dissertation submitted in partial fulfillment for degree of Masters of Science in Industrial Mathematics in the Department of Mathematics, University of NairobiThis study is concerned with the mathematical modeling for human immunodeficient virus (HIV) transmission epidemics. The mathematical models are specified by stochastic differential equations. The differential equations are solved by use of Generating Functions (GF).In the process of literature review, a conceptual framework is drawn which summarizes the literature on HIV/ AIDS transmission epidemic models. Models based on Mother to child transmission (MTCT) (age group 0-5 years), Heterosexual transmission (age group 15 and more years) and combined case (incorporating all groups and the two modes of transmission) are developed and the expectations and variances of Susceptible (S) persons, Infected (I) persons and AIDS cases found. It is shown from the combined model that MTCT and Heterosexual models are special cases of the combined model.General aspects of modeling HIV/ AIDS are described in chapter 1, Chapter 2 focuses on the literature review. MTCT model is formulated in chapter 3. Heterosexual model is developed in chapter 4, Chapter 5 focuses on the development of the Combined model. Chapter 6 concludes the study.This study is concerned with the mathematical modeling for human immunodeficient virus (HIV) transmission epidemics. The mathematical models are specified by stochastic differential equations. The differential equations are solved by use of Generating Functions (GF).In the process of literature review, a conceptual framework is drawn which summarizes the literature on HIV/ AIDS transmission epidemic models. Models based on Mother to child transmission (MTCT) (age group 0-5 years), Heterosexual transmission (age group 15 and more years) and combined case (incorporating all groups and the two modes of transmission) are developed and the expectations and variances of Susceptible (S) persons, Infected (I) persons and AIDS cases found. It is shown from the combined model that MTCT and Heterosexual models are special cases of the combined model.General aspects of modeling HIV/ AIDS are described in chapter 1, Chapter 2 focuses on the literature review. MTCT model is formulated in chapter 3. Heterosexual model is developed in chapter 4, Chapter 5 focuses on the development of the Combined model. Chapter 6 concludes the study

The mathematical modelling of the transmission dynamics of HIV/AIDS and the impact of antiviral therapies

Thesis presented for the degree of Doctor of Philosophy, Department of Electronics and Mathematics, Faculty of Science, Technology and Design, the University of LutonThis thesis is concerned with the structure, analysis and numerical solution of the mathematical models used to estimate the transmission dynamics of the Human Immunodeficiency Virus (HIV)) the causative agent of Acquired Immune Deficiency Syndrome (AIDS). Investigations show that the devised deterministic mathematical models in term of system of first-order non-linear ordinary differential equations (ODEs) follow the stochastic nature of the problem at any time. In this thesis a generic form of the deterministic mathematical models is introduced which mirrors the transmission dynamics of HIV/AIDS in populations with different states of affairs, which leads to the division of large-scale and complex mathematical models. When analysing and;or solving a large-scale system of ODEs numerically, the key element in speeding up the process is selecting the maximum possible time step. This work introduces some new techniques used to estimate the maximum possible time step, avoiding the appearance of chaos and divergence in the solution when they are not features of the system. The solution to these mathematical models are presented graphically and numerically, aiming to identify the effect of the anti-HIV therapies and sex education in controlling the disease. The numerical results presented in this thesis indicate that lowering the average number of sexual partners per year is more effective in controlling the disease than the current anti-HIV treatments. For the purpose of this study the mathematical software 'Mathematica 3.0' was used to solve the system of differential equations, modelling HIV/AIDS propagation. This package also provided the graphical detail incorporated in the thesis