Mathematical modelling of hepatitis C treatment for injecting drug users

doi:10.1016/j.jtbi.2010.12.041

Journal of Theoretical Biology

Volume 274, Issue 1, 7 April 2011, Pages 58-66

https://doi.org/10.1016/j.jtbi.2010.12.041 Get rights and content

Abstract

Hepatitis C virus (HCV) is a blood-borne infection that can lead to progressive liver failure, cirrhosis, hepatocellular carcinoma and death. In developed countries, the majority of HCV infections are transmitted via injecting drug users (IDUs). Despite effective antiviral treatment for HCV, very few active IDUs are treated. Reluctance to treat is partially due to the risk of reinfection. We develop a mathematical model of HCV transmission amongst active IDUs, and examine the potential effect of antiviral treatment. As most mathematical models of interventions utilise a treatment function proportional to the infected population, but many policy implementations set fixed yearly targets for specific numbers treated, we study the effects of using two different treatment terms: annually treating a proportion of infecteds or a fixed number of infecteds. We examine the behaviour of the two treatment models and find different bifurcation behaviours in each case. We calculate analytical solutions for the treatment level needed for disease clearance or control, and observe that achievable levels of treatment can result in control or eradication across a wide range of prevalence levels. Finally, we calculate the sensitivity of the critical treatment threshold to the model parameters, and find that for a given observed prevalence, the injecting duration and infection risk play the most important role in determining the treatment level needed. By contrast, the sensitivity analysis indicates the presence (or absence) of immunity does not alter the treatment threshold. We conclude by discussing the public health implications of this work, and comment on the importance and feasibility of utilising treatment as prevention for HCV spread amongst IDUs.

Introduction

Hepatitis C virus (HCV) is a blood-borne disease with an estimated global prevalence of 2–3%, or 130–170 million people, and is one of the leading causes of chronic liver disease (Shepard et al., 2005). If left untreated, about 7–18% of those infected will progress to liver disease within 20 years, which can result in progressive liver failure, cirrhosis, hepatocellular carcinoma and death (Seeff, 2009).

In developed countries, the primary mode of transmission is amongst injecting drug users (IDUs) through needle and syringe sharing, with over 80% of new cases in the UK attributed to injecting drugs (ACMD, 2009). HCV is easily transmitted amongst IDUs, with 15–90% of IDUs testing positive for HCV antibodies (Page-Shafer et al., 2008, Judd et al., 2005, Hahn et al., 2002). Current preventative measures to reduce HCV transmission such as health education and advice, needle and syringe exchange, and opiate substitution therapy aim to prevent transmission by reducing unsafe injecting (ACMD, 2009). However, public health surveillance indicates substantial decreases in prevalence have not been achieved (Palmateer et al., 2010).

HCV antiviral treatment (peginterferon- $α$ and ribavirin) is effective, resulting in viral clearance in 45–80% of cases, depending on HCV genotype (NICE, 2000). Prior to 2002, guidelines in the US and UK recommended against treating active IDUs. However, current guidelines now do not exclude IDUs from treatment eligibility, given mounting evidence that IDUs exhibit a similar response to treatment, and are just as compliant with treatment as ex- or non-IDUs (Hellard et al., 2009, NICE, 2006, Shepherd et al., 2007, NIH, 2002). Nevertheless, despite these recommendations and the high numbers of IDUs infected, very few ( $< 3 - 4 %$ ) active IDUs have ever been treated (Grebely et al., 2006, Seal et al., 2005). Studies on treatment barriers have indicated a reluctance to treat active IDUs due to the possibility of subsequent reinfection (Booth et al., 2001, Reimer et al., 2005, Foster, 2008).

We examine the potential of antiviral treatment as a prevention strategy for HCV amongst IDUs. By using antiviral treatment to reduce prevalence amongst active IDUs, the treatment can act to reduce the risk of infection for other IDUs. But to what extent? This paper examines the potential impact of HCV treatment on prevalence and transmission, including the possibility of reinfection. We incorporate two treatment scenarios (treating a proportion of infected IDUs, and a fixed number of IDUs) and examine the resulting dynamics and treatment needed for eradication. Treating a constant proportion of the population is the function most commonly used in infectious disease modelling. However, annually treating a fixed number of IDUs would be more likely in the initial stages of a treatment delivery programme, or in situations with budget constraints. Hence, we analyse both situations.

Section snippets

Background and assumptions for the model

Infection with HCV leads to a brief acute stage, which is relatively short (on the order of weeks to months) in comparison to the prolonged chronic stage (on the order of decades) (ECMDDA, 2004). In the first few weeks, viral levels may be undetectable, increasing but possibly remaining low during the remainder of the acute stage. A fraction of people (about 26%) spontaneously clear the acute infection (Micallef et al., 2006). The specifics of spontaneous clearance are not well known, although

Details and explanation of the model

We use a system of ordinary differential equations to describe the transmission of HCV amongst active IDUs. We utilise a four compartment model, tracking susceptible, chronically infected, treated, and immune IDUs. Susceptible IDUs become infected through sharing of needles with an infected IDU. About one quarter spontaneously clear the infection, and become susceptible or immune. The remaining three-quarters progress to chronic infection. Chronic infecteds can be treated, with a certain chance

Proportional treatment

At steady state, $N = θ / μ$ . Setting the left-hand side of Eqs. (7), (10) to zero (with $g (c)$ defined by (12) and solving for the equilibrium values we find two steady states. One is the trivial disease-free steady state, $FP 1 = (x_{1}^{*}, c_{1}^{*}, t_{1}^{*}, z_{1}^{*}) = (1, 0, 0, 0) .$ The second equilibrium is the infected endemic steady state, defined by FP2=(x₂^⁎, c₂^⁎, t₂^⁎, z₂^⁎), where $t_{2}^{*} = \frac{ϕ c_{2}^{*}}{ω + μ},$ $x_{2}^{*} = \frac{μ (ω + μ) + ω α σ ϕ c_{2}^{*}}{π (1 - δ + δ ξ) (ω + μ) c_{2}^{*} + μ (ω + μ)},$ $z_{2}^{*} = \frac{π δ ξ ω α σ ϕ (c_{2}^{*})^{2} + π δ ξ μ (ω + μ) c_{2}^{*}}{μ π (1 - δ + δ ξ) c_{2}^{*} + μ^{2}} + \frac{ω α (1 - α) ϕ c_{2}^{*}}{μ (ω + μ)},$ $c_{2}^{*} = - μ (ω - μ) [\frac{ω (1 - α) ϕ}{ω + μ} - ϕ - μ] - π (1 -$

Numerical methods

Numerical simulations of Eqs. (7), (11) were performed using the MATLAB ODE solver, ode45. Because HCV is not in the breakout epidemic phase in the majority of real-world settings, the simulations were run until steady state ( $τ = 600$ years), and then treatment initiated. The parameter values are given in Table 2. For specific predictions at untreated equilibrium infected prevalences of 20%, 40%, and 60%, the values of $π$ used were $π = 0.1468$ , $0.2033$ , and $0.3307$ , respectively.

The calculation of the

Discussion and conclusions

In this paper we analyse a mathematical model of HCV transmission amongst active IDUs, examining the potential for antiviral treatment to reduce HCV transmission. Despite guidelines stating that current IDUs should not be excluded from obtaining treatment, very few are treated, with the risk of reinfection used as justification for withholding treatment. However, our model indicates that antiviral treatment could act as a prevention measure for the wider IDU community by reducing prevalence

Acknowledgements

Grant support: This manuscript was supported by the Scottish Government Hepatitis C Action Plan, NCCRCD/NIHR CRDHB, and the MRC New Investigator Award.

References (32)

R.M. Granich et al.
Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: a mathematical model
The Lancet
(2009)
J. Grebely et al.
Uptake of hepatitis C virus (HCV) treatment among injection drug users (IDUS) in Vancouver, Canada
Journal of Hepatology
(2006)
J. Grebely et al.
Treatment uptake and outcomes among current and former injection drug users receiving directly observed therapy within a multidisciplinary group model for the treatment of hepatitis C virus infection
International Journal of Drug Policy
(2007)
A.H. Litwin et al.
Successful treatment of chronic hepatitis C with pegylated interferon in combination with ribavirin in a methadone maintenance treatment program
Journal of Substance Abuse Treatment
(2009)
S.H. Mehta et al.
Protection against persistence of hepatitis C
The Lancet
(2002)
C. Nordt et al.
Incidence of heroin use in Zurich, Switzerland: a treatment case register analysis
The Lancet
(2006)
C.W. Shepard et al.
Global epidemiology of hepatitis C virus infection
The Lancet Infectious Diseases
(2005)
ACMD. 2009. The primary prevention of Hepatitis C among injecting drug users. Technical...
S.M. Blower et al.
A tale of two futures: HIV and antiretroviral therapy in San Francisco
Science
(2000)
J.C.L. Booth et al.
Clinical guidelines on the management of Hepatitis C
Gut
(2001)

ECMDDA, 2004. Hepatitis C and injecting drug use: impact, costs and policy options. Office for Official Publications of...

G.R. Foster

Injecting drug users with chronic hepatitis C: should they be offered antiviral therapy?

Addiction

(2008)

J.A. Hahn et al.

Hepatitis C virus seroconversion among young injection drug users: relationships and risks

The Journal of Infectious Diseases

(2002)

M. Hellard et al.

Hepatitis C treatment for injection drug users: a review of the available evidence

Clinical Infectious Diseases

(2009)

M. Hickman et al.

Hepatitis C virus (HCV) prevalence and injecting risk behaviour in multiple sites in England in 2004

Journal of Viral Hepatitis

(2007)

M. Hickman et al.

Assessing IDU prevalence and health consequences (HCV, overdose and drug-related mortality) in a primary care trust: implications for public health action

Journal of Public Health (Oxford)

(2009)

Cited by (87)

Optimism and eternal vigilance: Gathering disease, responsible subjects and the hope of elimination in the new hepatitis C treatment era
2023, International Journal of Drug Policy
The advent of direct-acting antiviral hepatitis C medications has reshaped experiences of hepatitis C treatment and cure. Positioned as a treatment revolution, the new medications mean a world without hepatitis C has become imaginable, and this optimism is reflected in Australia's commitment to the WHO's target of ‘eliminating’ the virus as a public health threat by 2030. Alongside optimism about new treatments, Australia's current National Hepatitis C Strategy also emphasises the importance of partnerships with, and the ‘meaningful involvement’ of, priority populations for elimination to be achieved. We draw on Fraser and Seear's (2011) work on hepatitis C as a ‘gathering’ to examine these developments, and to approach hepatitis C as a disease in-the-making. Analysing 50 interviews conducted with people affected by the virus, we identify three key articulations that combine to trouble the distinction between old and new treatments: (1) the new treatment constitutes the disease as readily curable; (2) nevertheless, those who have been cured are responsibilised against acquiring it again by managing and monitoring their conduct; and (3) in the process, hepatitis C becomes re-constituted as an ongoing threat requiring continual post-cure medical and other monitoring. We argue that while treatment experiences have dramatically improved, responsibilising people affected by hepatitis C to attain cure in the context of an elimination agenda constitutes cure as valuable as much for the greater good as for self-care. This raises pressing ethical and political questions. Overall, we shed light on how, even in a context shaped by the availability of highly effective treatment, the hepatitis C-free body is never hepatitis C-free, but must be continually reproduced through regulatory practices.
Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model
2023, Mathematics and Computers in Simulation
The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare–Bendixson theorem in combination with the Bendixson–Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $R_{0} \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $R_{0} > 1$ . Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorous mathematical analysis, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.
A fractional order differential equation model for Hepatitis B virus with saturated incidence
2021, Results in Physics
This manuscript presents a nonlinear fractional order Hepatitis B virus (HBV) model with saturated incidence. Model governing equations are defined using Caputo fractional derivatives. We use the Adams–Bashforth-Moulton technique to compute numerical solutions of the presented epidemic model. Generally, fractional derivatives are used to model real-world phenomena that contains nonlocal effects, history and/or memory. The main objective is to develop and explore the dynamics of an HBV fractional derivative epidemic model via the Caputo definition. Firstly, we discuss the model basic properties which include positivity of solutions and the invariant regions where the solution set is bounded using the generalized fractional mean value theorem. The model dynamics are described based on the reproduction number $R_{0}$ . Thereafter, we use the fractional Routh-Hurwitz stability criterion as well as simple matrix algebra to determine sufficient conditions for the stability of the equilibrium points. Lastly, we present graphical representation of numerical results. The results indicate the importance of fractional order in modeling HBV transmission dynamics and ensures that by including the memory effects, the model is more appropriate and insightful for such a study.
The hepatitis C care cascade among people who inject drugs accessing harm reduction services in Catalonia: Major gaps for migrants
2021, International Journal of Drug Policy
This study aimed to describe the HCV cascade of care among people who inject drugs (PWID) in Catalonia, as well as to compare the observed gaps in care between Spanish-born and migrant PWID.
A cross-sectional study of PWID (N = 410) attending four harm reduction services (HRS) was performed in 2016–17 (HepCdetect II Study). Participants were tested for both HCV antibodies (rapid testing) and RNA (from dried blood spot samples). The HCV care cascade was estimated from HCV testing results combined with self-reported data on previous testing, diagnosis and treatment collected through a questionnaire. Logistic regressions were used to test for an association between migration status and the proportions observed in each step of the HCV care cascade adjusting for age, sex, years of injection, homelessness, and treatment for drug dependence.
Overall, 85.4% were men and 28.0% were migrants. Among Spanish-born (n = 295) and migrant (n = 115) PWID participants in the study, 96.6% vs. 88.6% had previously been HCV screened (AOR=3.11; 95% CI: 1.11–8.65), 79.3% vs. 80.9% were antibody positive, and 70.7% vs. 67.6% were HCV-RNA positive or cured with treatment; among the latter, 36.6% vs. 18.2% had started treatment (AOR=2.41; 95% CI: 1.09–5.34), and 20.6% vs. 9.1% had been cured by treatment, respectively. Unawareness of having hepatitis C was more common among migrants than Spanish-born PWID (46.0% and 31.5%, respectively; p<0.05).
This study estimates the HCV care cascade among Spanish-born and migrant PWID in Catalonia for the very first time, and highlights a higher attrition of migrant PWID in all HCV care cascade stages. The observed limited linkage to care and treatment by PWID that attend the HRS network warrants future implementation of decentralized diagnosis and antiviral treatment. Strategies focusing on migrants by increasing HCV screening coverage and treatment access will be especially relevant in our setting.
How to think with models and targets: Hepatitis C elimination as a numbering performance
2021, International Journal of Drug Policy
Citation Excerpt :
Not only are DAAs enacted to afford the promise of cure amongst infected individuals, they potentiate a future without hepatitis C at the level of populations, territories, and nation states (Rhodes &Lancaster, 2019b). This population-level impact is evidence-made through mathematical models of theoretical projection (Hickman et al., 2015; Pitcher et al., 2018), in which the prevention potential of hepatitis C treatment curative outcomes is projected at a scale sufficient to reduce the pool of infections, and thus transmissions, in the population at risk, including accounting for rates of re-infection amongst those treated (Martin et al., 2011). The publication of models theorising hepatitis C cure as having treatment-as-prevention effects is a foundational moment in evidence-making (Martin et al., 2011, 2013a; Zeiler et al., 2010).
The field of public health is replete with mathematical models and numerical targets. In the case of disease eliminations, modelled projections and targets play a key role in evidencing elimination futures and in shaping actions in relation to these. Drawing on ideas within science and technology studies, we take hepatitis C elimination as a case for reflecting on how to think with mathematical models and numerical targets as ‘performative actors’ in evidence-making. We focus specifically on the emergence of ‘treatment-as-prevention’ as a means to trace the social and material effects that models and targets make, including beyond science. We also focus on how enumerations are made locally in their methods and events of production. We trace the work that models and targets do in relation to three analytical themes: governing; affecting; and enacting. This allows us to situate models and targets as technologies of governance in the constitution of health, which affect and are affected by their material relations, including in relation to matters-of-concern which extend beyond calculus. By emphasising models and targets as enactments, we draw attention to how these devices give life to new enumerated entities, which detach from their calculative origins and take flight in new ways. We make this analysis for two reasons: first, as a call to bring the social and enumeration sciences closer together to speculate on how we might think with models and targets differently and more carefully; and second, to encourage an approach to science which treats evidencing-making interventions, such as models and targets, as performative and political.
Global Epidemiology of Viral Hepatitis
2020, Gastroenterology Clinics of North America

View all citing articles on Scopus

View full text

Mathematical modelling of hepatitis C treatment for injecting drug users

Abstract

Introduction

Section snippets

Background and assumptions for the model

Details and explanation of the model

Proportional treatment

Numerical methods

Discussion and conclusions

Acknowledgements

The Lancet

Journal of Hepatology

International Journal of Drug Policy

Journal of Substance Abuse Treatment

The Lancet

The Lancet

The Lancet Infectious Diseases

A tale of two futures: HIV and antiretroviral therapy in San Francisco

Science

Clinical guidelines on the management of Hepatitis C

Gut

Injecting drug users with chronic hepatitis C: should they be offered antiviral therapy?

Addiction

Hepatitis C virus seroconversion among young injection drug users: relationships and risks

The Journal of Infectious Diseases

Hepatitis C treatment for injection drug users: a review of the available evidence

Clinical Infectious Diseases

Hepatitis C virus (HCV) prevalence and injecting risk behaviour in multiple sites in England in 2004

Journal of Viral Hepatitis

Assessing IDU prevalence and health consequences (HCV, overdose and drug-related mortality) in a primary care trust: implications for public health action

Journal of Public Health (Oxford)