Elsevier

Mathematical Biosciences

Volume 263, May 2015, Pages 51-69
Mathematical Biosciences

Differential characteristics of primary infection and re-infection can cause backward bifurcation in HCV transmission dynamics

https://doi.org/10.1016/j.mbs.2015.02.002Get rights and content

Highlights

  • Backward bifurcation occurs in HCV (hepatitis C virus) transmission dynamics.

  • Four scenarios under which backward bifurcation does not occur have been identified.

  • Five parameters have been identified as the most significant drivers of the disease dynamics.

Abstract

Backward bifurcation, a phenomenon typically characterized by the co-existence of multiple stable equilibria when the associated reproduction number of the model is less than unity, has been observed in numerous disease transmission models. This study establishes, for the first time, the presence of this phenomenon in the transmission dynamics of hepatitis C virus (HCV) within an IDU population. It is shown that the phenomenon does not exist under four scenarios, namely (i) in the absence of re-infection, (ii) in the absence of differential characteristics of HCV infection (with respect to infectivity, progression, treatment and recovery) between re-infected individuals and primary-infected individuals, (iii) when re-infected and treated individuals do not transmit HCV infection and (iv) when the average infectivity-adjusted duration of re-infection is less than that of primary infection. This study identifies, using sensitivity analysis, five parameters of the model that have the most influence on the disease transmission dynamics, namely: effective contact rate, progression rate from acute to chronic infection, recovery rate from acute infection, natural death rate and the relative infectiousness of chronically-infected individuals. Numerical simulations of the model show that the re-infection of recovered individuals has marginal effect on the HCV burden (as measured in terms of the cumulative incidence and the prevalence of the disease) in the IDU community. Furthermore, treatment of infected IDUs, even for small rate (such as 4%), offers significant impact on curtailing HCV spread in the community.

Introduction

Hepatitis C, a blood-borne viral infectious disease caused by the single-strained RNA Hepatitis C virus (HCV) [2], [10], continues to pose major public health challenges globally. The World Health Organization (WHO) recently estimated the HCV prevalence to be between 2 and 3% (i.e., 130–170 million people currently live with HCV infection globally) [31], [38], [49], [50]. While countries in Africa and Asia have the highest reported prevalence, industrialized countries in North America, Northern and Western Europe and Australia have lower prevalence (Germany (0.6%), Canada (0.8%), France (1.1%), and Australia (1.1%) have relatively low rates of HCV seroprevalence, with the USA (1.8%), Japan (1.5–2.3%), and Italy (2.2%) having slightly higher seroprevalence rates) [38], [49].

The primary mode of HCV transmission is through blood contact [1], [50]. In particular, there are three main age-specific transmission patterns [1], [42]. The first is the 30–49 year age (middle age) group in developed countries (such as the United Kingdom and USA) [31], [42]. For this age group, injecting drug use (IDU) is the major cause of HCV infection (via needle and syringe-sharing; with over 80% of new cases attributed to injecting drugs use) [31], [42]. The second transmission pattern is for the elderly (as is the case in Japan [26], [35], [42], [48]). The third transmission pattern entails all age groups. For these two later categories, the causes (or risk factors) of HCV infection include unsafe therapeutic injections (performed by both healthcare professionals and non-professionals) and blood transfusion from unscreened donors. In addition to the aforementioned HCV risk factors, other factors, such as exposure to blood by the healthcare workers (mostly through contact with contaminated needles), mother-to-child transmission, sex with an infected partner, sex with multiple partners and other healthcare-related procedures, further contribute to HCV transmission [38].

The common symptoms of HCV infection include jaundice, dark urine, fatigue, nausea, vomiting, and abdominal pain [13], [25]. While majority of patients with acute HCV will progress to chronic infection [2], [10], [13], about 25% of cases clear the virus and build natural immunity against re-infection [18]. The mean incubation period for acute HCV infection is 7 weeks [24]. Unfortunately, up to 90% of HCV-infected individuals (acute or chronic) may not to be aware of their infection status (i.e., they are asymptomatically-infected). Consequently, if undetected and untreated, about 7–18% of these (asymptomatically-infected) individuals will progress to develop liver disease, such as liver fibrosis, cirrhosis, hepatocellular carcinoma, within 20 years (and about 5%–7% of these patients may ultimately die of HCV) [2], [10], [13], [15], [25], [31], [49], [50].

HCV-infected individuals can be treated using a combination therapy with pegylated interferon and ribavirin (having a response rate of 40%–80%) [13], [17] (furthermore, several new anti-HCV drugs have recently been approved and/or are undergoing various stages of clinical trials [29]). Although there is currently no safe and effective vaccine for use against HCV infection in humans, efforts are underway to develop one [13], [15]. Another intervention strategy for controlling HCV transmission among IDUs is increasing the access to unused syringes and needles, aimed at reducing the frequency of sharing/unsafe injection needles (it should, however, be mentioned that although this approach may have a positive effect on reducing HCV transmission, there is no evidence for substantial reductions in HCV prevalence) [32], [41], [43].

Several mathematical and statistical models have been developed and used to gain insight into the transmission dynamics of HCV in an IDU population [8], [9], [27], [31], [40], [45], [46], [52]. Corson et al. [9] developed a deterministic compartmental mathematical model for the spread of HCV in an IDU population that has been separated into two groups (naive and experienced) based on the time since the onset of injection (and includes measures that allow for the prevention of HCV infection). Sutton et al. [40] used statistical modelling to estimate the force of infection of HCV and hepatitis B virus in England and Wales (using saliva sample of IDUs) for 1998–2003.

Vickerman et al. [45] used a deterministic model to simulate the transmission of HCV in IDUs in London, England and assessed the impact of intervention measures that reduced syringe and needle sharing in some of the targeted IDU populations. Elbasha [13] introduced the effect of the re-infection of recovered primary infected individuals (and associated differential infection characteristics between re-infected individuals and primary infected individuals) on HCV transmission dynamics via the use of a deterministic model. Furthermore, Elbasha [13] provided a rigorous qualitative analysis of a special case of the model in [13] (where re-infected individuals behave in the same manner as primary infected individuals, with respect to disease infectivity, recovery, progression and treatment).

The main purpose of this study is to provide qualitative insight into the effect of treatment on the transmission dynamics of HCV in an IDU population. To achieve this objective, the treatment model developed in [13] will be considered (and fully analyzed, unlike the special case considered in [13]). Furthermore, the effect of uncertainties on the associated parameters of the model (on the overall simulation results obtained) will be assessed using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) [3], [30], [36]. The paper is organized as follows. A treatment model for HCV transmission dynamics in an IDU population is formulated in Section 2. The analogous model in the absence of treatment is analyzed in Section 3. The full treatment model is analyzed in Section 4. Uncertainty and sensitivity analyses of the treatment model are also carried out. Numerical simulation results are also presented.

Section snippets

Model formulation

The HCV transmission model to be considered in this section is developed in [13], and is formulated by splitting the total IDU population at time t, denoted by N(t), into mutually-exclusive compartments of susceptible, S(t), acutely-infected, I(t), chronically-infected, P(t), treated chronically infected, T(t), recovered with partial immunity, R(t), acutely re-infected, V(t), untreated chronically re-infected, W(t) and treated chronically re-infected, Q(t), individuals, so that N(t)=S(t)+I(t)+P(

Analysis of model without treatment

The special case of the treatment model (2.2) in the absence of treatment (referred to as treatment-free model), obtained by T(t) = Q(t) = χT = χQ = τ = ϕ = ζ = ρ = θ = 0 in (2.2), is given by the following deterministic system of non-linear differential equations [13]: dSdt=Λ+γRλwtSμS,dIdt=λwtS(σ+ɛ+μ)I,dPdt=ɛI(δ+μ)P,dRdt=σI+δP+ασV+ηδWψλwtR(γ+μ)R,dVdt=ψλwtR(ασ+κɛ+μ)V,dWdt=κɛV(ηδ+μ)W,where, λwt=β(I+πP+υV+ωπW)NwtandNwt(t)=S(t)+I(t)+P(t)+R(t)+V(t)+W(t).

It is worth noting that the

Analysis of the treatment model

Consider, now, the treatment model (2.2).

Acknowledgments

One of the authors (FN) acknowledges, with thanks, the support of the University of Manitoba and the Manitoba Graduate Scholarship. ABG acknowledges the support of NSERC of Canada. The authors are grateful to the anonymous reviewers for their constructive comments.

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