Rachel Hart-Malloy^{1†} and Gregory DiRienzo^{2†*}

*Correspondence: Gregory DiRienzo adirienzo@albany.edu

†These authors contributed equally to this work.

2. Department of Epidemiology and Biostatistics, University at Albany, State University of New York, Rensselaer, USA.

1. AIDS Institute, New York State Department of Health, Albany NY; Department of Epidemiology and Biostatistics, University at Albany, State University of New York, Rensselaer, USA.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Background: **The use of mathematic modeling to better understand the spread of Hepatitis C and
the impact of interventions can be invaluable for localities, states and countries with a large burden of
injection drug use. New York State (NYS) is estimated to be home to a large number of people who inject
drugs (PWID), however the burden of hepatitis C among this population and the impact of interventions
are not known and/or fully understood. Since accurate modeling can be complex and require costly data
analysis software to implement, the purpose of this paper was to derive a methodology to accurately model
the prevalence of hepatitis C at the state level that is tractable and easily implemented with free software.

**Methods:** A* non-stationary quasi-Markov* model, is proposed that is implemented using free software R^{©}. The
methodology aims to estimate hepatitis C prevalence and evaluate impacts of interventions on disease
burden among PWID in NYS. The approach is "quasi-Markov" because transition probabilities among
states change in time and depend on past information. Interventions evaluated aimed to: reduce sharing
needles and/or other drug use paraphernalia; increase needle disinfection rates; and increase availability of
clean needles and other drug use paraphernalia.

**Results:** The quasi-Markov model estimated hepatitis C prevalence among PWID reached an equilibrium
value of 63.6 percent after 50 years. In order to eliminate disease, all proposed interventions were needed,
resulting in an estimated prevalence less than 1.0 percent, 56 years after implementation. Using parameters
defined for an alternate study modeling hepatitis C prevalence found similar results, serving to reinforce
the validity of the proposed methods.

**Conclusion:** The results of this analysis demonstrate the feasibility of using a non-stationary quasi-Markov methodology to model the spread of hepatitis C among PWID using free programming software.
Coding provided can be used by other researchers to use and modify for their own purposes and could
further impact this field of research as well as inform and promote additional education prevention and
interventions for PWID.

**Keywords:** Hepatitis C virus, markov modeling, injection drug use, people who inject drugs

Hepatitis C virus (HCV) infection, estimated to be the most common chronic blood borne infection in the United States (US), has been shown to disproportionately affect people who inject drugs (PWID) [1-3]. Whereas prevalence estimates among the general US population ranges from 1.0 to 1.6 percent, prevalence estimates among PWID range from 40 to 90 percent [1-8]. HCV transmission among PWID can result from, but not limited to, sharing needles or other drug use paraphernalia (i.e., cookers, cotton, or rinse water; referred to as equipment) [9-11]. Individuals may be asymptomatic for decades and if left untreated, HCV infection can lead to cirrhosis of the liver, liver failure, hepatocellular carcinoma and death [12,13]. PWID, in particular, are at an even greater risk of the morbidity and mortality associated with untreated HCV because they often times do not have access to the healthcare system [3,8,14,15]. HCV prevention and interventions targeted towards PWID have included behavioral interventions, substance use treatment, syringe exchange or access programs, needle disinfection and multi-component interventions [8]. The impacts of these interventions on reducing transmission risk are not fully understood since corresponding studies have varying results [8,16,17]. One meta-analyses found that of the interventions examined, the use of a combination of prevention strategies, or multi-component interventions, were most effective, however; this was based on only two studies neither of which were conducted in the US [8].

Mathematical models have been used to estimate HCV incidence and prevalence among PWID as well as to model the effect of interventions and HCV treatment [18-21]. The use of modeling to better understand the spread of HCV and the impact of interventions can be invaluable for localities, states and countries with a large burden of injection drug use (IDU). New York State (NYS) is estimated to be home to a large number of PWID, however the burden of HCV among this population and the impact of interventions are not fully understood [22-28].

Given that modeling can be complex and require costly
data analysis software, the purpose of this paper was to
derive a methodology to model the prevalence of HCV that
is accurate, easily understood and implemented with free
software (R^{©}). The methodology is designed specifically to
better understand HCV disease burden among PWID, and
permit evaluation of the impact of interventions on the HCV
burden at a state level. In an effort to better serve the larger
population of PWID, a secondary purpose of this research was
to provide annotated R^{©} coding for other states and localities
to conduct similar research and thus broaden the reach of
this analysis. All coding for the model was written using R^{©}
version 2.14.1 (source code provided as a **Supplement Data**).

**Study design**

*Non-stationary quasi-Markov* modeling was utilized to estimate
HCV prevalence and to assess the impact of interventions on
disease elimination among PWID in NYS. The structure of the
model was motivated from those results of Corson, et al., 2012,
who utilized a deterministic compartmental model for HCV
transmission [19]. Although the results of Corson, et al., 2012 are
very valuable, their approach is somewhat sophisticated by way
of use of deterministic differential equations that require high
powered computing to implement, both of which may present
a barrier to general researchers in this field for a thorough
understanding and tractable implementation. Similar to Corson,
et al., we defined a model whereby individuals transition
between a set of possible predefined states through time.
However, our approach uses a tractable and somewhat simple
stochastic framework based on Markov modeling for governing
transitions between states and is easily implemented with free
software (**Supplement Data**). Possible routes of
HCV infection modeled were through needle and equipment
sharing only. The natural history of HCV was developed based
upon previous literature and includes basic states of the
disease which will be described in more detail below (Figure 1)
[20]. The transition probabilities between states of HCV were
dependent upon virologic parameters of the disease, such as
the probability of spontaneously clearing the infection, and
also parameters determined based upon characteristics of
the population, such as the prevalence of HCV among PWID.
For this reason, transition probabilities between states were
non-stationary, meaning they were updated at each time
unit to accommodate the changing parameters of the PWID
population. The time unit chosen for this analysis was days,
which we think realistically accommodates the real world
behaviors in this population, and estimates were modeled over
75 years (27,375 days), which in our opinion is an acceptable
time range based on expected human life span. Time zero is
a theoretical year in which the PWID population is defined
by the assumptions of the model, explained in more detail
below, and is not intended to reflect a specific calendar year

Figure 1 : **Natural history and transition matrix for hepatitis
C virus used for analysis.**

**Quasi-markov model transition probabilities and
parameters**

A graphic showing the possible states of the progression of
HCV and the allowable transitions between states is provided in
Figure 1. The probability that PWID transition from susceptible
state (*x*) (i.e., never infected with HCV) to acutely infected (state
*h _{1}* or

Table 1 **: Parameters to model hepatitis C virus (HCV) prevalence
among people who inject drugs (PWID) in New York State (NYS).**

The proportion of infected needles was dependent upon the average number of injections per day (2.8) [37] and the average number of needles in NYS: 1) dispensed through syringe exchange programs (SEPs) (from 2002-2011); and 2) sold through the expanded syringe access program (ESAP) (Table 1) [30,38]. See Corson et al., for derivation of the proportion of infected needles [19]. The proportion of infected equipment was dependent upon the amount of equipment dispensed from SEPs. For the purpose of this analysis, equipment was considered to include cotton, cookers and rinse water. A determination of this quantity is not available. Therefore, the ratio of the amount of equipment available to the number of needles dispensed or sold was assumed to be 3:4 based on feedback from SEPs. This was varied in a sensitivity analysis, the details of which are described below.

Not all persons who become acutely infected develop chronic HCV infection; a proportion of acutely infected persons
spontaneously clear the virus within six months [39]. Therefore,
based on previous work, two acute HCV states (*h _{1}*,

and for *k*=1,...,(27,375-180)

Some research has found that following spontaneous
clearance of an acute infection a proportion of individuals
may have protective immunity to re-infection; however,
findings regarding a conferring of a protective immunity are
equivocal [40-43]. In order to capture the possibility of some
level of a protective immunity, a corresponding state (*z*) was
included in the natural history model with a low transition
probability (0.25) [19].

Inherent in the model at each time unit is the probability
of leaving the PWID population (μ), represented by the state
exit *e _{x}*. This transition is possible from all states of the natural
history (including during the first 180 days of an incident case).
Research has shown that the probability of an individual
leaving the population is roughly equivalent to the probability
of new individuals entering the population [31,44]. Thus, the
proportion of the population that transitions to state

**Analysis**

The quasi-Markov model in Figure 1 was utilized to model
HCV prevalence among PWID through needle and equipment
sharing (model 1). To determine what interventions would be
needed to eliminate HCV from the population, modifications
using a basic reproductive number (*R _{0}*) were explored and the
initial model was altered to determine the effects different
interventions would have on HCV prevalence over time
(model 2).

**Modeling HCV prevalence: model 1**

Using the defined probabilities, the population distribution
of PWID in NYS at time *t*=0 was assumed to be 99.0 percent
susceptible (*x*) and 1.0 percent acutely infected only (state *h _{1}*).
This is to reflect an assumption that only a small proportion
of the population is infected at time

Where the transition matrix *M*(*t*) is defined in Figure 1.

HCV prevalence among the individuals was defined as
those who have ever been infected, or those who would
be defined as HCV antibody positive (anti-HCV), therefore
summing the prevalence of individuals in the following states:
*h _{1}*,

Several sensitivity analyses were conducted. First, the ratio
of available equipment to needles dispensed or sold in NYS
was varied from 1:1 to 1:10. Second, rather than assume 99.0
percent of the population was susceptible (*x*) at time t=0, the
proportion susceptible (*x*) was varied from as low as 70.0 to as
high as 99.9 to determine any differences in the prevalence
once reaching equilibrium. Third, given the equivocal nature
of protective immunity, the model was evaluated removing
this state in the natural history. Lastly, the model was run
assuming the proportion of individuals who spontaneously
clear the infection (probability of transitioning from susceptible
(*x*, *x _{1}*) to acutely infected (

**Modeling disease elimination through modifying
parameters: model 2**

Modifiable parameters assessed were: 1) reducing the amount
of needle and equipment sharing (λ_{N}, λ_{E}); 2) increasing the
proportion of PWID who properly sterilize needles (ϕ); 3)
increasing the number of needles dispensed or sold in a given
year and; 4) increasing the amount of equipment dispensed in
a given year. To determine to what degree specific parameters would need to be modified to eliminate HCV, defined here as a
prevalence of less than 1%, a formula for the basic reproductive
number (*R _{0}*) was used. The number

where1/σ=0.5, as defined by Corson, et al., represents the duration of an acute phase in years [19].

The number *R _{0}* was calculated for low, medium and high
risk PWID using the same behavioral and virologic parameters
utilized in model 1. Assuming

The parameters were then updated in model 1 to determine
what prevalence would be over time given the interventions
(model 2). The distribution of the population of PWID at time
*t*=0 was defined based upon the resulting distribution from
model 1. This population was then projected again through
75 years using the updated parameters. A sensitivity analysis
was employed in which the ratio of available equipment to
needles distributed or sold was varied from 1:1 to 1:10.

**HCV prevalence**

Estimated prevalence among PWID in NYS from model 1 rose
to over 50.0 percent within 25 years and reached equilibrium
at 64.1 percent within 50 years (Figure 2). Varying the ratio of
equipment to needles dispensed or sold from the original 3:4
to 1:1 and 1:10, respectively, resulted in prevalence ranging
from 63.0 to 67.6 percent (results not shown). Prevalence was not impacted when the proportion susceptible, *x*, at time
*t*=0 was varied. When removing the transition probability
for protective immunity, prevalence increased at a more
rapid rate and reached a higher prevalence at equilibrium
(65.8 percent) (Figure 2). When assuming the proportion that
spontaneously clears the infection was higher, prevalence
increased at a slower rate and the prevalence at equilibrium
was lower at 63.3 percent (Figure 2).

Figure 2 : **Hepatitis C virus antibody positivity (anti-HCV)
among persons who inject drugs and needles assuming current
behavioral and virologic characteristics of the population.**

**Disease elimination**

The basic reproductive number exceeded 1 for only the
medium (4.3) and high risk groups (7.3). These parameters,
referred to collectively as a multi-component approach,
included modifying the parameters as follows: 1) reducing
the probability of sharing needles in the high risk group by
half (from a probability of 0.22 per day to 0.11per day; the
probability in the other risk groups was 0), 2) reducing the
probability of sharing equipment in the high and medium
risk groups by a quarter (from 0.40 and 0.51 to 0.10 and 0.13,
respectively), 3) increasing the probability of cleaning ones
needle in the high risk group by two-fold (from 0.145 to
0.290), 4) increasing the amount of equipment in the high
and medium risk groups available by four-fold (from 3,812,583
to 15,250,332) and, 5) increasing the number of needles
dispensed in the high and medium risk groups two and a half
times (from 5,083,444 to 12,708,611). When prevalence was
modeled assuming the above mentioned multi-component
approach, the prevalence declined reaching 50.0 percent in
less than 5 years (Figure 3). Upon reaching equilibrium, the
prevalence was 0.2 percent. A visual display of the impact of
each modified parameter is displayed in Figure 3.

Figure 3 : **Modeling aims of interventions on disease
elimination of hepatitis C virus antibody positivity (anti-HCV).**

Varying the ratio of available equipment to dispensed needles impacted what is required to eliminate disease and reach equilibrium. As would be expected, when assuming a 1:1 ratio, less equipment was needed to reach equilibrium (a three-fold and two-fold increase among the high and medium risk groups were needed, respectively, compared to the fourfold increase for the initial analysis). When assuming a 1:10 ratio, to reach equilibrium, adjustments were needed for the high risk group only requiring an increase in both the amount of needles (needed to increase four-fold for the high risk group compared to 2.5 times for the initial analysis) and equipment (needed a six-fold for the high risk group compared to the four-fold increase for the initial analysis) and a reduction in the probability of sharing needles per day (reduced by onethird compared to one-half for the initial analysis).

**Model validation**

The prevalence of HCV among PWID in NYS is unknown and
therefore cannot be used to validate the model. However,
in an effort to validate the findings, our model was run on
previously published work modeling prevalence of HCV [19].
Here, Corson, et al., modeled HCV prevalence in Glasgow,
Scotland using a deterministic compartmental model using
the dynamic systems modeling software Berkeley Madonna™.
Contrary to our proposed methodology, the model used by
Corson, et al., did not account for equipment sharing.

Therefore, for comparison purposes, prevalence was modeled by our proposed methods using the parameter values implemented by Corson, et al., where equipment sharing was not accounted for. In addition, since Corson, et al., utilized shared injections per year in their model, the parameters were adjusted to reflect to the probability of sharing per day for our proposed methodology.

The model used by Corson, et al., determined that HCV prevalence among PWID in Glasgow rose steadily for the first decade reaching equilibrium around year 15 at an HCV prevalence of 68.9 percent. The resulting HCV prevalence results from our model using their parameters were similar with HCV prevalence rising steadily and reaching equilibrium later at year 35 at a slightly lower HCV prevalence of 62.1 percent. Although a similar prevalence was determined in Glasgow by Corson et al., as in NYS, the populations examined differed in several ways: the estimated number of PWID, the estimated number of needles distributed, the proportion of the population estimated to be sharing, and the probability of acute and chronic transmission per shared injection event. These differences thus resulted in differing PWID to needle ratios and needle turnover rate as well.

Modeling HCV prevalence when *R _{0}*<1, was also done in an
effort to provide some validation. Similar to results found by
Corson, et al., elimination was reached using the parameters
necessary to meet

In an effort to further validate the model, the estimate for HCV prevalence among PWID in NYS from this model was compared to other estimates among PWID. The prevalence of HCV among PWID has been cited to range between 40 and 90 percent, depending upon geography [8]. In addition, studies were done in the late 1990's and early 2000's in areas of New York City finding HCV prevalence ranging from 32-90 percent. The prevalence found in this study falls within the range of previous estimates [24,45,46].

Furthermore, the estimated number of secondary infections
resulting from PWID belonging to the medium and high risks
groups (4.3 and 7.3, respectively) is similar to other findings
by Magiorkinis et al., (2013), who calculated *R _{0}* in Greece for
HCV subtypes using epidemiologic (surveillance) data and
phylodynamic modeling [47]. For subtypes with a higher
proportion of PWID, they estimated that the number of
secondary infections were 3.4, 11.5 and 2.4 for subtypes 1a, 3a
and 4a, respectively; the subtype with the highest proportion
of PWID (47 percent) was subtype 3a [47]. The results from the
present analysis thus fall within the range found by Magiorkinis
et al., despite the differing methodologies.

Finally, two pathological cases were explored in an effort to validate the model in which the transmission probabilities were set to the unrealistic parameters of zero and 1. With the parameters set to zero, the steady-state prevalence went to zero; alternatively, with the parameters set to 1, likewise the steady-state prevalence went to 1 (i.e., all individuals would be infected).

This study demonstrated that HCV prevalence can be estimated at the state level using a non-stationary quasi-Markov model run entirely using free software. To the best of our knowledge, this is the first study using a quasi-Markov model to estimate HCV prevalence. In addition, the other available methodologies presented in the literature to estimate HCV prevalence require complex statistical analyses software packages that can be expensive. Although the model cannot be directly compared to existing NYS HCV prevalence among PWID due to lack of observed data, the results estimated an HCV prevalence within the range of other studies conducted in NYS and elsewhere among samples of PWID further supporting the validity of the method [8,24,45,46]. In addition, results from this methodology demonstrated that using a multi-component intervention model was needed to effectively eliminate HCV from the population. This supports other research on the effectiveness of such harm reduction strategies among PWID [8,48,49].

Although it is beyond the scope of the paper to assess or recommend interventions that would result in the modifications as specified to eliminate HCV among PWID in NYS, the parameters were intentionally chosen as "modifiable" in nature.

It must be noted that time in this analysis does not reflect prevalence over time in NYS among PWID. This model is used as a tool to see what prevalence could be assuming a small proportion of PWID are initially infected, accounting for current IDU behaviors, and needle and equipment distribution. The actual prevalence among PWID in NYS over time has likely been impacted by interventions implemented such as needle exchange and sale through SEPs and the ESAP, and the distribution of equipment by SEPs.

There are some limitations to this analysis. First, this study does not account for the possibility of dual infection, in which a person is infected with more than one HCV strain [50]. This could impact the probability of spontaneous clearance. Further analysis examining what impact this would have on disease spread and elimination are needed. Second, model 1, though taking into account other current interventions such as needle and equipment distribution, does not account for the possibility of HCV treatment. The proportion of active PWID who are treated is likely low, given low overall treatment in the general population, and would therefore only marginally lower the overall prevalence [51]. Third, parameters from localities other than NYS were used in this analysis. Although several studies conducted in NYS among PWID have been done, they have been restricted to areas in NY City and do not have the detailed level of information needed (daily probability of sharing needles and equipment, daily probability of cleaning ones needle, and daily rate that PWID exits population) to inform the parameters in our model. In addition, the prevalence of HCV among PWID in NYS is unknown and therefore cannot be used to validate the model. Although surveillance is currently conducted, risk factor data is not required for reporting purposes. Future studies conducted among PWID in NYS are needed and could be used to improve this study methodology. Fourth, the model does not specifically account for factors that may expedite the transition between disease stages such as alcohol use or HIV [13]. Given the large HIV epidemic in NYS, this could impact the model [52]. Further modeling incorporating these factors should be done, for example, by modeling HIV positive and HIV negative persons who consume alcohol to assess the difference. Lastly, the amount of equipment available was assumed to be a percentage of the number of needles dispensed or sold. Although sensitivity analyses explored varying this ratio, knowledge of a more precise amount would benefit the analysis.

The results of this analysis demonstrate the feasibility of using
non-stationary quasi-Markov modeling to model the spread
of HCV among PWID using free programming software. The R^{©}
code in the appendix has been provided for other researchers
to use and modify for their own purposes and could further
impact this field of research as well as inform and promote
additional education prevention and interventions for PWID.

The authors declare that they have no competing interests.

Authors' contributions |
RH |
GD |

Research concept and design | √ | √ |

Collection and/or assembly of data | √ | √ |

Data analysis and interpretation | √ | √ |

Writing the article | √ | √ |

Critical revision of the article | √ | √ |

Final approval of article | √ | √ |

Statistical analysis | √ | √ |

We would like to thank Colleen Flanigan, Director of the Viral Hepatitis Section at the AIDS Institute New York State Department of Health, for her contributions to the conception and interpretation of this study. We are grateful to Maxine Philips, Barbara Agatstein and the Division of Prevention at the New York State Department of Health for providing us pertinent information regarding syringe exchange programs and for their expertise in the field.

Editor: Guy Nathaniel Brock, University of Louisville, USA.

EIC: Jimmy Efird, East Carolina University, USA.

Received: 27-Jun-2014 Final Revised: 27-Sep-2014

Accepted: 02-Oct-2014 Published: 07-Oct-2014

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Volume 2

Hart-Malloy R and DiRienzo G. **A quasi-markov model for transmission and disease elimination: Hepatitis C among people who inject drugs**. *J Med Stat Inform*. 2014; **2**:8. http://dx.doi.org/10.7243/2053-7662-2-8

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