 Research article
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Transmission dynamics for Methicilinresistant Staphalococous areus with injection drug user
BMC Infectious Diseasesvolume 18, Article number: 69 (2018)
Abstract
Background
Methicillinresistant Staphylococcus aureus (MRSA) is a bacterial pathogen resistance to antibiotics including methicillin. The resistance first emerged in 1960 in a healthcare setting only after two years of using methicillin as a viable treatment for methicillinsusceptible Staphylococcus aureus. MRSA leads to infections in different parts of the body including the skin, bloodstream, lungs, or the urinary tract.
Methods
A deterministic model for methicillinresistant Staphylococcus aureus (MRSA) with injection drug users is designed. The model incorporates transmission of MRSA among noninjection drug users and injection drug users (IDUs) who are both lowand highrisk users. A reduced MRSA transmission model with only nonIDUs is fitted to a 20082013 MRSA data from the Agency for Healthcare and Research and Quality (AHRQ). The parameter estimates obtained are projected onto the parameters for the lowand highrisk IDUs subgroups using risk factors obtained by constructing a risk assessment ethogram. Sensitivity analysis is carried out to determine parameters with the greatest impact on the reproduction number using the reduced nonIDUs model. Change in risk associated behaviors was studied using the full MRSA transmission model via the increase in risky behaviors and enrollment into rehabilitation programs or clean needle exchange programs. Three control effectiveness levels determined from the sensitivity analysis were used to study control of disease translation within the subgroups.
Results
The sensitivity analysis indicates that the transmission probability and recovery rates within the subgroup have the highest impact on the reproduction number of the reduced nonIDU model. Change in risk associated behaviors from nonIDUs to lowand highrisk IDUs lead to more MRSA cases among the subgroups. However, when more IDUs enroll into rehabilitation programs or clean needle exchange programs, there was a reduction in the number of MRSA cases in the community. Furthermore, MRSA burden within the subgroups can effectively be curtailed in the community by implementing moderate and higheffectiveness control strategies.
Conclusions
MRSA burden can be curtailed among and within noninjection drug users and both lowand highrisk injection drug users by encouraging positive change in behaviors and by moderate and higheffectiveness control strategies that effectively targets the transmission probability and recovery rates within the subgroups in the community.
Background
Methicillinresistant Staphylococcus aureus (MRSA) is a bacterial pathogen resistance to antibiotics including methicillin. It first emerged in 1960 in a healthcare setting only after two years of methicillin being used as a viable treatment for methicillinsusceptible Staphylococcus aureus. MRSA has long been isolated in the community since its emergence in the hospital setting.
Transmission of MRSA is achieved either by direct contact with a colonized patient or indirect contact. Indirect transmission occurs when the bacteria is transferred from an infected person to a fomite where it can stay infectious for months [1].
The epidemic of MRSA among injection drug users (IDUs) began in 1981 and had since become endemic in the community [2]. A number of research studies have studied MRSA among IDUs by tracking registered IDUs [3–6]. For instance, Fleisch et al. [5] followed 31 MRSA infected IDUs and found that 19 individuals developed secondary, lifethreatening infections such as septic arthritis, endocarditis, pneumonia, and osteomyelitis. Another study by Binswanger et al. [4] found that out of a population of 169 IDUs, 29 individuals displayed subcutaneous inflammation and infection.
Injection drug users present a unique set of behavioral factors that all accumulate to an increased risk of MRSA transmission and infection. One of those risk factors is trauma to the skin which creates opportunities for MRSA bacteria to access their soft tissue during the act of injecting their drug of choice. The proceeding factors all vary depending on habits and behaviors associated with injection drug use that will be detailed in the “Parameter estimation” section.
Our goal was to understand the transmission dynamics of MRSA in a community with IDUs and to investigate the impact of drug rehabilitation programs, intervention, education, clean needle exchange programs, and so forth [7, 8] as part of control measures to curtail MRSA transmission in the community. Therefore, we formulate a deterministic model for MRSA transmission dynamics which include nonIDUs and two subgroups of IDUs with different risk associated behaviors (i.e., low and highrisk behaviors).
This paper is organized as follows: in “Model formulation” section, we present the MRSA transmission model and calculate the basic reproduction number. In the “Parameter estimation” section, we estimate the values of the model parameters. In the “Sensitivity analysis” section, we carry out a sensitivity analysis to identify the model’s parameters with the most impact on our response function. Using the results obtained from the sensitivity analysis, we investigate in “Control measures” section the impact of some control strategies on MRSA transmission in the community.
Method
Model formulation
To formulate the methicillinresistant Staphylococcus aureus transmission model with injection drug users, individuals in the community were divided into three subgroups, noninjection drug users (IDUs), lowrisk IDUs, and highrisk IDUs in order to understand the interactions between these populations. Each subgroup was subsequently divided into three compartments according to their disease status. Thus, we have uncolonized susceptible individuals (U_{ i }), colonized individuals (C_{ i }), and infected individuals (I_{ i }), where i=N,L,H for nonIDUs, lowrisk IDUs, and highrisk IDUs. These premises lead to the following total population
The MRSA transmission dynamics with IDUs is given by the following system of ordinary differential equations and depicted in Fig. 1. The associated variables and parameters are described in Table 1.
The parameter π_{ N } is the recruitment rate into the uncolonized noninjection drug users (U_{ N }) subgroup. The parameter μ is the natural death rate in each subpopulation. The force of infection of the noninjection drug users is given by
where β_{ N } is the probability of an uncolonized individual becoming colonized upon being exposed to MRSA bacteria through contact with colonized or infected individuals in either the noninjection drug users class or the lowrisk class or the highrisk classes. Once individuals in the colonized class are decolonized at the rate τ_{ N } [9], they move back to the uncolonized class. However, as the disease progresses, the colonized nonIDUs (C_{ N }) move to the infected class at rate σ_{ N }. They recover at the rate γ_{ N } and thus move from the infected compartment (I_{ N }) back to the uncolonized class (U_{ N }). The parameters and transitions for the lowrisk and highrisk populations are similarly defined (with the subscript N replaced by L and H respectively).
If nonIDUs engage in risky behavior resulting from injection drug use, we assume that this behavior initially involves a less risky use of drugs and as such these individuals leave the nonIDUs subgroup at rate ω_{ N } into the lowrisk IDU subgroup. However, if these individuals decrease their risky behaviors either by enrolling in drug rehabilitation programs [3, 10] or by other kinds of interventions they move back into the nonIDU subgroup at the rate α_{ L }.
Individuals in the lowrisk subgroup who further engage in increased risky behaviors move into a highrisk injection drug users subgroup at the rate ω_{ L }. As with the lowrisk individuals, these individuals may stop injecting drugs when they enroll in a drug rehabilitation or needle exchange programs [3, 10]. We assume these processes are not instantaneous. Hence they first enter the lowrisk subgroup at rate α_{ H }. Injection drug users harbor more S. aureus bacteria compare to non users [11], and required prolonged treatment [12, 13], thus we assume that these risky behaviors increases the rates of MRSA transmission [14, 15].
The basic reproduction number
The basic reproduction number (\(\mathcal {R}_{0}\)) of the MRSA model (1) with IDUs is given below; the theoretical study of the model basic properties is stated in Appendix A under ‘Analysis of the model’ and the calculations of \(\mathcal {R}_{0}\) are given in Appendix B.
where
Furthermore, the expression \(\mathcal {R}_{N}\) is the number of secondary infections among the noninjection drug users, \(\mathcal {R}_{L}\) is the number of secondary infections among the lowrisk injection drug users, \(\mathcal {R}_{H}\) is the number of secondary infections among the highrisk injection drug users. These expressions (\(\mathcal {R}_{N}, \mathcal {R}_{L}, \mathcal {R}_{H}\)) include the secondary infection in each subgroups due to both horizontal and vertical transitions of infectious individuals due to disease translation and risky behaviors within and between the subgroups.
The basic reproduction number \(\mathcal {R}_{0}\) is defined as the average number of new infections that is produced as a result of the introduction of one infectious individual into a population that is fully susceptible [16–19].
The basic reproduction number when ω _{ N }=ω _{ L }=α _{ L }=α _{ H }=0
Suppose the vertical upward and downward transition between the subgroups are absent, that is, ω_{ N }=0, ω_{ L }=0, α_{ L }=0, α_{ H }=0, then the basic reproduction number of the MRSA model (1) with IDUs, is given as:
where
It should be noted that the reproduction number stated in Eq. (2) gives the reproduction number in Eq. (3) in the absence of vertical downward and upward transition, that is, if ω_{ N }=ω_{ L }=α_{ L }=α_{ H }=0.
Parameter estimation
MRSA Demographic data from 20082013
Demographic data from 20082013 was obtained from the Agency for Healthcare Research and Quality (AHRQ) [20]. The AHRQ compiles international classification of diseases (ICD) data from hospitals throughout the United States. These data include patients who were discharged with MRSA and are identified with ICD9 code. To obtain the MRSA ICD9 data, AHRQ demographic data targets were set to look at large metro, large suburb, and rural areas. As expected, the large metro population produced the highest number of patients with MRSA listed on their medical records upon discharge. Figure 2 displays the obtained data in tens of thousands of hospital discharges from 20082013.
NonIDU subgroup parameter estimation
In order to parametrize MRSA model (1) we use patient data obtained from the AHRQ surveys. One of the limitations of this data is that it does not identify individuals with MRSA that are injection drug users. To overcome this short coming, we first assume that the population consist of only nonIDUs and then we reduce the MRSA model (1) to the following system of ordinary differential equations
For the reduced MRSA model (5) with nonIDUs, the total population N=U_{ N }+C_{ N }+I_{ N }, and the force of infection is given by \(\frac {\beta _{N} (C_{N}+I_{N})}{N}\). The basic reproduction number is given as:
where k_{1}=μ, k_{2}=τ_{ N }+σ_{ N }+μ, k_{3}=γ_{ N }+μ+δ_{ N }.
Next, we estimate the parameters of the reduced MRSA model (5) by using the ICD9 MRSA data for large metro, rural, and suburb regional areas. The results of the parameter estimation are given in Table 2; the model simulation profile and the fitted data are depicted in Fig. 3.
A number of studies have identified injection drug use as a significant risk factor for developing MRSA secondary to hospitalization rates and increased exposure to antibiotics arising from treatment of skin abscesses, bacteremias, and endocarditis [12–14]. Cohen [21] established a relationship between the use of methamphetamine and the occurrence of MRSA. Hence, we assume that the relationship between injection drug use and the risk of developing MRSA is linear. This enabled us project the nonIDUs parameters onto the low and highrisk IDUs parameters using some risk related parameters discussed below.
Therefore, in order to obtain the parameter estimates for the IDUs in the full MRSA model (1), we modify the transmission probability, disease progression and recovery rates in the low and highrisk subgroups by multiplying the nonIDUs subgroup parameter estimates β_{ N },σ_{ N }, and γ_{ N } with ε_{ L } and ε_{ H }. These parameters represent the associated risky behaviors in lowrisk and highrisk subgroups. The nonIDU disease induced mortality δ_{ N } is also modified by ε_{ L } and ε_{ H }, since IDUs have a higher mortality rate due to their drug use [22–24].
These modification parameters ε_{ L } and ε_{ H } are called risk factors; their values give us an estimate away (either in an increasing or decreasing form) from the nonIDUs parameters. How these parameter values are obtained for the IDUs in the lowrisk, and highrisk subgroups are discussed below.
Risk factors ε _{ L } and ε _{ H } formulation
The risk factor is designed to distinguish the parameter values in all the subgroups from one another. It incorporates qualitative characteristics of injection drug use in the lowrisk and highrisk subgroup transformed into quantitative integers that are used to impact the lowrisk and highrisk parameters described in the MRSA model (1) formulation.
Hence, the lowrisk IDU subgroup transmission probability (β_{ L }), disease progression rate (σ_{ L }), recovery rate (γ_{ L }), and MRSAinduced death (δ_{ L }) are all modified from the noninjection drug users parameter values by a risk factor (ε_{ L }) associated with the riskassociated behaviors. Hence,
The parameters β_{ L } and σ_{ L } are both increased by the introduction of a risk factor ε_{ L }. This increases the rate at which individuals move from the uncolonized class, U_{ L }, to the colonized class, C_{ L }, and subsequently, to the infected class, I_{ L }, compare to the individuals in the nonIDUs subgroup. On the other hand, γ_{ L } is decreased by the effect of ε_{ L }, thus decreasing recovery rate and holding individuals in the infected compartment for a longer period compare to nonIDUs.
The highrisk IDU subgroup parameters undergo an analogous treatment but with a larger, and higher impacting value ε_{ H } in place of ε_{ L } as shown in [6]. Thus,
To estimate the values of the risk factors ε_{ L } and ε_{ H }, we first construct an ethogram. The practice of using ethograms is common within the study of animal behavior [25, 26]. This technique allows the researcher to quantify behaviors observed in the subject(s) and their interaction with other organism or with their environment. This technique was used to construct Tables 3 and 4.
To construct Table 3, we classify individuals’ associated risk behavior from nondrug use to severe drug use, assigning a value from 05 based on the perceived level of risk. The ranking values in Table 3 is then translated to the rating points in the complimentary Table 4 by summing the ranking values of an injection drug user from their previous to current categories. For instance, a moderate IDU has a rating point of 6; this comes from them having been a limited to mild and moderate drug user (i.e., 0+1+2+3=6, obtained from ranking values in Table 3). Thus, a severe IDU will have a ranking of 15 points (i.e., 0+1+2+3+4+5+6=15).
Using these ranking points, we classify the individuals into lowrisk and highrisk IDUs (see Table 4). To obtain the risk factor percentages, we use a ratio of 1 rating point to 6% risk factor value (1 rating = 6% risk factor); the ratio of 16 is chosen since there are six risk categories under consideration. Thus, we have 636% for a lowrisk IDU and a 4290% highrisk IDUs thresholds. These represent the percentage that the nonIDU parameters will be amplified by (β_{ N },σ_{ N },δ_{ N }) or decreased by (γ_{ N }) due to associated IDU risky behaviors.
Sensitivity analysis
In order to determine the robustness of the model in relation to each parameter, a sensitivity analysis was performed. Analysis of parameters can give insight into the uncertainty an input may have and how this will affect the outcome of the model. To determine system sensitivity to its parameters, the normalized forward sensitivity index [27–29] given in Eq. (9) is used
where p represent the parameter of interest and \(\mathcal {R}_{N}\), the reproduction number \(\mathcal {R}_{N}\) of the reduce model (5); since the parameters of MRSA model (1) were obtained from fitting to data the reduced model (5) and modifying their values using the risk factors ε_{ L } and ε_{ H } to obtain the values for the IDUs parameters. Hence, the local sensitivity analysis was performed on the following parameters: the transmission probability (β_{ L }), disease progression rate (σ_{ L }), recovery rate (γ_{ L }), MRSAinduced death (δ_{ L }), and natural death (μ) based on their influence on \(\mathcal {R}_{N}\).
Control measures
The best strategy to preventing MRSA infections among IDUs would be for them to end their risky behaviors, many of these IDUs are not ready to stop or make a change in their behaviors [30]. However, by simply learning new hygiene skills or best practices may substantially decrease the risk of MRSA among individuals who are reluctant to stop injecting drugs [30, 31]. For instance, Hart et al. observed a reduction in the incidence of abscesses among IDUs registered in a needle exchange program in London.
In this section we considered two types of control strategies; the first strategy investigates the impact of vertical downward transition between the subgroups due to individuals in the community engaging in risky behaviors that promote injection drug use and the impact of vertical upward transition that can be achieved by enrolling in rehabilitation programs. The aim of this strategy is to investigate the impact of the parameters ω_{ N } and ω_{ L }, for the vertical downward transition and the parameters α_{ L } and α_{ H } for the vertical upward transition.
The second strategy uses results obtain from the sensitivity analysis to consider control within each subgroup. With this strategy, we target the parameters β_{ N } and γ_{ N } by implementing three different control strategies: loweffectiveness strategy, moderateeffectiveness strategy, and higheffectiveness strategy. The goal of each strategy is to reduce β_{ N } and increase γ_{ N }. The risk factors ε_{ L } and ε_{ H } are then used to determine the values of β_{ L }, β_{ H }, γ_{ L } and γ_{ H } in the lowand highrisk IDUs subgroups respectively.
Note that for the two different control strategies either between or within the subgroups, we used the intermediate risk factor values ε_{ L }=21% and ε_{ H }=66 %.
Control of risky injection drug use behaviors
In this section, we investigate the impact of vertical downward and upward transitions between the compartments due to risky behaviors that promotes (downward transitions) and discourages (upward transitions) injection drug use in the community. The aim of this control strategy is to investigate the impact of vertical downward transition using the parameters ω_{ N } and ω_{ L }, and impact of the vertical upward transition using the parameters α_{ L } and α_{ H }. Note in this section we are controlling the risky behaviors and not MRSA transmission within the subgroups.
No vertical downward and upward transitions
First, suppose that there are no transition between the subgroups (i.e., nonIDUs, lowrisk IDUs and highrisk IDUs); in other words, there are no vertical downward transitions due to risky behaviors nor are there vertical upward transitions due to enrollment in rehabilitation programs, that is ω_{ N }=0,ω_{ L }=0,α_{ L }=0,α_{ H }=0.
Vertical downward transitions only
Next, we consider the situation with only vertical downward transitions and no upward transitions. Suppose individuals in the nonIDUs slowly engage in the risky behavior of injection drug use, but individuals in the lowrisk IDUs quickly engage in these risky behaviors either due to peer pressure or due to individuals physiology [32, 33]. Thus, we set ω_{ N }=0.05825,ω_{ L }=0.116,α_{ L }=0,α_{ H }=0.
Vertical upward transitions only
Next, we investigate the impact of the vertical upward transitions that can be achieved by enrolling in rehabilitation programs [3, 6]. Suppose more individuals in the highrisk IDUs enter rehabilitation programs either due to referrals from those around them or due to increased access to such programs [34] or through family interventions [35]. Further, suppose fewer lowrisk IDUs enter rehabilitation program because they either do not see the need for the program or they believe they could handle the problem without the rehabilitation program or they believe in their ability to control their risky behaviors [36]. For this scenario, we set ω_{ N }=0,ω_{ L }=0,α_{ L }=0.0112,α_{ H }=0.0560.
Vertical downward and upward transitions
Lastly, we investigate the impact of both downward and upward transitions due to changes in risky behaviors; that is, we set ω_{ N }=0.05825,ω_{ L }=0.116,α_{ L }=0.0112,α_{ H }=0.0560.
Control of MRSA transmission among the subgroups
According to the Centers for Disease Control and Prevention (CDC), maintaining adequate personal hygiene is vital for the control of MRSA in the community [37]. In hospital setting, healthcare workers are required to constantly clean their clothing, laundry, medical equipment and the entire hospital environment to prevent and minimize contact and transmission of the bacteria [37].
In this section, we investigate the impact of controlling MRSA transmission within the subgroups, we will not consider decolonization as a control strategy. Thus, to determine the impact of controlling MRSA within the subgroups, we use results obtained from the sensitivity analysis. From the sensitivity analysis, we observed that control strategies that reduce β_{ N } and increases γ_{ N } would impact MRSA in the community since these parameters have strong positive (β_{ N }) and negative (γ_{ N }) impact on the reproduction number \(\mathcal {R}_{N}\). The same holds for the parameters β_{ L },β_{ H },γ_{ L }, and γ_{ H }. Hence, we implement three different strategies: loweffectiveness strategy, moderateeffectiveness strategy, and higheffectiveness strategy. Note that these strategies are only for theoretical purpose to illustrate the impact of these control interventions.
Loweffectiveness control strategy
Our aim in this section is to investigate the control strategies that reduces MRSA transmission in the community. To achieve this in both the nonIDUs and IDUs populations, we assumed that the IDUs do not forfeit any of their risk associated behaviors. And they also do not alter their medical circumstances by actively seeking out medical treatment.
Thus, for this loweffective strategy, we set β_{ N }=0.27780 and γ_{ N }=0.27780 (the values obtained from the data fitting above). Using the modifying risk factors ε_{ L } and ε_{ H }, we obtain the values for β_{ L },γ_{ L },β_{ H } and γ_{ H } respectively. That is, β_{ L }=β_{ N }(1+ε_{ L }),β_{ H }=β_{ N }(1+ε_{ H }),γ_{ L }=γ_{ N }(1−ε_{ L }), and γ_{ H }=γ_{ N }(1−ε_{ H }).
Moderateeffectiveness control strategy
We reiterate that unlike in the previous section, our aim in this section is not to control the risky behavior but to control the transmission of MRSA within the subgroups. Thus, to implement the moderateeffectiveness control strategy, we assume that if an IDU ceases to share their drug paraphernalia with other IDUs [6], this will reduce the physical contact with other IDUs which invariably reduces the transmission probabilities β_{ N }, β_{ L }, and β_{ H }.
Hence, for the moderateeffectiveness control strategy we set
Note that recovery rate (γ_{ N }) in the nonIDUs have been increased by 0.7%.
Higheffectiveness control strategy
We assume that the IDUs in the community enroll in a drug abuse program (DAP) [3, 6]. Note that the voluntary or involuntary enrollment into a DAP will not stop the use of injection drugs, but it will significantly decrease the practice of riskassociated behaviors. Education and accountability given to individuals enrolled in the DAP will effectively reduce the riskassociated behaviors listed in Table 3 and by extension will lead to a reduction in the disease transmission probabilities β_{ N }, β_{ L }, and β_{ H } [5, 6].
Additional benefits gained by the IDUs who participate in the program include access to medical professionals and to resources such as symptom and treatment education. These benefits can empower MRSA infected IDUs with tools necessary to recognize and treat their infections. Bassetti et al. found that the addition of medical care to MRSA infected IDU patients significantly reduced the use of drugs [3]. As explained in [3], a major benefit of DAPs is access to quality health care. Such access will not only provide a better understanding of disease transmission and hygiene techniques but will also provide timely diagnosis and a more precise consumption of medicine. This improved access to health care invariably leads to increase in the recovery rate of the IDUs, that is, increase in γ_{ L } and γ_{ H }.
Thus, for the higheffectiveness control strategy we set
In this case, the recovery rate (γ_{ N }) in the nonIDUs have been increased by 0.9%.
Results
In this paper, we have designed and studied a deterministic model for methicillinresistant Staphylococcus aureus (MRSA) transmission in the community with injection drug users (IDUs). Unique to our model is the incorporation of two IDUs with low and highrisk associated behaviors. Our goal in this paper is to understand the transmission dynamics of MRSA in a community with IDUs and to investigate the impact of different control strategies in curtailing MRSA transmission in the community. Thus, the intervention strategies analyzed include the impact of drug rehabilitation, intervention and education, and clean needle exchange programs on MRSA transmission in the community (i.e., the vertical upward and downward transition) as well as three control effectiveness strategies at each subgroup levels (i.e., loweffectiveness, moderateeffectiveness, and higheffectiveness strategies).
The model theoretical results indicate that the diseasefree equilibrium of the model is locallyasymptotically stable whenever the related reproduction number is less than unity and stable otherwise. The implication of this result is that the disease will die out or be curtailed whenever the reproduction number is less than unity and will spread whenever the reproduction is greater than unity.
Parameter estimation
We parameterized the model using the 20082013 Agency for Healthcare Research and Quality (AHRQ) dataset; these data include patients who were discharged with MRSA. A major limitation with the use of this dataset is that it includes individuals who may have acquired MRSA infection due to their hospital stay. As a result, the dataset does not truly capture the transmission of the bacteria in the community which is more appropriate for evaluating MRSA burden amongst IDUs in the community. To overcome this limitation, we assumed that the population consist of only nonIDUs and reduce the MRSA model (1). This enabled us determine parameters that are projected onto the low and high riskIDUs using risk factors ε_{ L } and ε_{ H }.
Having obtained the estimate for the risk factors ε_{ L } and ε_{ H }, we investigated the impact of varying these modification parameters on disease transmission in the community. For the lowrisk factor ε_{ L }, we used the lower and upper bound values (6% and 36%) given in Table 4 and their intermediate value (21%). Similarly, for the highrisk factor, we used 42%, 66%, and 90%.
Figure 4 depict the results of simulating the full MRSA model (1) using the parameter estimates given in Table 2 and the low, intermediate, and high bounds of the risk factors ε_{ L } and ε_{ H } in Table 4. At the lower bounds of the risk factors (ε_{ L }=6% and ε_{ H }=42%), the nonIDUs have the highest number of colonized individuals, this is followed by lowrisk IDUs, while the highrisk IDUs have the least number of colonized individuals (not shown here in the figure). Within the infected compartment, the highrisk IDUs have the highest number of individuals followed by the lowrisk then nonIDUs. The disparity between each subclass increases when the risk factor is increased such as the highrisk IDUs with a greater number of infected compared to the lowrisk and nonIDUs.
A similar trend is observed in Fig. 5 for the suburb and rural regional areas using intermediate risk factors estimates ε_{ L }=21% and ε_{ H }=66%.
For the rest of the paper, we will use the parameter estimates for the large metro area since the number of patients in the ICD9 data for this area is the largest; moreover, the results for the large suburb and rural areas are not expected to deviate from the results obtained for large metro areas.
Sensitivity analysis
The outcome of the local sensitivity analysis using parameters estimated in Table 2 is shown in Table 5; the parameter β_{ N } and γ_{ N } are both shown to have the largest impact on the reproduction number (\(\mathcal {R}_{N}\)). The implication of this result is that any control strategy which target these two parameters will give the greatest impact on \(\mathcal {R}_{N}\). For instance a control strategy that decreases β_{ N } by 10% will lead to a 10% reduction in \(\mathcal {R}_{N}\), similarly, a strategy that increases γ_{ N } by 10% will lead to a 7.2% decrease in \(\mathcal {R}_{N}\). In the next section, we addressed control measures that targets β_{ N } and γ_{ N } with the goal of reducing \(\mathcal {R}_{N}\).
Control measures
We investigated the impact of the different control strategies to curtail MRSA transmission within and between each subgroup in the community by first controlling the risky behaviors within the groups and secondly by controlling the transmission of the bacteria among the groups using parameter values given in Tables 6 and 7.
Control of risky injection drug use behaviors
Under this control strategy, we consider four scenarios involving transitions between the groups.
No vertical downward and upward transitions
We observed in Fig. 6a that the nonIDUs have more colonized individuals followed by the lowrisk IDUs and the highrisk IDUs have the least number of colonized individuals. These dynamics is due to the fact that disease progression rate is highest in the highrisk IDUs and smallest in the nonIDUs. This explains the result of Fig. 6b where the highIDUs have the most number of infected individuals followed by the lowrisk IDUs, and the nonIDUs have the least number of infected individuals.
Vertical downward transitions only
We observed in Fig. 7 an increase in the number of colonized and infected highrisk individuals and a decrease in the colonized and infected compartments of the other two subgroups. The result of this simulation can be thought of as a funnel that sifts individuals down through the model into more risky behaviors and thus an increase in transmission of MRSA for the highrisk IDUs.
Vertical upward transitions only
We observed more colonized individuals in the nonIDUs subgroup in Fig. 8a, followed by the lowrisk IDUs and the highrisk IDUs have the least number of colonized individuals. However, in the infected compartment (see Fig. 8b) the lowrisk IDUs have more individuals for about 55 days due to the surge from the highrisk individuals changing their behavior. But over time, the nonIDUs have more infected individuals as individuals move upwards between the subgroups.
Vertical downward and upward transitions
We observed in Fig. 9 similar effect as seen in Fig. 7 but in this case, fewer nonIDUs individuals were colonized and infected; this is due to the transitions to the other two subgroups. Thus, there are more individuals in the colonized and infected compartments in the highrisk, and lowrisk IDUs subgroups in Fig. 9 compare to those in Fig. 7.
Hence, we have explored in this section the impact of the vertical downward and upward transitions among noninjection drug users and injection drug users who are both lowand highrisk users due to change in risky behaviors and found that as individual engage in these risky behaviors, MRSA cases in the community increase. However, with more IDUs enrolling into rehabilitation, intervention, and education, and clean needle exchange programs MRSA cases reduces in the community. This control is intended to simulate the decline in risky behavior which could be achieved by rehabilitation programs, intervention and education, clean needle exchange programs, and so forth [7, 8].
Control of MRSA transmission among the subgroups
The result of the sensitivity analysis was used to study the horizontal translation within the subgroups due to disease transmission by implementing three different strategies: loweffectiveness strategy, moderateeffectiveness strategy, and higheffectiveness strategy with the goal of reducing the number of colonized and infected individuals.
The total number of colonized and infected individuals in each of the different subgroups is simulated for the three levels of effectiveness for the loweffectiveness, moderateeffectiveness, and higheffectiveness control strategy and depicted in Fig. 10 are the solution profiles for the infected.
Figure 10 shows a reduction in each of the subgroups under the moderate and higheffectiveness levels of the control strategy. It is worth noting that the higheffectiveness level of the control strategy, as expected, is far more effective in curtailing MRSA burden in the community. The moderateeffectiveness level of the strategy also resulted in a significant decline in the number of cases in comparison to the loweffectiveness level of the control strategy.
Specifically, the comparison of the three effectiveness strategies at t=100 days in each of the different subgroups (see Table 8), shows that the higheffectiveness control strategy led to a considerable reduction in the total number of colonized and infected individuals. This is followed by the moderateeffectiveness level, and the loweffectiveness level which produced the most number of colonized and infected cases.
Thus, in this section, we found that the higheffectiveness control strategy is more effective in curtailing MRSA burden in the community, this is followed by the moderateeffectiveness strategy, and the loweffectiveness strategy performed the least; furthermore, we found that the moderateeffectiveness strategy is also an effective control strategy.
These simulations clearly shows that MRSA is controllable in a community with IDUs using the control measures, such as the moderate and higheffectiveness levels of the control strategy described above.
Discussion
MRSA nasal carriage is shown to be in 33% of the population in the United States [38]. This high number of carriers have the potential to infect individuals they come into physical contact or share intimate materials with [39]. In communities of people in close contact such as inmates and sports teams, the transmission of MRSA is increased due to increased physical contact and increased handling of mutual fomites [40–43]. The dynamics between these communities can be applied to injection drug users who exhibit similar habits as those listed in Table 3.
In today’s IDU communities, the spread and consequences of MRSA infections are of the greatest concern. The resistant bacteria’s severely virulent nature and the associated risk factors of IDU behaviors create an ideal habitat for MRSA. Living in drug houses, sharing needles or saliva are all routine practices for IDUs and all lead to an increase in the risk of MRSA infection [3]. For instance, Cohen et al. [21] found that the use of methamphetamine resulted in an increased number of MRSA skin and soft tissue infections than in those who were not using the drug. MRSA transmission is often times unknown due to asymptomatic colonization [39]. This combined with the increase in risky behaviors contributes to the increase in infected individuals in both IDU subgroups. The known and unique risks IDUs possess contribute to the high infection rate of not only MRSA but well studied HIV transmission [44, 45].
The National Survey on Drug Use and Health (NSDUH) broadens their definition of a risk factor to also include social and mental contributors such as the ideation of a low risk of harm from illicit drug use and availability of the drug [46]. In large metropolitan areas illicit drugs are readily available and according to NSDUH 2015 survey, over 1.5 million people in large metropolitan areas perceived access to illicit drugs as being fairly or very easy versus only 405,000 in nonmetropolitan areas [36, 46]. Hence, it is imperative to reduce the risk factor acting on individuals and improve protective factors or decreased likelihood of substance abuse [36, 46]. And these, according to NSDUH, should be the goals of any prevention programs [46].
These are the goals that we aimed to achieve in this study and the results obtained from the control strategies implemented in our model align well with these goals. Furthermore, the results of our model numerical exploration shows that as the risk factor increases, the number of colonized and infected individuals increases as expected. This increase is observed in all the three subgroups under consideration.
In parameterizing our model we have used large metro, large suburb, and rural areas data from the Agency for Healthcare Research and Quality (see Table 2 and Fig. 3). However, we have only used the results for the large meteropolitan areas in our numerical explorations. MRSA is also of concern in suburbs and rural areas due in part to the presence of IDUs [47] and we clearly observed this from the data. Furthermore, the NSDUH 2015 survey found that lifetime use of illicit drugs in large and small metropolitan areas was 113,967 while nonmetropolitan areas were 16,644. These statistics bring to light the great need for action in drug abuse interventions in not only metropolitan areas but also suburbs and rural areas. In future work, we will study the impact of movement between large metro, suburb and rural regional areas on the transmission of MRSA among nonIDUs and both lowand highrisk IDUs.
The Agency for Healthcare Research and Quality reported a 21% reduction in hospital acquired conditions (HAC) from 20102015 [48], this reduction includes HAMRSA. During this fiveyear period, there was a 3 million reduction in the number of cases and $28 billion dollars saved in fighting HACs [48]. Unfortunately, this same results cannot be said of CAMRSA. To achieve similar significant results as seen with HACs, more research and community efforts will be needed. Nevertheless, these results can be seen with a bright outlook for the future prevention and treatment of MRSA in communities such as IDUs.
Conclusions
In this section, we summarize some of the main theoretical and epidemiological findings of this study:

The diseasefree equilibrium of the model (1) is locallyasymptotically stable whenever the associated reproduction number (\({\mathcal R}_{0}\)) is less than unity.

The model was parametrized by

parameterizing the reduced nonIDUs model (5) using AHRQ ICD6 MRSA regional data for large metro, suburb and rural areas;

using risk factors obtained from a constructed ethogram, the nonIDUs parameters are projected onto low and highrisk IDUs related model parameters.


The sensitivity analysis of the parameter variations using the associated reproduction number (\({\mathcal R}_{N}\)) as response function of the reduced model (5) with nonIDUs show the most dominant parameters are the transmission probability (β_{ N }) and the recovery rate (γ_{ N }).

The effect of the risk factor was studied, and we found as expected, that as the risk factor increases, the number of colonized and infected increases.

The study of the vertical downward and upward transitions among nonIDUs and IDUs who are both lowand highrisk users due to change in risky behaviors shows an increase in MRSA cases in the community as individual engage in these risky behaviors. However, with more IDUs entering rehabilitation programs (such as intervention, education, and clean needle exchange programs) MRSA cases reduces.

The horizontal translation within the subgroups due to disease transmission was also studied by implementing three different strategies: loweffectiveness strategy, moderateeffectiveness strategy, and higheffectiveness strategy. We found that higheffectiveness control strategy is more effective in curtailing MRSA burden in the community; further, we found that the moderateeffectiveness strategy is also an effective control strategy.
Appendix A: Analysis of the model
Basic qualitative properties
Positivity and boundedness of solutions
For the MRSA transmission model (1) with IDUs to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time. In other words, solutions of the model system (1) with nonnegative initial data will remain nonnegative for all time t>0.
Lemma 1
Let the initial data F(0)≥0, where F(t)=(U_{ N },C_{ N },I_{ N },U_{ L },C_{ L },I_{ L },U_{ H },C_{ H },I_{ H }). Then the solutions F(t) of the MRSA model (1) with IDUs are nonnegative for all t>0. Furthermore
where π_{ H }=π_{ N }+π_{ L }+π_{ H } and
Proof
Let t_{1}=sup{t>0:F(t)>0∈[0,t]}. Thus, t_{1}>0. It follows from the first equation of the system (1), that
which can be rewritten as
where k_{1}=ω_{ N }+μ. Hence,
so that,
Similarly, it can be shown that F>0 for all t>0.
For the second part of the proof, note that 0<C_{ N }(t)≤N(t), 0<I_{ N }(t)≤N(t), 0<U_{ L }(t)≤N(t), 0<C_{ L }(t)≤N(t), 0<I_{ L }(t)≤N(t), 0<U_{ H }(t)≤N(t), 0<C_{ H }(t)≤N(t), 0<I_{ H }(t)≤N(t).
Adding the human and mosquito component of the MRSA model (1) with IDUs gives
where π_{ H }=π_{ N }+π_{ L }+π_{ H }.
Hence,
as required. □
Invariant regions
The MRSA model (1) with IDUs will be analyzed in a biologicallyfeasible region as follows. Consider the feasible region
with,
Lemma 2
The region \(\Omega \subset \mathbb {R}^{9}_{+}\) is positivelyinvariant for the MRSA model (1) with IDUs with nonnegative initial conditions in \(\mathbb {R}^{9}_{+}\).
Proof
It follows from summing equations of model (1) that
Hence, \(\frac {dN(t)}{dt}\leq 0\), if \(N(0)\geq \frac {\Pi }{\mu } \). Thus, \(N(t)\leq N(0)e^{\mu t}+\frac {\Pi }{\mu }\left (1e^{\mu t}\right)\). In particular, \(N(t)\leq \frac {\Pi }{\mu }\).
Thus, the region Ω is positivelyinvariant. Furthermore, if \(N(0)>\frac {\Pi }{\mu }\), then either the solutions enters Ω in finite time, or N(t) approaches \(\frac {\Pi }{\mu }\) asymptotically. Hence, the region Ω attracts all solutions in \( \mathbb {R}^{9}_{+}\). □
Appendix B: Stability of diseasefree equilibrium (DFE) and the basic reproduction number \(\mathcal {R}_{0}\)
The conditions for stability of the equilibria of the model (1) are stated in this section.
The MRSA model (1) with IDUs has a diseasefree equilibrium (DFE), obtained by setting the righthand sides of the equations in the model to zero, given by
where
with g_{1}=ω_{ N }+μ, g_{2}=ω_{ L }+α_{ L }+μ, g_{3}=α_{ H }+μ.
The linear stability of \(\mathcal {E}_{0}\) can be established using the next generation operator method on the system (1). Taking, C_{ N },I_{ N },C_{ L },I_{ L },C_{ H },I_{ H }, as the infected compartments, then using the notation in [19], the Jacobian matrices F and V for the new infection terms and the remaining transfer terms are respectively given by,
and
where k_{1} = ω_{ N }+μ, k_{2} = τ_{ N }+σ_{ N }+ω_{ N }+μ, k_{3} = γ_{ N }+ω_{ N }+μ+δ_{ N }, k_{4} = ω_{ L }+α_{ L }+μ, k_{5} = τ_{ L }+ω_{ L }+α_{ L }+σ_{ L }+μ, k_{7} = α_{ H }+μ, k_{6} = ω_{ L }+α_{ L }+γ_{ L }+μ+δ_{ L }, k_{8} = τ_{ H }+α_{ H }+σ_{ H }+μ, k_{9} = α_{ H }+γ_{ H }+μ+δ_{ H }.
It follows that the basic reproduction number of the MRSA model (1) with IDUs, is given by:
where ρ is the spectral radius and
Furthermore, the expression \(\mathcal {R}_{N}\) is the number of secondary infections among the noninjection drug users, \(\mathcal {R}_{L}\) is the number of secondary infections among the lowrisk injection drug users, \(\mathcal {R}_{H}\) is the number of secondary infections among the highrisk injection drug users. These expressions (\(\mathcal {R}_{N}, \mathcal {R}_{L}, \mathcal {R}_{H}\)) further show the secondary infection in each subgroups due to both horizontal and vertical transition of infectious individuals. Hence, using Theorem 2 in [19], the following result is established.
Lemma 3
The diseasefree equilibrium (\({\mathcal E}_{0}\)) of the MRSA model (1) with IDUs is locally asymptotically stable (LAS) if \(\mathcal {R}_{0} < 1\) and unstable if \(\mathcal {R}_{0} >1\).
The basic reproduction number \(\mathcal {R}_{0}\) is defined as the average number of new infections that result from one infectious individual in a population that is fully susceptible [16–19]. The epidemiological significance of Lemma 3 is that MRSA will be eliminated from the community if the reproduction number (\({{\mathcal R}}_{0}\)) can be brought to (and maintained at) a value less than unity.
The basic reproduction number when ω _{ N }=ω _{ L }=α _{ L }=α _{ H }=0
Suppose the vertical upward and downward transition between the subgroups are absent, that is, ω_{ N }=0, ω_{ L }=0, α_{ L }=0, α_{ H }=0, then the DFE (10) becomes
and the V matrix (11) is given as
The basic reproduction number of the MRSA model (1) with IDUs, in this case is given by:
where
It should be noted that the reproduction number stated in Eq. (12) gives the reproduction number in Eq. (13) in the absence of vertical downward and upward transition, that is if ω_{ N }=ω_{ L }=α_{ L }=α_{ H }=0.
Abbreviations
 AHRQ:

Agency for Healthcare and research and quality
 CAMRSA:

Communityacquired MRSA
 DAP:

Drug abuse program
 HAC:

Hospitalacquired conditions
 HAMRSA:

Hospitalacquired MRSA
 ICD:

International classification of diseases
 IDUs:

Injection drug users
 MRSA:

Methicillinresistant Staphylococcus aureus
 NSDUH:

National survey on drug use and health
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Keywords
 Methicillinresistant
 Injection drug users
 Sensitivity analysis
 Risk factors
 Control strategies
AMS Subject Classification
 92B05
 93A30
 93C15