Intervened 2-Tier Poisson Distribution for Understanding Hospital Site Infectivity

Problem statement: Whether it is a surgical site or medical treatment at a hospital site, the nurses in particular and the entire medical team in cluding surgeons/physicians in general undergo a risk of being infected ironically by the patients w hom they intend to disinfect. There are 2-tiers of patients at the site. One type consists of patients who are internally and well pre-disinfected, not t o be sourced for infecting the medical team. The second type consists of patients who are influx to the hospital site and are not well pre-disinfected enou gh. It is the second type which is a source of hosp ital site infection for the medical team in general and for the nurses in particular. In other words, the n urses who have to deal with the second type of patients g et more exposed to the virus from the patients themselves. This is named the nurses’ exposure expo sure rate. Independently, there is an inactivity ra e in general for anyone. To reduce such an infectivit y, the hospital management makes an intervention with preventive efforts to reduce the infection rat e and the impact of such preventive intervention efforts is captured by a parameter in our model. Us ing a maximum likelihood estimate of the intervention parameter with the data information, w e assess the significance of the intervention effor ts. Approach: For this concept to work, there is a need to devel op an appropriate model as none exists in the literature to be suitable. The model is an abst r ction of the reality in the hospital set up. Such a needed, new probability count model is introduced. It is named an Intervened 2-Tier Poisson (I2TP) distribution in this article. Several statistical p ro erties of the I2TP distribution are derived and illustrated to explain the inactivity rate, θ>0 during the treatments of contagious patients in a hospital. Not all nurses are exposed to the virus, while 0 ( π≤1 is their exposure rate towards infection. The physicians/surgeons, nurses and staffs undergo a ri sk of being infected during their treatment of infection or surgery on patients in spite of precau tions to avoid infection. The hospital management intervenes with several precautions to minimize, if not eliminate the health care personnel’s risk of being infected. In this article, a statistical meth odology is developed to estimate and test the significance of the management’s intervention effec t, ρ≥0. Results: The methodology is illustrated using the number of exposed and infected nurses dur ing their healthcare of SARS patients in a Toronto hospital as reported in http://wwwnc.cdc.gov/. Ther e were 32 nurses in the Toronto hospital working with the Severe Acute Respiratory Syndrome (SARS) p atients. The sixteen activities of the nurses included in our analysis are administration of medi cation, intubation, bathing, manipulation of bipap mask, radiology procedures among others. In all these activities, the nurses are well train ed to use disinfected gloves, nasal masks. As part of the pre ventive measures to avoid infection from the SARS patients. The exposure rates for the nurses in thes e activities to SARS patients varied from 0.13 to 0,81. The infectivity ranged from 1.26 to 8 in thes e activities. The impact of the intervention effort s ranged from 0.25 to 206.3 in all these sixteen acti vities. The impact of the intervention efforts was insignificant in the activities: endotracheal aspir ate, integration of a peripheral,, intravenous cath eter Intubation, manipulation of bipap mask, Manipulatio n f bipap mask, manipulation of commodes or bedpans, Nebulizer treatment and Suctioning before intubation . The impact of the intervention was significant in the activities: administration of me dication, assessment of patient, bathing or patient transfer, manipulation of oxygen mask, mouth or den tal care, performing an electrocardiogram, radiology procedures, suctioning after intubation and venipuncture . Conclusion/Recommendations: It is interesting to notice that the preventive interv ention efforts by the hospital management for the nurses to be disinfected from the SARS patients wor ked in some activities but not in others. This distinction could be made because of the intervened 2-tier Poisson distribution which is introduced in this article. Clues for successful intervention in some but not in other activities perhaps hid in Intl. J. Res. Nursing 3 (1): 8-14, 2012 9 covariates. Currently, the author is not able to ac cess such data on covariates. The future research w ork would proceed in this direction using regression co cepts.


INTRODUCTION
Infection is a colonization of a virus leading towards a disease. Hosts do normally fight infections via their immune system. The physicians/surgeons, nurses and staffs undergo a risk of being infected during their treatment of infections or surgery patients in spite of precautions to avoid infection. Ironically, the source of infection for physicians/surgeons, nurses and supportive staffs is the patients who are helped by them. This serious phenomenon occurs in surgical or hospice situations. Viable prevention strategies are necessary, though not sufficient, to avoid being infected. Techniques like hand washing, wearing gowns and wearing face masks among others help to prevent infections from being passed on to the healthcare workers from the patients. This article examines the issue and develops a new model. This new model is named Intervened 2-Tier Poisson (I2TP) distribution. The subtitle "2-tier" is appropriate to suit the theme that not all healthcare workers of the contagious patients get exposed in the first place and not all exposed health care workers do end up with an infection.
The statistical properties of I2TP distribution are derived to explain health care workers' infectivity rate during a surgery or treatment of patients in hospital. The results are illustrated later in the article using the number of exposed and infected cases nurses during the treatment of SARS patients in a Toronto hospital as reported in Loeb et al. (2004). The final thoughts are stated for future research direction in the end.

Main results: Intervened 2-tier poisson distribution
Let 0<π≤1, θ>0 and ρ≥0 denote respectively the exposure rate, infectivity rate and intervention effect to minimize (if not eliminate) the infections during the activities rendered to the patients by healthcare workers in general and by nurses in particular. To be specific, let the random variable (rv), N of healthcare workers have encountered with the contagious patients in a hospital. The rv N usually follows a Poisson distribution. That is: x Pr(N ) e / n!;n 0,1, 2,3,...; 0 Realizing that not all healthcare workers are exposed, let I k = 1if the k th healthcare worker is exposed with the probability π and I k = 0 otherwise. Consider X = l 1 + l 2 +…+l n . Notice that X follows a binomial distribution with parameter π conditional on N = n. That is: Unconditionally, the rv X follows a Poisson distribution because Eq. 1: The Poisson distribution in (1) with parameter πθ is a 2-tier type. However, the event X = 0 is not usually observed. The data collection apparatus is activated only when X≥1. In other words, medical intervention takes place to control the infectivity only when a nonzero X = 1, 2, 3… incidence is noticed. This probability pattern of the non-zero incidence of X is then a positive Poisson (PP) distribution (2). That is Eq. 2: 1 x Pr(X , ) (e 1) ( ) / x!; x 1,2,3,...; 0 At this stage, the healthcare management intervenes to control and/or eliminate the infectivity by resorting to various preventive actions including training healthcare workers to be disinfected. The effectiveness of this intervention is not observable but an unknown parameter ρ≥0. Let Z be the number of additional healthcare workers with infection since the time of interventions. The general infectivity rate θ>0 is now modified to ρθ>0 because of the intervention efforts. Consequently, Pr (Z = z) = e-ρθ (ρθ) z /z! . The recorded number of infected healthcare workers is not Z but rather Y = X + Z. The rv Y follows an intervened 2tier Poisson (I2-TP) distribution Eq. 3. That is: where, y = 1, 2, 3,… Is the expression (3) a bona-fide probability distribution? The answer is affirmative because Pr(Y = yπ,θ,ρ)>0 and y 1 Pr(Y y , , ) 1 After algebraic simplifications, its mean and variance are obtained. The mean of I2-TP distribution is in Eq. 4: and the variance is a quadratic function of the mean as in Eq. 5. That is: The survival function of the I2-TP distribution is: The odds of having no more than r infected healthcare workers is then: r 2 2 1 2 r 2 r odds 1 S(r , , ) The odds are popular in healthcare studies. The epidemiologists are fond of the odds. In particular, the odds of having one infected healthcare worker in a hospital where contagious patients are treated by healthcare workers is given in Eq. 6: An estimation procedure is necessary for the model parameters based on a collected sample y 1 , y 2 …, y n of size n from I2-TP distribution in (3). The Maximum Likelihood Estimates (MLE) are preferable because the MLE possess invariance property. That is the MLE of a function is the function of MLE. First, the MLE of π is the solution of the score functions ∂ π In L(x 1 , x 2 ,…,x n n) = 0 where ∂ a is the derivative with respect to a. It is in Eq. 7: m l e x n π = Secondly, the conditional score functions ∂ θ In L(θ, ρπ) = 0 and ∂ ρ In L(θ, ρπ) = 0 need to be solved. To obtain the conditional score functions, the log likelihood function is in Eq. 8: . The log likelihood function in (8) is differentiated with respect to the parameters θ and ρ to obtain the score functions. The score functions are in Eq. 9 through Eq. 10: and: mlê ln L( , ) 0 n ny n(1 ) The simultaneous and conditional MLE of the parameters are therefore in Eq. 11 and 12: and: For administrative reasons, the healthcare management might want to assess the significance of the implemented intervention effect ρ. This task amounts to perform a hypothesis testing of against H ο :ρ = 0 the alternative hypothesis H α : ρ = ρ * # 0. It is possible to develop a procedure based on Wald (1943) criterion. For this purpose, the log-likelihood ratio-In Λ ρ is first obtained in Eq. 13. It is:  Johnson et al. (1997) and Stuart and Ord (2009) for definition and properties of the non-central chi squared distribution. Recall that the variance-covariance matrix of the MLE of the parameters is the inverse of the information matrix: with and a significance level a ∈(0, 1). We now write the p-value for rejecting the null hypothesis in favor of an alternative hypothesis and it is in Eq. 14.
The statistical power of our test statistic is now examined with a selection of a specific attainable value for ρ * in the alternative hypothesis The statistical power is the probability of rejecting the null hypothesis H ο : ρ = 0. in favor of an alternative hypothesis H 1 : ρ = ρ * = 1. After algebraic simplifications, we find that as stated in Eq. 15:  Illustration using sars infections: Severe Acute Respiratory Syndrome (SARS) patients were treated by nurses who worked in two Toronto critical care units. Some nurses were infected. Chen et al. (2008) and Poutanen et al. (2003) for details about SARS. McKibben et al. (2005), Understanding Infectious Diseases, 2010 and Preventing Infections Adequately, 2010. for details about the recommended preventive actions to be disinfected. The Table 1 provides infected data during services to contagious patients on treatment care activities and the results for the methodology in this article. We considered only sixteen healthcare activities. The excluded healthcare activities had either missing entry or just one infected nurse which is not enough for modeling. Our methodology is suitable for the activities which had two or more infected nurses. The Fig. 1 confirms an upward relationship between the number of exposed nurses (X) and the number of infected nurses (Y). More exposed nurses resulted in more infected nurses.
The MLE of exposure rate, infectivity rate and intervention effect are indicated by notations mle mlê , π ρ and mlê θ respectively in Table 1.
The notations df0 and df1 denote the degrees of freedom under the null hypothesis H ο : ρ = 0 and alternative hypothesis H α : ρ ≠ 0 respectively. The pvalue is the chance for the null hypothesis to be true meaning that its smaller value refers rejection of the null hypothesis. An interpretation is that the intervention was effective only in the healthcare activities: administration of medication, assessment of patients, bathing or patient transfer, manipulation of oxygen mask, mouth or dental care, performing an electro cardiogram, radiology procedures, suctioning after intubation and vein puncture. The expression (15) is used to perform the hypothesis testing. In most of the above healthcare activities, the statistical power to accept the specific alternative H α : ρ = ρ * = 1 is excellent as they are shown in Table 1.
The intervention was not effective in healthcare activities: Endotracheal aspirate, insertion of a peripheral, intubation, manipulation of BiPAP mask, manipulation of commodes or bedpans, nebulizer treatment and suctioning before intubation. In these activities, the statistical power to accept the specific alternative H α : ρ = ρ * = 1 is poor as noticeable so in Table 1.
The odds for a nurse to get infected are given in Table 2 for all sixteen activities. Note the odds are high in endotracheal aspirate, insertion of a peripheral, intubation, manipulation of BiPAP mask, manipulation of commodes or bedpans and nebulizer treatment. The other activities have lesser odds as shown in Table 2.  The Figure 2 illustrates that when the exposure rate increase, the intervention effect has also increased as one would expect. The Fig. 3 illustrates that when the intervention effect is high, the infectivity rate is lower as one would expect. The Fig. 4 illustrates that when the exposure rate is high, the infectivity is low and it is not quite intuitive. It is so because the intervention effect has an impact on both of them.   The Fig. 5 illustrates that the odds for one nurse to be infected is high only when the intervention effect is low and vice versa. The importance of considering the intervention effect in analyzing exposure versus infection data could not be overstated.