This study is a hospital-based study examining the EAR in nosocomial infections caused by E. coli using data related to the Nosocomial Infection Care System of Iran . Patients who suffer from nosocomial infections in hospitals covered by this care system, follow-up and their characteristics such as date of hospitalization, duration of hospitalization, date of infection, type of infection, type of pathogen causing the infection, patient age, patient sex, type of anti-infection test Biogram and its result are registered in the system of this care system. In 1396, 107670 cases of nosocomial infections were registered in this system, of which 15165 cases were related to nosocomial infections caused by Escherichia coli pathogen. In this study, data related to these infections were used. The observed cases of antibiotic-resistant Escherichia coli were extracted from this data and compared with the predicted cases, which were predicted by setting up a compartment model (separately for the desired antibiotics). Thus, EAR in this pathogen was investigated.
The study population in this study were patients who had a nosocomial infection caused by E. coli in Iranian hospitals between April 1 and March 20, 2017. In this study, all patients admitted to hospitals located in the centers of 31 provinces (57 hospitals) who suffered from nosocomial infections caused by E. coli and underwent antibiogram test (a total of 12,954 antibiogram tests) were included in the study. The reason for choosing the university hospitals was their greater cooperation in performing antibiogram tests and the greater number of samples required in these hospitals. Also, public hospitals are on the same level in respect of patient admission and accept a wide range of patients in different wards.
Estimation of required parameters and validation of the model used in this study.The data related to the results of these 12,954 tests were divided into two parts for some reasons as following:
Part 1) Data related to the first half of the year, with 5,701 tests performed.
Part 2) Data related to the second half of the year, with 7,253 tests performed.
To evaluate EAR in this study, the Standardized Infection Ratio (SIR) index was used. This index was calculated by dividing the number of observed cases of antibiotic-resistant E.coli by the number of expected cases. If this index is greater than one, it may indicate EAR. A 95% confidence interval was created to check the significance of this index. If the 95% confidence interval included the number one, it indicated that the index was not statistically significant; otherwise, and if it did not include the number one, it indicated the statistical significance of the index. To predict the expected cases of antibiotic-resistant E. coli, a compartmental model  was developed for each of the 11 antibiotics. The population in this type of model is classified into sections (compartments) that actually create the structure of the model. In this study, the SIR structure has been used. In this structure, the subjects were divided into three sections called Susceptible (S), Infectious (I) and Remove (R). S denotes patients with E.coli nosocomial infections. I denotes nosocomial infections due to E.coli that developed antibiotic resistance, and R denotes resistant cases of E. coli that were discharged from the hospital for any reason (death, treatment). The model predicts the transfer of people between these segments with pre-determined coefficients. In this study, the probability of transferring people from section S to section I, with the incidence rate (ir) and transferring people from section I to section R, with the remove rate (rr). The "Difference Equations" approach was used to set up the model. In this approach, the transfer of items between sections is described using "Discrete time steps" (such as days). In fact, this method is to estimate the number of cases at a particular time . In this study, with the help of these equations, the transition from part S to part I and from part I to part R in the time unit of the day was described. Thus, the number of new cases of antibiotic-resistant E. coli in the days in question was estimated. To set up these equations, " Euler" method was used in Berkeley Madonna software version 8.3.23. This method is specifically used to set up this type of equation . We set up the models by drawing their diagrams. Thus, instead of manually typing the equations, the model was created graphically, followed by textual equations. In this study, by drawing a model diagram, the model was developed separately for each of the 11 antibiotics (Figure 1).
S: Total number of nosocomial infections caused by E. coli; ir: The incidence of resistant E.a coli per person-day hospitalization; New I: The number of cases that have become resistant between t and t+1 time; I: Number of cases of resistant E. coli at time t; rr: The ratio of resistant E. coli discharged from the hospital per day; New r: The number of cases of resistant E. coli that have been discharged between t and t+1 time; R: The number of cases of resistant E. coli that were discharged from the hospital at time t.
The equations (Difference equations) generated by the model are shown below:
S(t+dt): Number of cases of nosocomial infections caused by E. coli at t+1 time
I(t+dt): Number of resistant E. coli at t+1 time
R(t+dt): Number of resistant E. coli removed at t+1 time
t: times used in the model (times between start time and end time)
dt : Size of time in dialy units, in this study, the number one is considered.
When developing these models, several parameters were required (INIT S, INIT I, INIT R, incidence rate (ir), remove rate (rr), start time and end time. INIT S, INIT I and INIT R were known parameters, and their values were available at the beginning of the study. INIT S were all cases of E. coli nosocomial infections for which antibiogram test was performed. INIT I, the number of patients with resistant E. coli was at the beginning of the study, which is considered "1" in these models. INIT R, cases of resistant E. coli were discharged from the hospital at the beginning of the study, which are considered "zero" in these models. The rest of the parameters including ir, rr, start time and end time were among the unknown parameters of the model, the values of which should have been determined by examining the texts. Since this study was the first study to investigate EAR in nosocomial infections caused by E. coli with the help of the compartmental model, it was not possible to extract the unknown parameters of this model by reviewing the literature. In other words, by examining the texts, there was no similar study. As a result, these parameters were estimated using data from the first half of the year. When only one "data set" is available, it can be randomly divided into two parts, the model developed in one part and its validity evaluated in the other part . The model was developed in the second half of the year to predict the expected cases of resistant E. coli. The validity and performance of the model were evaluated by performing the model in the first half of the year. If necessary (difference between model predictions and reality), the model was optimized.
Then all the new cases of predicted resistant E. coli were added together. Thus, the total number of predicted cases of E. coli resistant to 11 antibiotics was estimated. In this study, cross-validation method was used to evaluate the validity of the model. In this method, before the model was implemented to investigate resistant E. coli outbreak in the second half of the year, it was first performed to evaluate its validity in the first half of the year. The output of the model implemented in the first half of the year was compared with the cases observed in the same half. In other words, the predicted cases of resistant E. coli were compared by the model with the observed cases of resistant E. coli in the first half of the year.
If the resistance cases predicted in the first half of the year were similar to the resistance cases observed in this half, and there was no significant difference, it was a sign of good performance of the model and its validity. In cases where the difference between the two was statistically significant, the incidence parameter value was changed to create the smallest distance between the number of predicted cases of resistant E. coli and the number of observed cases of resistant E. coli. This change was made with the help of a function called "Optimize"  in the Berkeley Madonna program.
This function determines the distance for the incidence rate parameter (50% less than the incidence rate and 50% more than the incidence rate), then using the values specified in this interval, the model performed. Output of each model closer to the observed data, was selected as the optimized incidence rate. Since in this study the available data sets were divided into two parts based on time, the validity of the model is considered as the intermediate of the internal and external validity. .