Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling

Ulas Hurdoganoglu; Bilgen Kaymakamzade; Nazife Sultanoglu; Emrah Guler; Evren Hincal; Kaya Suer

doi:10.1515/phys-2022-0054

Open Access Published by De Gruyter Open Access June 22, 2022

Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling

Ulas Hurdoganoglu , Bilgen Kaymakamzade , Nazife Sultanoglu , Emrah Guler , Evren Hincal and Kaya Suer

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0054

Abstract

Antibiotic resistance is becoming one of the threats to global health. This crisis has been attributed to the over-the-counter and overuse of antibiotics leading bacteria to gain the ability to resist and survive even in the presence of antimicrobial agents. Escherichia coli (E. coli) is one of the major gram-negative bacteria that are the representative indicators of antibiotic resistance. One of the mechanisms of gaining antibiotic resistance is the ability of E. coli to gain the production of extended-spectrum beta-lactamases (ESBL). In this study, Near East University Hospital data from 2016 to 2019 were used to study the dynamics of ESBL-producing ( ESBL + ) and non-ESBL-producing ( ESBL − ) E. coli infections by using a mathematical model. In our study, the aim was to evaluate the distribution of infections caused by resistant E. coli strains in later years and to increase the success of treatment in patients infected with E. coli by reducing the problem of antibiotic resistance. By using the mathematical model and data of the patients, basic reproduction number (R ₀) values were calculated to study epidemiologic dynamics of the disease. The R 0 E − and R 0 E + values for ESBL − and ESBL + E. coli infections were calculated, respectively. According to the model and the data used within the study, it was calculated that R 0 E − to be 1,266,403 and R 0 E + to be 2,096,747. Since the values for R 0 E − and R 0 E + were equal or greater than 1, this suggests that currently the ESBL − and ESBL + E. coli infections are in epidemic character for Cyprus. Furthermore, when simulation analyses were carried out for the model, it was predicted that in 2042 the ESBL − and ESBL + E. coli infection trends will equalize. After 2042, the ESBL − E. coli infections will indicate a descending pattern whereas ESBL + E. coli infection will increase constantly.

Keywords: Escherichia coli ; mathematical modeling; antibiotic; resistance; ESBL

1 Introduction

Microorganisms in the Enterobacteriaceae family are the most frequently isolated gram-negative bacteria group from clinical specimens. One example is Escherichia coli (E. coli), which is a member of the human intestinal normal flora but can also be the cause of infections like urinary tract infections, hemolytic uremic syndrome, pneumonia, sepsis, meningitis, diarrhea, etc. [1].

In the last decade, bacteria developing resistance to available antibiotics has become a global threat worldwide. One of the biggest reasons for this is the inappropriate and over-the-counter use of antibiotics. Resistance development in gram-negative bacteria against beta-lactam antibiotics is due to the widespread inappropriate use of new beta-lactam antibiotics. Bacteria acquire resistance by gaining the property of producing the beta-lactamase enzyme. These enzymes destroy the antibacterial effect by breaking the amide bonds in the beta-lactam ring of beta-lactam antibiotics [2,3].

One of the most important resistance mechanisms against beta-lactam group antibiotics is the production of extended-spectrum beta-lactamase (ESBL). ESBL-producing Enterobacteriaceae isolates are common human pathogens that pose serious problems for individual and public health [4]. Infections originating from ESBL producing E. coli strains can lead to long-term hospitalization, high treatment costs, and high mortality rates [5,6].

The first choice of antibacterial agent in the treatment of ESBL producing gram-negative bacterial infections is the carbapenem antibiotic group. Similar to beta-lactam group antibiotic, resistance development in the carbapenem antibiotic group is seen frequently due to the widespread and inappropriate use of carbapenems [7]. Carbapenem-resistant Enterobacteriaceae isolates can also be resistant to many other antibiotics and are considered virulent pathogens. Therefore, serious measures must be taken to prevent the spread of these microorganisms [8].

The distribution of microorganisms that cause infectious diseases and their resistance to antibiotics have changed over the years. For this reason, changes in the causative microorganism and antibiotic resistance status should be constantly monitored by each laboratory to guide the application of empirical treatment [9].

Mathematical modeling is often used to analyze the dynamics of infectious diseases such as influenza, childhood infections, HIV, or vector-borne infections. It has been a widely used technique in recent years for the generation of heath policies as well as supporting the control strategies development of infectious diseases [10,11]. Also, simulations applied in a mathematical model are used to forecast ongoing epidemic spread within the studied population [12]. In addition, with the emergence of coronavirus disease 2019 (COVID-19) pandemic, many studies have successfully used mathematical modeling to control and develop country-based health policies [12,13]. The applied mathematical models generally aim to predict the recurrence, spread, and mortality rates of the disease and to explain their causes [14,15].

In this study, culture samples of patients who were admitted to the Near East University (NEU) Hospital between 2016 and 2019 were evaluated. A mathematical model was created with the data determined and obtained retrospectively. The mathematical model used in this study is S ( t ) I ( t ) type model, where S ( t ) stands for people who are likely to be infected at time t d and I ( t ) stands for infected individuals who can spread the disease [16].

The very important threshold quantity of the mathematical model is the basic reproduction number denoted as R ₀ which is an epidemiologic metric used to analyze the dynamics of infectious diseases. The R ₀ is generally reported as a single numeric value and the magnitude of R ₀ value is indicative of a potential size of an outbreak or epidemic within the studied population. Interpretations of the R ₀ value are made as follows: if the value is equal to or above 1 an outbreak is expected to continue. On the other hand, if R ₀ value is below 1, an outbreak is expected to end [17,18].

The created SI type of mathematical model within this study aimed to analyze the antibiotic resistance patterns of E. coli infections and the rate of encountering non-ESBL and ESBL producing E. coli infections. This study also aims to reveal the emergence rates of E. coli strains in patients from the cumulative data of patients visiting NEU Hospital from 2016 to 2019.

Before starting this study, research related to mathematical modeling in the literature was examined. It has been seen that mathematical models are used in the analysis of HIV (human immunodeficiency virus) dynamics [11], in different fields for many purposes during the COVID-19 pandemic [19], in the follow-up of parasitic infections like Hookworm infection [20], in modeling the measles epidemic [21], and in studies related to the investigation of immunological tumor dynamics in cancer patients [22]. In addition, this study was aimed because there are almost no modeling studies on bacteria and antibiotic resistance. The absence of any modeling studies on E. coli and antibiotic resistance patterns in Northern Cyprus is one of the main reasons for planning this study.

There are very few articles in the literature about following or predicting the bacteria that cause common diseases and the resistance patterns they develop with mathematical modeling. Our article is especially important in terms of following up E. coli infections taking precautions by monitoring resistance strains, and therefore, reducing mortality and morbidity rates. The main goals of this study were to: (i) emphasize the inappropriate use of antibiotics, which is the main reason for the development of resistance in bacteria, (ii) evaluate the status of ESBL producer E. coli strains in the future, (iii) make assumptions by discussing the increasing problem of antibiotics, and (iv) estimate the ways to combat these problems and to increase the success rates in the treatment of patients infected with E. coli.

The remainder of the article proceeds as follows. In Section 2, collection of E. coli data and applied models are described. In Section 3, the values of the model in numerical simulation and sensitivity analysis of the dynamics of resistant E. coli strains are explained in the light of the data obtained as a result of the modeling. In the last sections, a discussion of the results and concluding remarks are presented.

2 Experimental procedures

2.1 Collection of data

E. coli strains detected in patients who visited the NEU Hospital between 2016 and 2019 were taken from the Nucleus data system in the hospital with their associated pre-determined parameters. These parameters included age, gender, and culture samples (urine, blood, aspirate, etc.). Also, the parameters included the departments where these samples arrived from such as internal medicine, intensive care, etc. Furthermore, the strains of E. coli were grouped into ESBL negative and ESBL producing E. coli strains according to their antibiotic resistance profile. In this study, to simplify the flow of the text, ESBL producing E. coli strains are referred to as ESBL + and non-ESBL producing E. coli strains as ESBL − .

2.2 Identification of strains and antimicrobial susceptibility tests

The samples taken from the relevant services and polyclinics of the NEU Hospital during the study period were delivered to the microbiology laboratory. The delivered samples were cultured on blood agar and eosin methylene blue (EMB) agar. These media were kept in an incubator at 35°C for 24–48 h depending on the growth status.

For the samples with gram-negative bacterial growth, McFarland bacterial suspensions in the range of 0.50–0.63 MFU were prepared based on the manufacturer’s recommendations. These prepared suspensions were loaded into the Biomerieux VITEK^® 2 Compact (bioMerieux, Inc. Durham, USA) device for identification of the bacterial species and analysis of antimicrobial susceptibility tests. VITEK^® 2 GN (bioMerieux, Inc. Durham, USA) cards were used to identify the gram-negative bacterial species. The antimicrobial resistance of the detected bacteria was determined by the type of sample (urine, sputum, etc.) by using VITEK^® 2 AST-N327 (bioMerieux, Inc. Durham, USA), VITEK^® 2 AST-N325 (bioMerieux, Inc. Durham, USA), and VITEK^® 2 AST-N326 (bioMerieux, Inc. Durham, USA) cards.

2.3 Application of obtained data to the mathematical model

The study used SI type of mathematical model. Basic reproduction numbers for E. coli non-ESBL and ESBL producing infections were calculated and represented as R 0 E + and R 0 E − , respectively. The R ₀ values were calculated using the next-generation matrix method. In addition, local and global sensitivity analyses were performed to illustrate the impact of each parameter on R ₀ value within the mathematical model.

In the constructed model, populations within the hospital were grouped into three mutually exclusive ways: The susceptible S ( t ) , infected with ESBL + E. coli named as E + ( t ) , and infected with ESBL − E. coli named as E − ( t ) . It was assumed that individuals entered one of these classes with the constant variable Λ when they first entered the hospital. However, for the study, the model was continued only with E. coli ESBL + individuals.

Within the created model, the rate of admission to the hospital with infected ESBL − E. coli was referred to as m 1 and individuals infected with ESBL + E. coli was m 2 . Furthermore, the individual’s discharge from hospital by death was stated as μ or hospital discharge as d. The susceptible class can be infected with ESBL − E. coli ( δ 1 ) or ESBL⁺ E. coli ( δ 2 ). The recovery rate from ESB L − was β 1 and the recovery rate from ESBL⁺ was given by β 2 . The rate of transmission from ESBL − E. coli to ESBL⁺ E. coli is β 3 . With these assumptions, the model is given by the system of differential equation in (1), where the definitions and the values of each parameter are given in Table 1.

(1) d S d t = τ ( 1 − m 1 − m 2 ) − δ 1 E − S − δ 2 E + S + α 1 K + α 2 T + c β 1 E − − ( μ + d ) S , d E − d t = τ m 1 + δ 1 E − S − ( c β 1 + ( 1 − c ) β 3 ) β 1 E − − μ E − , d E + d t = τ m 2 + δ 2 E + S + ( 1 − c ) β 3 E − − β 2 E + − ( k + p ) E + − μ E + , d K d t = k E + − α 1 K , d T d t = p E + − α 2 T .

Table 1

Descriptions and the values of the parameters within the mathematical model

Parameters	Descriptions	Values
S	Susceptible to E. coli	17,579
E ⁻	ESBL negative E. coli	1,279
E +	ESBL positive E. coli	601
K	Carbapenems	518
T	Piperacillin-tazobactam	472
Λ	The number of hospital admissions	67.064
m 1	The fraction of patients admitted with ESBL −	0.5099
m 2	The fraction of patients admitted with ESBL +	0.0239
c	The probability that a person takes drug one and be resistant to the drug	0.7
δ 1	The transmission rate of a susceptibility patient infected with ESBL −	0.00000045
δ 2	The transmission rate of susceptible to ESBL⁺ E. coli	0.00000052
β 1	Removed from ESBL − E. coli to susceptible class	0.007819
β 2	Transmission rate from ESBL + E. coli to susceptible class	000655
k	Rate of individuals that can be treated with K	0.000861897
p	Rate of individuals that can be treated with T	0.000078536
β 3	Transmission rate from ESBL − to ESBL +	0.00218
μ	Natural death	0.0002

The most often used treatment options for ESBL + E. coli stated as E + ( t ) within the model were indicated as K (carbapenems) and T (piperacillin-tazobactam). In this study, the model used in [23] was extended by adding two treatment classes K and T . The constructed mathematical model can be visualized in Figure 1.

$Figure 1 The flowchart representing compartments and their interaction with ESBL + {\text{ESBL}}^{+} and ESBL − {\text{ESBL}}^{-} E. coli strains within the model.$

Figure 1

The flowchart representing compartments and their interaction with ESBL + and ESBL − E. coli strains within the model.

It is assumed that the individual with ESBL⁺ can be treated with K (carbapenems) and T (piperacillin-tazobactam) with the rate of k and t , respectively. With these assumptions, newly created differential equation in (2) were used for the model, with the parameters presented in Table 1 [24].

(2) d S d t = τ ( 1 − m 1 − m 2 ) − δ 1 E − S − δ 2 E + S + α 1 K + α 2 T + c β 1 E − − ( μ + d ) S , d E − d t = τ m 1 + δ 1 E − S − ( c β 1 + ( 1 − c ) β 3 ) β 1 E − − μ E − , d E + d t = τ m 2 + δ 2 E + S + ( 1 − c ) β 3 E − − β 2 E + − ( k + p ) E + − μ E + , d K d t = p E + − α 1 K , d T d t = t E + − α 2 T .

2.4 Stability of disease-free equilibrium (DFE) point and basic reproduction ratio

With equalizing to zero of each equation in the system (2) and with the assumption m 1 = m 2 = 0 , the DFE was obtained as E 0 = Λ μ + d , 0 , 0 .

By using the next-generation matrix method, R 0 E + and R 0 E − values were calculated as follows:

F = Λ m 1 + δ 1 S E − Λ m 2 + δ 2 S E + , V = c β 1 E − + ( 1 − c ) β 3 E − + μ E − − ( 1 − c ) β 3 E − + β 2 E + + μ E + .

f = δ 1 Λ μ 0 0 δ 2 Λ μ , v = c β 1 + ( 1 − c ) β 3 + μ 0 − ( 1 − c ) β 3 β 2 + μ .

f v − 1 = S δ 1 c β 1 + ( 1 − c ) β 3 + μ 0 S δ 2 ( 1 − c ) β 3 ( c β 1 + ( 1 − c ) β 3 + μ ) ( β 2 + μ ) S δ 2 β 2 + μ .

Then, Λ δ 1 μ ( c β 1 + ( 1 − c ) β 3 + μ ) and Λ δ 2 μ ( β 2 + μ ) are the dominant eigenvalues of f v − 1 . Therefore, the basic reproduction numbers are given as:

(3) R 0 , E − = Λ δ 1 μ ( c β 1 + ( 1 − c ) β 3 + μ ) ,

(4) R 0 , E + = Λ δ 2 μ ( β 2 + μ ) .

Then, parameters represented in Table 1 were integrated into equations. (3) and (4) and the below values of R 0 , E − and R 0 , E + were obtained:

R 0 , E − = 1 , 266 , 403 ,

R 0 , E + = 2 , 096 , 747 .

Theorem 1

For model (2), the disease-free equilibrium was locally asymptotically stable when R 0 , E − < 1 and R 0 , E + < 1 .

Proof

The Jacobian matrix at the DFE point of the model (2):□

J = − ( μ + d ) c β 1 − δ 1 S 0 − δ 2 S 0 α 1 α 2 0 δ 1 S 0 − ( β 1 + μ ) 0 0 0 0 ( 1 − c ) β 1 δ 2 S 0 − ( β 2 + k + p + μ ) 0 0 0 0 k − α 1 0 0 0 t 0 − α 2 .

The eigenvalues of the Jacobian matrix were: λ 1 = δ 1 S 0 − ( β 1 + μ ) ,

λ 2 = δ 2 S 0 − ( β 2 + k + p + μ ) ,

λ 3 = − ( μ + d ) ,

λ 4 = − α 1 ,

λ 5 = − α 2 .

The λ 3 , λ 4 , and λ 5 were always less than zero and λ 1 was less than zero when R 0 , 1 was less than zero and λ 2 was less than zero when R 0 , 2 was less than zero. Therefore, DFE was locally asymptotically stable when both R 0 , 1 and R 0 , 2 were less than zero.

Sensitivity analysis of R 0

Local sensitivity analyses were applied in order to highlight the sensitivity of some key associated parameters to the value of R 0 in the model.

This was performed to analyze the sensitivity of R 0 in relation to the important parameters used throughout the model. The computation of the expression of R 0 was carried out using the technique of next-generation matrix and it is denoted by

R 0 = R 0 , E − = Λ δ 1 μ ( c β 1 + ( 1 − c ) β 3 + μ ) , R 0 , E + = Λ δ 2 μ ( β 2 + k + p + μ ) .

Then, computation of the normalized local sensitivity was performed which indicates that R 0 is proportional to the parameters in connection to the model.

By using the R 0 indexes, γ = { Λ , μ , c , β 1 , β 2 , β 3 , δ 1 , δ 2 , k , t } , R 0 , the normalized local sensitivity was defined as shown below:

(5) Ω γ R 0 = ∂ R 0 ∂ γ γ R 0 .

Using the above definition (5), the model was computed by the following indices for the output R 0 concerning every parameter presented in

(6) Ω Λ R 0 , ESBL − = δ 1 μ ( c β 1 + ( 1 − c ) β 3 + μ ) μ ( c β 1 + ( 1 − c ) β 3 + μ ) δ 1 Λ = 1 ,

(7) Ω δ 1 R 0 , ESBL − = Λ μ ( c β 1 + ( 1 − c ) β 3 + μ ) μ ( c β 1 + ( 1 − c ) β 3 + μ ) Λ δ 1 = 1 ,

(8) Ω μ R 0 , ESBL − = − Λ δ 1 [ c β 1 + ( 1 − c ) β 3 + 2 μ ] μ 2 ( c β 1 + ( 1 − c ) β 3 + μ ) 2 × μ 2 ( c β 1 + ( 1 − c ) β 3 + μ ) δ 1 Λ = − [ c β 1 + ( 1 − c ) β 3 + 2 μ ] c β 1 + ( 1 − c ) β 3 + μ ,

(9) Ω c R 0 , ESBL − = − Λ δ 1 [ β 1 − β 3 ] μ ( c β 1 + ( 1 − c ) β 3 + μ ) 2 μ c ( c β 1 + ( 1 − c ) β 3 + μ ) δ 1 Λ = − c [ β 1 − β 3 ] c β 1 + ( 1 − c ) β 3 + μ ,

(10) Ω β 1 R 0 , ESBL − = − Λ δ 1 c μ ( c β 1 + ( 1 − c ) β 3 + μ ) 2 × μ β 1 ( c β 1 + ( 1 − c ) β 3 + μ ) δ 1 Λ = − β 1 c c β 1 + ( 1 − c ) β 3 + μ ,

(11) Ω β 3 R 0 , ESBL − = − Λ δ 1 ( 1 − c ) μ ( c β 1 + ( 1 − c ) β 3 + μ ) 2 × μ β 3 ( c β 1 + ( 1 − c ) β 3 + μ ) δ 1 Λ = − β 3 ( 1 − c ) c β 1 + ( 1 − c ) β 3 + μ ,

(12) Ω Λ R 0 , ESBL + = δ 2 μ ( β 2 + k + p + μ ) μ ( β 2 + k + p + μ ) Λ δ 2 = 1 ,

(13) Ω δ 2 R 0 , ESBL + = Λ μ ( β 2 + k + p + μ ) μ ( β 2 + k + p + μ ) Λ δ 2 = 1 ,

(14) Ω μ R 0 , ESBL + = − Λ δ 2 [ β 2 + k + p + 2 μ ] μ 2 ( β 2 + k + t + μ ) 2 μ μ ( β 2 + k + p + μ ) Λ δ 2 = − [ β 2 + k + p + 2 μ ] ( β 2 + k + p + μ ) ,

(15) Ω β 2 R 0 , ESBL + = − Λ δ 2 μ μ 2 ( β 2 + k + p + μ ) 2 β 2 μ ( β 2 + k + p + μ ) Λ δ 2 = − β 2 ( β 2 + k + p + μ ) ,

(16) Ω k R 0 , ESBL + = − Λ δ 2 μ μ 2 ( β 2 + k + p + μ ) 2 k μ ( β 2 + k + p + μ ) Λ δ 2 = − k ( β 2 + k + p + μ ) ,

(17) Ω p R 0 , ESBL + = − Λ δ 2 μ μ 2 ( β 2 + k + p + μ ) 2 t μ ( β 2 + k + p + μ ) Λ δ 2 = − p ( β 2 + k + p + μ ) .

3 Results

In this model, the patients who were diagnosed from 2016 to 2019 solely with E. coli strains were used to study E. coli infections and estimate the simulations on the antibiotic resistance analysis. Based on the collected data, sensitivity analysis was applied to each of the parameters indicated in Table 1 within the model. The parameters in Table 1 were integrated into Eqs. (6–17) and the sensitivity analysis for each parameter in R 0 is represented in Table 2.

Table 2

Sensitivity analysis of R 0 to the designed model

Sensitivity	Sensitivity values
Ω Λ R 0 , ESBL −	1
Ω δ 1 R 0 , ESBL −	1
Ω c R 0 , ESBL −	–0.9599
Ω β 1 R 0 , ESBL −	–0.98545
Ω β 3 R 0 , ESBL −	–0.01095
Ω μ R 0 , ESBL −	–1.0036
Ω Λ R 0 , ESBL +	1
Ω δ 2 R 0 , ESBL +	1
Ω β 2 R 0 , ESBL +	–0.87212
Ω μ R 0 , ESBL +	–0.87745
Ω k R 0 , ESBL +	–0.09817
Ω p R 0 , ESBL +	–0.01046

For ESBL − E. coli, increasing the parameters Λ and δ 1 to 10% also increased the R 0 by 10%. On the other hand, when increasing the c , β 1 , β 3 , and μ parameters to 10% within the model, R 0 value decreased. For the ESBL + E. coli analysis, when Λ and δ 2 parameters increased by 10% the R ₀ value also increased. Whereas, when the parameters β 1 , μ , k, and t were increased by 10%, the R ₀ value decreased.

Furthermore, dynamics of the E. coli ESBL + and ESBL − diagnosed patient numbers with their corresponding years have been studied. Figure 2 indicates the ESBL + and ESBL − patient distribution according to years simulated with the parameters used in the model.

$Figure 2 Dynamics of ESBL + {\text{ESBL}}^{+}\text{ } and ESBL − \hspace{.25em}{\text{ESBL}}^{-} E. coli infections.$

Figure 2

Dynamics of ESBL + and ESBL − E. coli infections.

With the present data, non-ESBL E. coli ( ESBL − E. coli) infection numbers collected in NEU Hospital for the study period (2016–2019) were higher compared to the ESBL producing ( ESBL + ) E. coli infections. For the future predictions, the mathematical model simulation indicated that ESBL + E. coli infections will increase steadily. According to the data, the ESBL + and ESBL − E. coli infections number will become equal in the 23rd year. After the 23rd year, the ESBL + E. coli infections will start to rise with an increasing pattern than the ESBL − E. coli.

3.1 Sensitivity analysis of R 0 within the model for E S B L + E. coli patients

From here, the rest of the study has emphasized the ESBL + E. coli infection analysis. Five parameters: k , p , Λ , δ ₂ and β ₂ used in the model were further increased by 10% to study the effect on the R 0 value for E. coli ESBL + infection, referred to as the sensitivity analysis.

The parameter represents the carbapenemase treatment. Figure 3a represents the original data and Figure 3b represents the 10% increase in carbapenemase treatment to the original data. In this way, the model sensitivity could be analyzed. According to our application, when k is increased within the model, the ESBL + patient number decreases.

The parameter t represents the piperacillin-tazobactam antibiotic treatment in the model. Figure 4a represents the original pattern of t applied to the collected data and Figure 4b represents the 10% increased piperacillin-tazobactam antibiotic treatment to the original data. When t was increased by 10%, the ESBL + patient indicated a decreasing pattern.

The parameter Λ is the total number of hospital admissions to NEU Hospital in the studied period from 2016 January to 2019 December in the model. The original data is depicted in Figure 5a; when Λ is increased by 10%, the number of hospital admissions will increase, the number of ESBL + patients treated with both k (carbapenemase treatment) and t (piperacillin-tazobactam treatment) will also increase as indicated in Figure 5b and c, respectively.

The parameter δ 2 is the transmission rate of susceptible to ESBL + E. coli as shown in Figure 6a. When δ 2 is increased by 10%, the number of E. coli ESBL + patients treated with k (carbapenemase treatment) and t (piperacillin-tazobactam treatment) will also increase as indicated in Figure 6b and c, respectively.

In the model β 2 is the transmission rate of ESBL + E. coli to the susceptible class as shown in Figure 7a. Hence, when β 2 is increased by 10%, the ESBL + E. coli patients treated with k (carbapenemase treatment) and t (piperacillin-tazobactam treatment) will indicate a decreasing pattern as shown in Figure 7b and c, respectively.

In the model, μ is the natural death rate, any death that is not related to E. coli ESBL + or ESBL − infections. In the case when μ is increased by 10%, the number of E. coli ESBL + patients will decrease.

4 Discussion

This study aimed to investigate the near future possibility of the development of antibiotic resistance to E. coli producing ESBL infections (ESBL⁺). To carry out this analysis S I -type model was constructed to determine the basic reproduction number ( R 0 ). Next-generation matrix method was used to calculate the values of R 0 . In this model, two separate values were calculated for ESBL − and ESBL + E. coli infections denoted as R 0 E − and R 0 E + , respectively.

Moreover, local sensitivity analysis was carried out to evaluate the impact of each parameter used in the model to study the impact on R 0 value. R 0 can be used as an indicator of the transmissibility of infectious diseases. It is often used in epidemiological and public health studies to analyze the dynamics of infectious diseases. The calculation of R 0 is generally presented as a single numeric value. If the value of R 0 is greater than or equal to 1 (≥1), the studied outbreak/epidemic is expected to continue whereas if the R 0 value is less than 1 (<1) the outbreak/epidemic is expected to end. Thus, the magnitude of the R 0 represents the potential size of an epidemic or an outbreak within the studied population [18].

For this study, ESBL − is the E. coli infection that can be treated with cephalosporin and other antibiotic groups. Whereas, ESBL + is the infection that cannot be treated with the cephalosporin group of antibiotics. For this type of infection, carbapenem indicated or piperacillin-tazobactam is preferred for its treatment. For the first choice of treatment, carbapenem antibiotic group is used. If the ESBL + E. coli is resistant to this type of antibiotic group, piperacillin-tazobactam is one of the few options available.

This study was conducted using a total of 17,579 hospital admissions to NEU Hospital in the period between January 2016 and December 2019. Amongst the 17,579 hospital admissions, it was found that 1,871 E. coli infections related to hospital admissions occurred in the study period. According to the analysis of the E. coli infection, 1,270 were found to be ESBL − E. coli infections and 601 were found to be ESBL + E. coli infections.

By using the S I , a type of mathematical model, and the parameters depicted in Table 1, R 0 E − and R 0 E + values were calculated, which were found to be 1,266,403 and 2,096,747, respectively. This indicated that with the current data used, both the E. coli infections are in the characteristics of an outbreak. However, the ESBL + E. coli infections have a higher R 0 value than the ESBL − E. coli infections, suggesting that there is a higher possibility for susceptible individuals to acquire ESBL + E. coli than the non-ESBL producing strains of the E. coli.

Furthermore, simulation analysis was performed using the mathematical model and data for NEU hospital for the study period. The dynamics of ESBL − and ESBL + E. coli infections are represented in Figure 2. Within the present time, the ESBL + E. coli infection number is lower than ESBL − E. coli infections. However, with the simulation results, it can be easily seen that although the ESBL + E. coli infection number starts with lower infection rates, as time passes it will follow an increasing trend. The increasing trend of ESBL + E. coli infections will increase constantly and reach the number of ESBL − E. coli infections in the 23rd year of simulation, which is in accordant to the year 2042. After the 23rd year, ESBL − E. coli infections showed descending pattern whereas ESBL + E. coli infection indicated an increasing pattern.

From the simulation results, the increasing pattern predicted in the ESBL + E. coli infections with no doubt will lead to a major public health problem in the future. This is because for ESBL + E. coli infections, there are a few available antibiotic groups to treat patients as most ESBL + E. coli are resistant to many commonly used antibiotics such as the cephalosporins. To avoid the development of resistance to antibiotics that are currently used, it is very important to use antibiotics after antibiotic susceptibility tests are performed. Unconscious and over-the-counter antibiotic use is the biggest cause of the development of antibiotic resistance. Hence, prescribed antibiotics should be used in the correct dose and within the advised period.

Another study carried out in 2014 in Cyprus, also emphasized the arising problem of ESBL + E. coli in urinary tract infections. The conducted study revealed that the 389 strains of ESBL + E. coli were isolated from urine samples in Cyprus among the 53% of hospitalized and 44% of out-patients [25].

Furthermore, sensitivity analysis was used to determine which parameters affect the R 0 value and at what magnitude. Determination of the parameters and their influence in either increasing or decreasing the R 0 value is a key factor in the generation of the infectious disease control policies. To determine the key parameters influencing the R 0 values in this study, five parameters: k , p , Λ , δ ₂ and β ₂ were increased by 10% to perform sensitivity analysis. For this study, sensitivity analyses were only performed for ESBL + E. coli infections. From this analysis it was observed that increasing the parameters of hospital admissions Λ (Figure 3) and transmission of ESBL + E. coli infections to susceptible individuals δ 2 (Figure 4) by 10%, the R 0 value can be increased, indicating that when hospital admissions and ESBL + E. coli infections to susceptible individuals increase, the R 0 value of the ESBL + E. coli will be in an epidemic character.

$Figure 3 Impact of the variation in k k on the number of infectiveness of ESBL + {\text{ESBL}}^{+}\text{ } E. coli patients.$

Figure 3

Impact of the variation in k on the number of infectiveness of ESBL + E. coli patients.

$Figure 4 Impact of the variation in p p on the number of infectiveness of ESBL + {\text{ESBL}}^{+}\hspace{.25em} E. coli patients.$

Figure 4

Impact of the variation in p on the number of infectiveness of ESBL + E. coli patients.

On the other hand, increasing the death rate μ , any death that is not related to E. coli infections, by 10%, the R 0 value will decrease (Figure 5). Increasing 10% of the individuals that can be treated with k (carbapenemase treatment) (Figure 6) and t (piperacillin-tazobactam treatment) (Figure 7) will also decrease the R 0 value by 10%. However, if any drug resistance develops to these groups of antibiotics, which is a high chance when considering the limited available treatment options, it may lead to the development of persistent infections. Increasing the transmission rate from ESBL + E. coli to susceptible class β 2 by 10% will decrease the R 0 value by 10% (Figure 8). Accomplishment with correct antibiotic choice and adherence to the treatment within the correct period is a critical factor at this point.

$Figure 5 Impact of the variation in Λ \Lambda on the number of infectiveness of ESBL + {\text{ ESBL}}^{+} E. coli patients.$

Figure 5

Impact of the variation in Λ on the number of infectiveness of ESBL + E. coli patients.

$Figure 6 Impact of the variation in δ 2 {\delta }_{2} on the number of infectiveness of ESBL + {\text{ESBL}}^{+}\text{ } E. coli patients.$

Figure 6

Impact of the variation in δ 2 on the number of infectiveness of ESBL + E. coli patients.

$Figure 7 Impact of the variation in β 2 {\beta }_{2} on the number of infectiveness of ESBL + \hspace{.25em}{\text{ESBL}}^{+}\text{ } E. coli patients.$

Figure 7

Impact of the variation in β 2 on the number of infectiveness of ESBL + E. coli patients.

$Figure 8 Impact of the variation in μ \mu on the number of infectiveness of ESBL + \hspace{.25em}{\text{ESBL}}^{+} E. coli patients.$

Figure 8

Impact of the variation in μ on the number of infectiveness of ESBL + E. coli patients.

5 Conclusion

In conclusion, ESBL + E. coli strains are following an increasing trend in Cyprus and the limited number of options for treatment will be inadequate in near future to treat simple infections. Also, the model predicts that the number of ESBL + E. coli infections will equalize to ESBL − E. coli infections in 2042. After this point, the ESBL + E. coli infections rise constantly. Hence, an immediate rearrangement of the current therapy guidelines should be established. These facts raise the demand for the discovery of a new class of antimicrobial agents and the currently available alternatives should be used efficiently.

The use of over-the-counter antibiotics should be prevented all over the world. Deterrent measures should be strengthened by law to prevent the sale of antibiotics without a prescription. The problems that may be caused by unconscious antibiotic use in the following years should be tried to be shown to people through such studies. In addition, the effective and active functioning of the infection control committee in each hospital and the timely training of all personnel is an important step in slowing down the rate of all infections. This study shows us that when all these are not done, increasing antibiotic resistance will continue to be a major public health problem affecting the whole world.

Selling antibiotics without a prescription may cause bacterial resistance, such as ESBL or carbapenem, in the community. However, it is clear that the spread of resistant bacteria within the hospitals, significantly increases mortality and morbidity rates in patients. For these reasons, effective surveillance studies should be conducted which will provide us with information about these types of infections and give clues for the precautions to be taken in the future. The acceptance and implementation of the concept of “one health” in all countries has an important place among the measures that can be taken. In recent years, the use of mathematical models in health fields gives us the chance to obtain more precise information and to prevent or take early precautions against infections caused by resistant bacteria.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2022-02-22

Revised: 2022-05-10

Accepted: 2022-05-24

Published Online: 2022-06-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling

Abstract

1 Introduction

2 Experimental procedures

2.1 Collection of data

2.2 Identification of strains and antimicrobial susceptibility tests

2.3 Application of obtained data to the mathematical model

2.4 Stability of disease-free equilibrium (DFE) point and basic reproduction ratio

Theorem 1

Proof

3 Results

3.1 Sensitivity analysis of R 0 within the model for E S B L + E. coli patients

4 Discussion

5 Conclusion

References

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