Fecal carriage of ESBL-producing <i>Escherichia coli</i> in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University

Amira ElBaradei; Dalia Ali Maharem; Ola Kader; Mustafa Kareem Ghareeb; Iman S. Naga; Amira ElBaradei; Dalia Ali Maharem; Ola Kader; Mustafa Kareem Ghareeb; Iman S. Naga

doi:10.3934/microbiol.2020025

AIMS Microbiology

2020, Volume 6, Issue 4: 422-433. doi: 10.3934/microbiol.2020025

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Research article

Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University

1.
Department of Microbiology and Immunology, Faculty of Pharmacy, Pharos University in Alexandria, Alexandria, Egypt
2.
Alexandria University Hospital, Alexandria University, Alexandria, Egypt
3.
Department of Internal Medicine, Medical Research Institute, Alexandria University, Alexandria, Egypt
4.
Department of Microbiology, Medical Research Institute, University of Alexandria, Alexandria, Egypt
5.
Bachelor of Medical Laboratory Technology, Al-Yarmouk University, Iraq

Received: 25 July 2020 Accepted: 29 October 2020 Published: 03 November 2020

Commensal ESBL-producing E. coli represent a reservoir for resistance genes therefore, their detection is crucial to restrain the spread of beta-lactam resistance. Hence, the aim of the present study was phenotypic and genotypic characterization of commensal ESBL-producing E. coli obtained from the stool of patients at the time of admission and at the time of discharge from the Medical Research Institute hospital. A total of 70 E. coli isolates were collected from 35 patients and were categorized into Group A (samples obtained on admission) and Group B (samples obtained at the time of discharge). Phenotypically, 30 isolates were ESBL producers (40% of E. coli isolates collected on admission and 45.7% of the strains obtained at the time of discharge were ESBL producers). Most of them harbored one to three plasmids with sizes ranging from one kbp to ten kbp. Upon genotypic investigation, bla_CTX-M was the most detected gene in 80% of ESBL strains, followed by bla_TEM in 53.3% and the least detected was bla_SHV in only 13.3%. By comparing group A and group B, ten patients were found to carry commensal ESBL-producing E. coli, in two patients these isolates carried ESBL genes that were identical on admission and on discharge. However, in eight patients, these isolates carried different ESBL genes, which were newly harbored during hospital stay. The high abundance of MDR commensal E. coli 48.57% together with the presence of 42.86% ESBL-producing commensal E. coli among our isolates represents an alarming threat, as they are frequently associated with the increased risk of infection, higher costs and longer hospital stay.

Keywords:

Citation: Amira ElBaradei, Dalia Ali Maharem, Ola Kader, Mustafa Kareem Ghareeb, Iman S. Naga. Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University[J]. AIMS Microbiology, 2020, 6(4): 422-433. doi: 10.3934/microbiol.2020025

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Abstract

1. Introduction

Nonlinear evolution equations (NEEs) are mathematical equations that describe the behavior of complex physical systems. These equations are often used in various fields of science, including material and biological sciences, physics, mathematics, and mechanics ^[1,2,3,4,5]. One of the most commonly used NEE is the nonlinear Schrödinger equation, which is used in quantum mechanics to describe the behavior of wave packets ^[6,7]. Another example is the Klein-Gordon equation, which is used in particle physics to describe the behavior of spinless particles. The Cahn-Hilliard equation is another important equation used in materials science to describe the phase separation behaviour of binary mixtures. This equation has applications in the fields of metallurgy, polymer science, and biomaterials. The Navier-Stokes equation is a well-known NEE that describes the behavior of fluids, and is used in fluid mechanics to study the flow of fluids in various applications, including aerospace engineering, civil engineering, and oceanography ^[8,9,10,11].

NEEs are important in many fields of science because they can provide a mathematical model to describe the behavior of complex physical systems. By studying these equations and developing new numerical techniques to solve them, scientists can gain a better understanding of the physical phenomena they describe, and develop new technologies and materials that are based on this knowledge. To study the behavior of the physical problem (see ^{[12,13,14,15]}), either analytical or numerical solutions are needed. However, finding an analytical solution for NNEs is not always simple. As a result, developing numerical methods for such problems is useful. In this article, we investigate the nonlinear Klein-Gordon and sinh-Gordon equations, which are given as follows:

$\begin{equation} u_{tt}+\alpha \Delta u+ \psi(u) = g(x, t), \end{equation}$

(1.1)

where $u = u({\bf x}, {\bf{t}}),$ $\alpha$ is constant, and $\psi$ is some nonlinear function. For

$\psi(u) = \sinh(u),$

the Eq (1.1) becomes the sinh-Gordon equation ^[16]. When

$\psi(u) = \beta u+\gamma u^2,$

the Eq (1.1) becomes the Klein-Gordon equation, which is given as follows:

$\begin{equation} u_{tt}+\alpha \Delta u+ \beta u+\gamma u^2 = g(x, t), \end{equation}$

(1.2)

with the following initial and boundary conditions:

$\begin{align} u(x, 0)& = g_1(x), \; \; u_t(x, 0) = g_2(x), \; \; a\leq x \leq b, \end{align}$

(1.3)

$\begin{align} u(a, t)& = f_{1}(t), \; \; u(b, t) = f_2(t), \; \; t > 0. \end{align}$

(1.4)

The Klein-Gordon equation has numerous applications in different areas of physics and engineering and plays a crucial role in our understanding of the behavior of particles and fields in the universe. The Klein-Gordon equation is a relativistic wave equation that describes the behavior of particles with a zero spin, such as the Higgs boson particle. It has several applications in different fields of physics; for example, the Klein-Gordon equation is fundamental in the quantum field theory. It describes the behavior of scalar fields, which are fields that have a single value at each point in space-time ^[17,18]. Additionally, it can be used to describe the behavior of phonons, which are the quantized vibrations of a crystal lattice, and is used to describe the behavior of the inflaton field, which is thought to be responsible for the rapid expansion of the universe during the Big Bang ^[19]. Due to the wide range of applications, numerous researchers have investigated the solution of this equation utilizing various methodologies, including collocation points with the thin-plate splines radial basis function ^[20], the sine-cosine and tanh methods for travelling wave solutions ^[19], and the generalized auxiliary equation methods in ^[21]. Pseduospectral formulation with a perfectly matched layer has been suggested for the non-relativistic Klein-Gordon equation in ^[22]. Three different numerical methods were introduced to solve the rotating Klein-Gordon equation from relativistic regimes to non-relativistic regimes in polar and Cartesian coordinates ^[23]. Various numerical methods, including finite difference time domain methods, the time-splitting method, an exponential wave integrator, a limit integrator, a multiscale time integrator, the two-scale formulation method, and an iterative exponential integrator, were compared in terms of their accuracy while solving the Klein-Gordon equation (see ^[24]). The authors used a finite element technique based on cubic B-splines, the decomposition method ^[25], the fourth-order implicit Runge-Kutta-Nystrom, the fourth-order compact finite difference method ^[26], the boundary integral equation method ^[27], and the cubic B-spline collocation approach ^[28] to solve the generalized form of the nonlinear Klein-Gordon equation. For the solution of this model, a variety of numerical techniques have been reported in ^[29,30]. The two-dimensional sinh-Gordon equation is an essential partial differential equation (PDE) model with applications in a variety of fields.

The following is a concise review of the sinh-Gordon equation. In ^[31], the (G'/G)-expansion scheme was proposed for the generalized form of the double sinh-Gordon equations. For the analytical solutions to the sinh-Gordon equation, the tanh method was used in ^[32]. The element-free Galerkin technique was suggested in ^[33] to solve the generalized sinh-Gordon equation. The radial basis function was utilized by the authors in ^[34] to solve the sinh-Gordon equation. In ^[35], the author employed an implicit Lie-group iterative technique for the solution of the sinh-Gordon equation, whereas the Pseudo spectral and Kansa's methods based on radial basis functions were used in ^[36] to solve the two-dimensional sinh-Gordon equation. In ^[37], two effective approaches, namely the generalized leap-frog and the method of lines, were utilized for the sinh-Gordon equation, whereas the polynomial differential quadrature method was examined in ^[38].

In this work, we utilize the Fibonacci polynomials combined with Störmer's method to compute the numerical solutions of the one-dimensional Klein-Gordon and the two-dimensional sinh-Gordon equations. One of the key advantages of the suggested method is the ease implementation in higher-order derivatives using the relation of the Lucas and Fibonacci polynomials. Furthermore, the proposed approach improves the accuracy even for a small number of collocation points, thus lowering the computing costs. These polynomials have a wide range of applications in differential equations; for instance, the relationship between Chebyshev and Lucas polynomials was discussed in ^[39], and accurate solutions for the boundary value problems were obtained. Higher-order differential equations were solved using the Lucas polynomial technique in ^[40], while the Volterra-Fredholm integral differential equations were solved using a Fibonacci polynomial approach in ^[41]. Delay difference equations were solved using a hybrid Taylor-Lucas polynomial approach ^[42]. The Fibonacci wavelet approach was implemented for non-linear reaction diffusion equations of a Ficher-type, non-linear Hunter-Saxton and time fractional telegraph equation in ^[43,44]. The author and his co-worker used the Gegenbauer wavelet method for time fractional PDEs in ^[45,46]. For the first time, the authors proposed a hybrid Lucas and Fibonacci polynomial scheme to solve the time-dependent PDEs in ^[16,47]. More publications using Lucas polynomials and finite-differences to find efficient numerical solutions for different types of PDE models can be found in ^{[48,49,50,51]}.

The rest of the paper is organized as follows: in Section 2, fundamental concepts and definitions are discussed; in Section 3, the proposed methodology of the scheme is described; the method is verified with the help of numerical experiment in Section 4, and finally, the concluding remarks are drawn at the end.

2. Fundamental concepts

In this section, we will go over several key definitions and properties related to the Fibonacci and Lucas polynomials.

2.1. Fibonacci polynomials

The Fibonacci polynomial is an extension of Fibonacci numbers described by the linear recurrence relation, which is shown as follows ^[52]:

$\begin{equation} F_k(x) = kF_{k-1}(x)+F_{k-2}(x), \; \; k\; \geq 2, \end{equation}$

(2.1)

with the initial values

$F_0(x) = 0 \; \; \text{and}\; \; F_1(x) = 1.$

Their explicit form is as follows:

$\begin{equation*} F_{k}(x) = \sum\limits_{n = 0}^{\lfloor \frac{k}{2}\rfloor}\binom{k-n}{n} x^{k-2n}, \end{equation*}$

where

$\binom{m}{n} = \frac{m!}{n!(m-n)!},$

and $\lfloor{k}\rfloor$ denotes the integer floor function. Equation (2.1) generates a sequence for $x = 1$ of the Fibonacci numbers.

2.2. Lucas polynomials

The Lucas polynomials can be defined as follows ^[52]:

$\begin{equation} L_k(x) = kL_{k-1}(x)+L_{k-2}(x), \; \; k\; \geq 2, \end{equation}$

(2.2)

with

$L_0(x) = 2\; \; \text{and}\; \; L_1(x) = x.$

Their explicit form is

$\begin{equation*} \label{S3} L_{k}(x) = \sum\limits_{n = 0}^{\lfloor \frac{k}{2}\rfloor}\frac{k}{k-n}\binom{k-n}{n} x^{k-2n}, \end{equation*}$

when $x = 1$ . Equation (2.2) generates a sequence of the Lucas numbers.

2.3. Function approximation

Let $u(x)$ be a continuous function that may be approximated in terms of the Lucas series in the following manner:

$\begin{equation} u(x) = \sum\limits_{k = 0}^{M}\lambda_{k}^{n+1} L_k(x), \end{equation}$

(2.3)

where $L(x)$ are the Lucas polynomials and $\lambda_k$ are the unknown coefficients. Utilizing the finite terms of the Lucas series, the $mth$ -order derivative of a function $u(x)$ can be approximated as follows:

$\begin{equation} u^{(m)}(x) = \sum\limits_{k = 0}^{M}\lambda_{k}^{n+1} L^{(m)}_k(x). \end{equation}$

(2.4)

Utilizing the differentiation matrix ${\bf D}$ and Fibonacci polynomials, $L^{(m)}_k(x)$ can be approximated ^[52] as follows:

$\begin{equation} L^{(m)}_k(x) = kF_{k}(x)D^{m-1}, \end{equation}$

(2.5)

where

$\begin{equation*} D = \begin{bmatrix} 0 & 0 & \dots & 0 \\ 0 \\ \vdots & \ & d\\ 0 \end{bmatrix}, \end{equation*}$

and $d$ is a $N\times N$ matrix

$\begin{equation} d_{m, n} = \begin{cases} m\sin\frac{(n-m)\pi}{2}, & \text{if}\ n > m , \\ 0, & \text{otherwise}. \end{cases} \end{equation}$

(2.6)

3. Proposed methodology

To describe the proposed hybrid scheme, we consider Eq (1.1)

$\begin{equation} u_{tt}+\alpha \Delta u+ \psi(u) = g(x, t) \end{equation}$

(3.1)

with the following conditions:

$\begin{align} u(x, 0)& = g_1(x), \; \; u_t(x, 0) = g_2(x), \; \; a\leq x \leq b, \end{align}$

(3.2)

$\begin{align} u(a, t)& = f_{1}(t), \; \; u(b, t) = f_2(t), \; \; t > 0. \end{align}$

(3.3)

First, re-arrange Eq (3.1)

$\begin{equation} u_{tt} = g(x, t)-\alpha \Delta u- \psi(u). \end{equation}$

(3.4)

Now, using Störmer's method for the time discretization of Eq (3.4), we have the following:

$\begin{equation} u^{n+1} = 2u^{n}-u^{n-1}+{dt}^2(g(x, t)-\alpha \Delta u^{n}- \psi(u)^{n}), \end{equation}$

(3.5)

where

$u^{n+1} = u(t^{n+1}, x),$

$dt$ is time stepsize for $t\in[0, T]$ .

Then, let us discretize the spacial domain $[a, b]$ into the $M+1$ number of the nodes $x_i, \; i = 0, 1, 2, \ldots, M$ , where $x_0, \; x_M$ are the boundary points while the remaining (i.e., $x_i, \; i = 1, 2, \ldots, M-1$ ) are interior points of the domain, which are computed as follows:

$x_i = a+i*h,$

where

$h = (b-a)/M$

is spatial step size. Here, we approximate the unknown function $u(x)$ by Lucas polynomials at the $nth$ time level, which is denoted as $u^{n}(x)$ :

$\begin{equation} u^{n}(x) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}(x) = {\bf L}^{T}(x){\bf C}, \end{equation}$

(3.6)

where ${\bf C}$ is the vector of unknown coefficients dependent on time, and ${\bf L}$ is the $(M+1)\times 1$ vector of Lucas polynomials, such that

$\begin{equation} {\bf L}(x) = [L_{0}(x), L_{1}(x), ..., L_{M}(x)]^{T}. \end{equation}$

(3.7)

Now, for the collocation points $x_{i}$ ,

$\begin{equation} u^{n}(x_i) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}(x_i) = {\bf L}(x_i){\bf C}. \end{equation}$

(3.8)

Let

$u^{n}(x_i) = u_{i};$

then, in the matrix notation,

$\begin{equation} {\bf U} = {\bf AC}, \end{equation}$

(3.9)

where

${\bf U} = [u_{0}, u_{1}, ..., u_{M}]^{T}, \; \; \; {\bf C} = [c_{0}, c_{1}, ..., c_{M}]^{T}$

and

$\begin{align} {\bf A} & = \begin{bmatrix} L_{0}(x) \\ L_{1}(x) \\ \vdots \\ L_{M}(x) \end{bmatrix} = \begin{bmatrix} L_{0}(x_{0}) & L_{0}(x_{1}) & \cdots & L_{0}(x_{M}) \\ L_{1}(x_{0}) & L_{1}(x_{1}) & \cdots & L_{1}(x_{M}) \\ \vdots & \vdots & \ddots & \vdots \\ L_{M}(x_{0}) & L_{M}(x_{1}) & \cdots & L_{M}(x_{M}) \end{bmatrix}. \end{align}$

(3.10)

Similarly, the $kth$ order derivative of unknown functions are approximated by the $kth$ order derivatives of Lucas polynomials

$\begin{equation} u^{(k)}(x_i) = \sum\limits_{j = 0}^{M} c_{j}^{n}(t)L_{j}^{(k)}(x_i) = {\bf L}^{(k)}(x_i){\bf C}. \end{equation}$

(3.11)

In the matrix form,

$\begin{equation} {\bf U}^{(k)} = {\bf BC}, \end{equation}$

(3.12)

where

${\bf U}^{(k)} = [u_{0}^{(k)}, u_{1}^{(k)}, ..., u_{M}^{(k)}]^{T}, \; \; \; {\bf C} = [c_{0}, c_{1}, ..., c_{M}]^{T}$

and

$\begin{align} {\bf B} & = \begin{bmatrix} L_{0}^{(k)}(x) \\ L_{1}^{(k)}(x) \\ \vdots \\ L_{M}^{(k)}(x) \end{bmatrix} = \begin{bmatrix} L_{0}^{(k)}(x_{0}) & L_{0}^{(k)}(x_{1}) & \cdots & L_{0}^{(k)}(x_{M}) \\ L_{1}^{(k)}(x_{0}) & L_{1}^{(k)}(x_{1}) & \cdots & L_{1}^{(k)}(x_{M}) \\ \vdots & \vdots & \ddots & \vdots \\ L_{M}^{(k)}(x_{0}) & L_{M}^{(k)}(x_{1}) & \cdots & L_{M}^{(k)}(x_{M}) \end{bmatrix}. \end{align}$

(3.13)

Using the above approximation in Eq (3.5), we have the following:

$\begin{equation} {{\bf u}^{n+1} = 2{\bf u}^{n}-{\bf u}^{n-1}+{dt}^2\left({\bf g}^{n}-\alpha {\bf D}{\bf u}^{n}- \psi({\bf u}^{n})\right)}, \end{equation}$

(3.14)

where

${\bf g}^{n} = g(x_i, t^{n})\ \; (i = 0, 1, \cdots, M)$

and

${\bf D} = {\bf B}{\bf A}^{-1}.$

For $n$ = 1, ${\bf u}^{1}$ and ${\bf u}^{0}$ can be obtained from the initial conditions. To satisfy the boundary conditions we collocated the boundary conditions that corresponded to each of the following:

$x_i \in \Gamma = \{a, b\}.$

That is, for

$\begin{equation*} u(x_i, t) = f(x_i, t), \; \forall x_i \in \Gamma, \end{equation*}$

we then update the solution vector at boundary points to have

$\begin{equation*} {\bf u}^{n+1}_{\Gamma} = {\bf f}(t^{n+1}). \end{equation*}$

Remark 3.1. The stability condition for the second order Störmer method is as follows:

$\begin{equation*} \alpha \left(\frac{dt}{dx} \right)\leq 1, \end{equation*}$

where $\alpha$ is the coefficient of the Laplace operator. Then the method is then said to be stable, by the rule of thumb, if all the eigenvalues, $\lambda$ , of the discretized operator,

${\bf D} = {\bf B}{\bf A}^{-1},$

scaled by ${dt}^2$ lies within the stability region of the time-stepping method ^[53], i.e.,

$\begin{equation*} -4\leq \Re(\lambda {dt}^2 )\leq 0. \end{equation*}$

4. Numerical results and discussion

This section demonstrates the numerical solution of one dimension Klien-Gordon and two dimension sinh-Gordon equations utilizing the proposed scheme. All the computations are performed using MATLAB (R2012a) on Dell PC Laptop with an Intel(R) Core(TM)i5-2450M CPU 2.50 GHz 2.50 GHz 8 GB RAM. To check the accuracy and convergence, we used the following error norms:

$\begin{align*} L_{\infty}& = max{\mid E_{k}-u_{k}}\mid_{k = 1}^{M+1}, \\ L_{2}& = \left(dx{\sum\limits_{k = 1}^{M+1} \left(E_{k}-u_{k}\right)^{2}}\right)^{1/2}, \\ L_{rms}& = \left(1/M{\sum\limits_{k = 1}^{M+1} \left(E_{k}-u_{k}\right)^{2}}\right)^{1/2}, \\ C-rate& = \frac{L_{\infty}({dt}_{\ell})}{L_{\infty}({dt}_{\ell+1})}, \end{align*}$

where $E$ is the exact solution.

4.1. (1+1)-dimensional case

Test Problem 4.1. Consider Eq (1.2) with $\alpha = \beta = -1$ , $\gamma = 0$ , $g(x, t) = 0$ ; we have

$\begin{align} u_{tt}-\Delta u-u = 0, \end{align}$

(4.1)

with the following initial and boundary conditions:

$\begin{align*} u(x, 0)& = 1+\sin(x), \ \ u_{t}(x, 0) = 0, \ \ {x \in [0, 1]}, \\ u(0, t) & = \sin(0)+\cosh(t), \ u(1, t) = \sin(1)+\cosh(t), \ t > 0, \end{align*}$

and the actual solution is as follows:

$\begin{align*} u(x, t) = sin(x)+cosh(t). \end{align*}$

Using the proposed methodology for this particular equation the iterative scheme (3.14) becomes the following:

$\begin{equation} {\bf u}^{n+1} = 2{\bf u}^{n}-{\bf u}^{n-1}+{dt}^2({\bf g}^{n}+\mathit{\boldsymbol{D}}{\bf u}^{n}+{\bf u}^{n}). \end{equation}$

(4.2)

The above equation is used to generate numerical results, which are provided in in terms of the $L_{\infty}$ and $L_2$ error norms. These findings are obtained using nodal points $M = 10$ and various values of the time step size $dt$ while preserving the final time $T = 0.1$ , and for various values of $T$ and fixing $dt = 0.0001$ . The accuracy increases as the value of $dt$ decreases, as shown in this table. Based on the comparison with ^[29], it is obvious that the suggested scheme is more accurate than those reported in the recent literature. The numerical results in terms of $L_{\infty}$ , $L_2$ , and $L_{rms}$ , as well as central processing unit (CPU) times in seconds, are computed using the suggested approach for various values of $M$ in Table 2. The table illustrates that as the number of collocation points rises, the current approach provides an improved accuracy. Furthermore, the proposed technique may be seen to be quite efficient, whereas the comparison between the exact and approximate solutions are visualized in Figure 1.

Table 1. Results of the proposed method for Test Problem 4.1 with

$M = 10$ and

$T = 0.1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	5.65E-04	4.92E-04	5.06E-04	4.46E-04	1.36E-03	1.26E-03
0.001	5.50E-05	4.86E-05	5.55E-05	4.74E-05	5.08E-05	4.43E-05
0.0005	2.75E-05	2.43E-05	4.04E-05	2.61E-05	2.57E-05	2.21E-05
0.0001	5.49E-06	4.85E-06	2.83E-05	1.12E-05	6.80E-06	5.05E-06
T	$L_\infty$	$L_2$	$L_\infty$		$L_\infty$
0.01	5.00E-07	4.78E-07	7.57E-07	-	5.35E-07	-
0.02	1.00E-06	9.49E-07	2.02E-06	-	1.13E-06	-
0.1	5.49E-06	4.85E-06	2.83E-05	-	6.80E-06	-
0.5	4.19E-05	3.42E-05	8.61E-05	-	4.63E-05	-
1	4.37E-04	2.14E-04	3.95E-05	-	4.20E-04	-

| Show Table

DownLoad: CSV

Table 2. Results of the proposed method for Test Problem 4.1 with

$dt = 0.0001$ and

$T = 0.1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	5.03E-06	4.45E-06	3.15E-07	0.037078
7	4.32E-06	3.73E-06	2.95E-07	0.045391
10	2.49E-06	1.85E-06	1.85E-07	0.051433
15	9.03E-07	8.56E-07	9.81E-08	0.071921
18	7.12E-07	6.42E-07	7.88E-08	0.109775
20	5.31E-07	5.13E-07	6.36E-08	0.155763
22	2.90E-07	1.65E-07	3.38E-08	0.330819
25	2.35E-07	1.09E-07	1.47E-08	1.366762

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Figure 1. Comparison of exact and approximate solutions for Test Problem 4.1.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.2. Now assume

$\alpha = -1, \; \beta = -2, \; \gamma = 0$

and

$g(x, t) = -2\sin{x} \sin{t}$

in Eq (1.2) with the following conditions:

$\begin{align*} u(x, 0)& = 0, \quad u_{t}(x, 0) = \sin{x}, \quad {x \in [0, \pi/2], }\\ u(0, t)& = 0, \quad u(\pi/2, t) = \sin t, \quad t > 0. \end{align*}$

The exact solution of Eq (1.2) for the above parameter is given in ^[29], which is

$u(x, t) = \sin{x}\sin{t}.$

In Test Problem 4.2, the numerical results are computed utilizing the proposed technique. These results are shown in Table 3 using nodal points

$M = 10$

and various values of $dt$ while preserving the final time

$T = 0.1,$

and for various values of $T$ and fixing

$dt = 0.0001.$

In this situation, as indicated in the table, the method's accuracy improves as the value of $dt$ declines. Furthermore, in comparison with ^[29], we came to know that the current method is more accurate than other articles published in the recent literature. Similarly to the previous test problem, the numerical results are computed in the form of $L_\infty$ , $L_2$ , and $L_{rms}$ , as well as the CPU times in seconds, using the suggested approach for various values of $M$ in Table 4. As the number of collocation points increases, the current approach efficiently improves the accuracy, as seen in the table whereas profiles of the exact and approximate solutions are given in Figure 2.

Table 3. Results of the proposed method for Test Problem 4.2 with

$M = 10$ and

$T = 0.1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	1.59E-06	1.23E-03	2.50E-03	1.04E-03	2.44E-03	9.86E-04
0.001	1.59E-08	1.23E-04	1.53E-04	6.58E-05	5.05E-04	2.09E-04
0.0005	3.97E-09	6.17E-05	7.65E-05	3.24E-05	3.87E-04	1.63E-04
0.0001	1.57E-10	1.23E-05	1.50E-05	2.59E-05	2.92E-04	1.26E-04
T	$L_\infty$	$L_2$	$L_\infty$		$L_\infty$
0.01	1.65E-11	3.91E-06	1.58E-07	-	5.45E-07	-
0.02	3.29E-11	5.53E-06	6.29E-07	-	3.32E-06	-
0.1	1.57E-10	1.23E-05	1.50E-05	-	2.92E-04	-
0.5	6.47E-10	2.27E-05	8.84E-05	-	1.07E-02	-
1	1.16E-09	1.39E-05	5.20E-05	-	2.34E-02	-

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Table 4. Results of the proposed method for Test Problem 4.2 with

$dt = 0.0001$ and

$T = 1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	4.23E-04	3.32E-03	1.87E-04	0.051816
7	2.99E-05	2.90E-04	3.65E-05	0.051183
10	9.76E-06	8.90E-05	7.09E-06	0.082319
15	6.51E-06	2.40E-05	2.34E-06	0.09778
18	1.64E-06	2.73E-05	6.92E-07	0.143512
20	8.01E-07	8.93E-06	1.01E-07	0.184772
22	9.55E-08	5.80E-06	8.68E-08	0.338581
25	7.12E-08	7.42E-07	7.70E-08	1.288726

| Show Table

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Figure 2. Exact versus approximate solution for Test Problem 4.2.

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Test Problem 4.3. Consider Eq (1.2) with the exact solution $u(x, t) = x^3t^3$ , when

$\alpha = -1, \; \beta = 0, \; \gamma = 1$

and

$g(x, t) = 6xt(x^2-t^2)+x^6t^6,$

with associated conditions

$\begin{align*} & u(x, 0) = 0, \quad u_{t}(x, 0) = 0, \quad {x \in [0, 1]}, \\ & u(0, t) = 0, \quad u(1, t) = 0, \quad t > 0. \end{align*}$

and show the numerical results for Test Problem 4.3 for various values of $dt$ and $M$ , respectively. We compared our results with the methods in ^[29] for

$M = 10\; \; \mathit{\text{and}}\; \; T = 1$

in . We can see from this table that the method's accuracy improves as the number of iterations increases, with a very good accuracy of $10^{-8}$ for

$dt = 0.0001.$

In comparison to the method in ^[29], the proposed method produces more accurate results. In , the results are computed in terms of $L_{\infty}$ , $L_2$ , and $L_{rms}$ for

$dt = 0.001$

and time $T = 1$ , as well as the CPU time in seconds. The suggested procedure is accurate and efficient, as shown in this table. The profile of the exact and approximate solutions are shown in Figures 3 and 4, respectively.

Table 5. Results of the proposed method for Test Problem 4.3 with

$M = 10$ and

$T = 1$ .

	Present method		MQ-RBF ^[29]		GA-RBF ^[29]
dt	$L_\infty$	$L_2$	$L_\infty$	$L_2$	$L_\infty$	$L_2$
0.01	3.35E-04	1.78E-04	2.24E-02	9.59E-03	4.66E-02	1.96E-02
0.001	3.24E-06	1.64E-06	2.46E-03	1.05E-03	2.27E-03	9.71E-04
0.0005	8.08E-07	4.09E-07	1.36E-03	5.79E-04	1.17E-03	5.00E-04
0.0001	3.17E-08	1.66E-08	4.85E-04	2.07E-04	2.94E-04	1.25E-04

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Table 6. Results of the proposed method for Test Problem 4.3 with

$T = 1$ and

$dt = 0.001$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	2.19E-04	1.37E-04	9.65E-05	0.050013
10	3.24E-05	6.36E-06	1.64E-06	0.102678
15	4.84E-06	4.88E-06	6.82E-07	0.088209
20	1.07E-06	1.3606E-06	1.92E-07	0.185567
22	8.11E-07	5.05E-07	7.13E-08	0.370034
25	6.08E-07	2.72E-07	4.21E-08	1.372918

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Figure 3. Comparison of exact and approximate solutions for Test Problem 4.3.

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Figure 4. Exact versus approximate solution for Test Problem 4.3.

DownLoad: Full-Size Img PowerPoint

Test Problem 4.4. Consider

$\alpha = -1, \ \beta = -1, \ \gamma = 0$

and

$g(x, t) = -\cos x \sin t$

in Eq (1.2) with the following conditions:

$\begin{align*} u(x, 0)& = 0, \quad u_{t}(x, 0) = \cos x, \quad {x \in [0, 1], }\\ u(0, t)& = \sin t, \quad u(1, t) = \cos 1 \sin t, \quad t > 0. \end{align*}$

The exact solution of Eq (1.2) for the above parameter given as follows:

$u(x, t) = \cos x \sin t.$

The solution is computed for the fixed point $T = 0.1$ and $M = 20,$ while decreasing the time step. The results are reported in , which shows that the accuracy improves as the number of iteration increases. In , the results are recorded for various numbers of nodal points $M$ , and notice that as number of collocation point increases, the numerical convergence accuracy improves. Along with, maximum error, the CPU time in seconds is also reported in the tables. The solution profiles of the exact and approximate solutions are illustrated in Figure 5, which shows that both the solutions are well matched.

Table 7. Results of the proposed method for Test Problem 4.4 with

$M = 15$ and

$T = 0.1$ .

dt	$L_\infty$	$L_2$	$L_{rms}$	CPU
0.01	4.28E-05	3.55E-05	4.79E-05	2.03E-01
0.001	4.88E-06	4.05E-06	1.80E-06	2.04E-01
0.0005	2.46E-06	2.04E-06	6.45E-07	1.77E-01
0.0001	4.97E-07	4.23E-07	5.98E-08	1.92E-01

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Table 8. Results of the proposed method for Test Problem 4.4 with

$dt = 0.0001$ and

$T = 0.1$ .

M	$L_\infty$	$L_2$	$L_{rms}$	CPU
5	4.97E-07	4.88E-07	3.74E-08	0.04553
7	4.80E-07	3.94E-07	3.53E-08	0.048321
10	4.90E-07	3.76E-07	3.40E-08	0.063045
15	4.64E-07	3.66E-07	3.39E-08	0.081496
18	4.54E-07	3.68E-07	3.36E-08	0.115411
20	4.40E-07	3.43E-07	3.33E-08	0.167235
22	4.36E-07	3.36E-07	3.32E-08	0.372988
25	4.24E-07	3.18E-07	2.82E-08	1.378855

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Figure 5. Exact versus approximate solution for Test Problem 4.4.

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Test Problem 4.5. (1+2)-dimensional case: Finally, consider the two dimensional sinh-Gordon equation

$\begin{equation} u_{tt}-\Delta u+\sinh(u) = g(x, y, t) \end{equation}$

(4.3)

with initial conditions

$\begin{align*} & u(x, y, 0) = u_{t}(x, y, 0) = \sin(x+y), \quad {(x, y) \in [0, 1]^2, } \end{align*}$

and boundary conditions

$\begin{align*} & u(0, y, t) = e^t\sin(0+y), \quad u(1, y, t) = e^t\sin(1+y), \quad t > 0, \\ & u(x, 0, t) = e^t\sin(x+0), \quad u(x, 1, t) = e^t\sin(x+1). \end{align*}$

The exact solution

$u(x, y, t) = e^t\sin(x+y)$

is considered, as given in ^[47]. Additionally the source term is as follows:

$\begin{equation} g(x, y, t) = 3e^t\sin(x+y)+\sinh(e^t\sin(x+y)). \end{equation}$

(4.4)

The results of the suggested method for Test Problem 4.5 are shown in for various values of $dt$ with $M = 4$ and $T = 1$ . This table shows that the method delivers a good accuracy for very coarse nodes, and that the method's accuracy improves to some extent as the number of iterations increases. Additionally in this table, we also compared the computed results to the approach in ^[47]. The results of the proposed approach in contrast to the method presented in ^[47] are shown in for various values of $M$ for the same test problem. This comparison reveals that the proposed approach is more accurate than ^[47]. The approximate solution of the suggested approach is compared to the exact solution in Figure 6. Although the absolute errors are indicated in Figure 7, the numerical solution is in a good agreement with the exact solution, as demonstrated in Figure 6.

Table 9. Results of the proposed method for Test Problem 4.5 with

$M = 4$ and

$T = 1$ .

	Present method			^[47]
dt	$L_\infty$	$L_2$	C-rate( $L_\infty$ )	$L_\infty$	$L_2$
0.100	1.91E-02	8.66E-03	-	2.20E-02	4.19E-02
0.050	9.19E-03	3.93E-03	2.08E+00	1.15E-02	1.47E-02
0.025	4.52E-03	1.93E-03	2.03E+00	7.53E-03	1.23E-02
0.013	2.24E-03	9.51E-04	2.02E+00	4.88E-03	9.57E-03
0.006	1.11E-03	4.68E-04	2.02E+00	2.90E-03	6.04E-03
0.003	5.45E-04	2.27E-04	2.04E+00	1.60E-03	3.40E-03
0.002	2.63E-04	1.08E-04	2.07E+00	8.49E-04	1.80E-03
0.001	1.22E-04	5.13E-05	2.15E+00	4.43E-04	9.30E-04

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Table 10. Results of the proposed method for Test Problem 4.5 with

$dt = 0.001$ and

$T = 1$ .

	Present method		^[47]
M	$L_\infty$	$L_2$	$L_\infty$	$L_2$
2	5.85E-03	2.93E-03	5.53E-03	5.53E-03
3	2.22E-03	1.47E-03	1.22E-03	1.74E-03
4	1.62E-04	6.66E-05	5.59E-04	1.18E-03
5	1.63E-04	6.75E-05	5.38E-04	9.99E-04
6	1.23E-04	6.37E-05	-	-
7	1.32E-04	7.41E-05	-	-

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Figure 6. Exact versus approximate solution at

$M = 15$ ,

$T = 1$ , for Test Problem 4.5.

DownLoad: Full-Size Img PowerPoint

Figure 7. Error for Test Problem 4.5.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

A numerical method based on Fibonacci polynomials combined with Störmer's method was developed to solve the nonlinear Klein/Sinh-Gordon equation. On a variety of test cases, numerical evaluation exhibited an improved performance of the proposed scheme against the available results in the literature. As evidenced by the comparison, the proposed technique is simple to execute and faster in convergence. Furthermore, it was observed that by reducing the spatial and time steps size, the errors were reduced. With slight modifications, the proposed method can be applied to a variety of PDEs. This modification is mainly dictated by the problem at hand with the associated initial and boundary conditions that are dictated by the particular problem of interest. As such, we are interested in applying the proposed method to time-dependent Neumann and/or mixed boundary value problems with possible discontinuous initial condition(s). The limitations, as is the case in spectral methods, was the high-condition numbers of the collocated/system matrices. However, this can be alleviated by the state-of-the-art pre-conditioned algorithms developed for spectral methods. In the future, the proposed method can be expanded for applications to complex fractal-fractional problems.

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Wang CH, Hsieh YH, Powers ZM, et al. (2020) Defeating antibiotic-resistant bacteria: exploring alternative therapies for a post-antibiotic era. Int J Mol Sci 21: 1061. doi: 10.3390/ijms21031061
[2]	Draenert R, Seybold U, Grutzner E, et al. (2015) Novel antibiotics: are we still in the pre-post-antibiotic era? Infection 43: 145-151. doi: 10.1007/s15010-015-0749-y
[3]	Pandit R, Awal B, Shrestha SS, et al. (2020) Extended-Spectrum beta-Lactamase (ESBL) genotypes among multidrug-resistant uropathogenic Escherichia coli clinical isolates from a teaching hospital of Nepal. Interdiscip Perspect Infect Dis 2020: 6525826. doi: 10.1155/2020/6525826
[4]	Aires-de-Sousa M, Lopes E, Goncalves ML, et al. (2020) Intestinal carriage of extended-spectrum beta-lactamase-producing Enterobacteriaceae at admission in a Portuguese hospital. Eur J Clin Microbiol Infect Dis 39: 783-790. doi: 10.1007/s10096-019-03798-3
[5]	Goncalves D, Cecilio P, Ferreira H (2016) Nursing homes and long-term care facilities: Reservoirs of CTX-M-15-producing Escherichia coli O25b-ST131 in Portugal. J Glob Antimicrob Resist 7: 69-71. doi: 10.1016/j.jgar.2016.08.001
[6]	Valverde A, Coque TM, Sanchez-Moreno MP, et al. (2004) Dramatic increase in prevalence of fecal carriage of extended-spectrum beta-lactamase-producing Enterobacteriaceae during nonoutbreak situations in Spain. J Clin Microbiol 42: 4769-4775. doi: 10.1128/JCM.42.10.4769-4775.2004
[7]	Baljin B, Baldan G, Chimeddorj B, et al. (2016) Faecal carriage of gram-negative multidrug-resistant bacteria among patients hospitalized in two centres in Ulaanbaatar, Mongolia. PLoS One 11: e0168146. doi: 10.1371/journal.pone.0168146
[8]	Cornejo-Juarez P, Suarez-Cuenca JA, Volkow-Fernandez P, et al. (2016) Fecal ESBL Escherichia coli carriage as a risk factor for bacteremia in patients with hematological malignancies. Support Care Cancer 24: 253-259. doi: 10.1007/s00520-015-2772-z
[9]	(2018) Clinical and Laboratory Standards InstitutePerformance Standards for Antimicrobial Susceptibility Testing, 28th ed. CLSI supplement M100. Wayne, PA: Clinical and Laboratory Standards Institute.
[10]	Japoni A, Alborzi A, Kalani M, et al. (2006) Susceptibility patterns and cross-resistance of antibiotics against Pseudomonas aeruginosa isolated from burn patients in the South of Iran. Burns 32: 343-347. doi: 10.1016/j.burns.2005.10.017
[11]	Yang JL, Wang MS, Cheng AC, et al. (2008) A simple and rapid method for extracting bacterial DNA from intestinal microflora for ERIC-PCR detection. World J Gastroenterol 14: 2872-2876. doi: 10.3748/wjg.14.2872
[12]	Paterson DL, Hujer KM, Hujer AM, et al. (2003) Extended-spectrum beta-lactamases in Klebsiella pneumoniae bloodstream isolates from seven countries: dominance and widespread prevalence of SHV- and CTX-M-type beta-lactamases. Antimicrob Agents Chemother 47: 3554-3560. doi: 10.1128/AAC.47.11.3554-3560.2003
[13]	Grobner S, Linke D, Schutz W, et al. (2009) Emergence of carbapenem-non-susceptible extended-spectrum beta-lactamase-producing Klebsiella pneumoniae isolates at the university hospital of Tubingen, Germany. J Med Microbiol 58: 912-922. doi: 10.1099/jmm.0.005850-0
[14]	Grimm V, Ezaki S, Susa M, et al. (2004) Use of DNA microarrays for rapid genotyping of TEM beta-lactamases that confer resistance. J Clin Microbiol 42: 3766-3774. doi: 10.1128/JCM.42.8.3766-3774.2004
[15]	Zhang S, Cahalan MD (2007) Purifying plasmid DNA from bacterial colonies using the QIAGEN Miniprep Kit. J Vis Exp 6: 247.
[16]	Pitout JD, Laupland KB (2008) Extended-spectrum beta-lactamase-producing Enterobacteriaceae: an emerging public-health concern. Lancet Infect Dis 8: 159-166. doi: 10.1016/S1473-3099(08)70041-0
[17]	Freeman JT, McBride SJ, Nisbet MS, et al. (2012) Bloodstream infection with extended-spectrum beta-lactamase-producing Enterobacteriaceae at a tertiary care hospital in New Zealand: risk factors and outcomes. Int J Infect Dis 16: e371-374. doi: 10.1016/j.ijid.2012.01.008
[18]	Wielders CCH, Van Duijkeren E, Van Den Bunt G, et al. (2020) Seasonality in carriage of extended-spectrum beta-lactamase-producing Escherichia coli and Klebsiella pneumoniae in the general population: a pooled analysis of nationwide cross-sectional studies. Epidemiol Infect 148: e68. doi: 10.1017/S0950268820000539
[19]	Magiorakos AP, Srinivasan A, Carey RB, et al. (2012) Multidrug-resistant, extensively drug-resistant and pandrug-resistant bacteria: an international expert proposal for interim standard definitions for acquired resistance. Clin Microbiol Infect 18: 268-281. doi: 10.1111/j.1469-0691.2011.03570.x
[20]	Huang IF, Lee WY, Wang JL, et al. (2018) Fecal carriage of multidrug-resistant Escherichia coli by community children in southern Taiwan. BMC Gastroenterol 18: 86. doi: 10.1186/s12876-018-0807-x
[21]	Islam MA, Amin MB, Roy S, et al. (2019) Fecal colonization with multidrug-resistant E. coli among healthy infants in Rural Bangladesh. Front Microbiol 10: 640. doi: 10.3389/fmicb.2019.00640
[22]	Hagel S, Makarewicz O, Hartung A, et al. (2019) ESBL colonization and acquisition in a hospital population: The molecular epidemiology and transmission of resistance genes. PLoS One 14: e0208505. doi: 10.1371/journal.pone.0208505
[23]	Janvier F, Merens A, Delaune D, et al. (2011) Fecal carriage of third-generation cephalosporins-resistant Enterobacteriaceae in asymptomatic young adults: evolution between 1999 and 2009. Pathol Biol (Paris) 59: 97-101. doi: 10.1016/j.patbio.2010.07.012
[24]	Abdallah HM, Alnaiemi N, Reuland EA, et al. (2017) Fecal carriage of extended-spectrum beta-lactamase- and carbapenemase-producing Enterobacteriaceae in Egyptian patients with community-onset gastrointestinal complaints: a hospital -based cross-sectional study. Antimicrob Resist Infect Control 6: 62. doi: 10.1186/s13756-017-0219-7
[25]	Sunde M, Simonsen GS, Slettemeas JS, et al. (2015) Integron, plasmid and host strain characteristics of Escherichia coli from humans and food included in the Norwegian Antimicrobial Resistance Monitoring Programs. PLoS One 10: e0128797. doi: 10.1371/journal.pone.0128797
[26]	Sharma J, Ray P, Sharma M (2010) Plasmid profile of ESBL producing Gram-negative bacteria and correlation with susceptibility to beta-lactam drugs. Indian J Pathol and Microbiol 53: 83-86. doi: 10.4103/0377-4929.59190
[27]	Wu PC, Wang JL, Hsueh PR, et al. (2019) Prevalence and risk factors for colonization by extended-spectrum beta-lactamase-producing or ST 131 Escherichia coli among asymptomatic adults in community settings in Southern Taiwan. Infect Drug Resist 12: 1063-1071. doi: 10.2147/IDR.S201086

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Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University

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Abstract

1. Introduction

2. Fundamental concepts

2.1. Fibonacci polynomials

2.2. Lucas polynomials

2.3. Function approximation

3. Proposed methodology

4. Numerical results and discussion

4.1. (1+1)-dimensional case

5. Conclusions

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Fecal carriage of ESBL-producing Escherichia coli in Egyptian patients admitted to the Medical Research Institute hospital, Alexandria University

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Abstract

1. Introduction

2. Fundamental concepts

2.1. Fibonacci polynomials

2.2. Lucas polynomials

2.3. Function approximation

3. Proposed methodology

4. Numerical results and discussion

4.1. (1+1)-dimensional case

5. Conclusions

Conflict of interest

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