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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+{\text{ESBL}}^{+}) and non-ESBL-producing (ESBL−{\text{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 R0E−{R}_{0}^{{E}^{-}} and R0E+{R}_{0}^{{E}^{+}}values for ESBL−{\text{ESBL}}^{-} and ESBL+ {\text{ ESBL}}^{+}\text{ } E. coli infections were calculated, respectively. According to the model and the data used within the study, it was calculated that R0E−{R}_{0}^{{E}^{-}}to be 1,266,403 and R0E+{R}_{0}^{{E}^{+}}to be 2,096,747. Since the values for R0E−{R}_{0}^{{E}^{-}} and R0E+{R}_{0}^{{E}^{+}} were equal or greater than 1, this suggests that currently the ESBL−\hspace{.25em}{\text{ESBL}}^{-} and ESBL+ {\text{ ESBL}}^{+}\text{ } 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−{\text{ESBL}}^{-} and ESBL+{\text{ ESBL}}^{+} E. coli infection trends will equalize. After 2042, the ESBL− {\text{ESBL}}^{-}\text{ } E. coli infections will indicate a descending pattern whereas ESBL+ {\text{ ESBL}}^{+}\text{ } E. coli infection will increase constantly.

The global population has been and will continue to be severely impacted by the COVID-19 epidemic. The primary objective of this research is to demonstrate the future impact of COVID-19 on those who suffer from other fatal conditions such as cancer, heart disease, and diabetes. Here, using ordinary differential equations (ODEs), two mathematical models are developed to explain the association between COVID-19 and cancer and between COVID-19 and diabetes and heart disease. After that, we highlight the stability assessments that can be applied to these models. Sensitivity analysis is used to examine how changes in certain factors impact different aspects of disease. The sensitivity analysis showed that many people are still nervous about seeing a doctor due to COVID-19, which could result in a dramatic increase in the diagnosis of various ailments in the years to come. The correlation between diabetes and cardiovascular illness is also illustrated graphically. The effects of smoking and obesity are also found to be significant in disease compartments.Model fitting is also provided for interpreting the relationship between real data and the results of thiswork.Diabetic people, in particular, need tomonitor their health conditions closely and practice heart health maintenance. People with heart diseases should undergo regular checks so that they can protect themselves from diabetes and take some precautions including suitable diets. The main purpose of this study is to emphasize the importance of regular checks, to warn people about the effects of COVID-19 (including avoiding healthcare centers and doctors because of the spread of infectious diseases) and to indicate the importance of family history of cancer, heart diseases and diabetes. The provision of the recommendations requires an increase in public consciousness. [ABSTRACT FROM AUTHOR]

This paper presents the generalized Pell p-numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell p-numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys. [ABSTRACT FROM AUTHOR]

This research paper focused on the solution of systems of fractional integro-differential equations (FIDEs) of the Volterra type with variable coefficients. The proposed approach combined the tau method and shifted Gegenbauer polynomials in a matrix form. The investigation of the existence and uniqueness of solutions for these systems was carried out using Krasnoselskii's fixed point theorem. The equations employed Caputo-style derivative operators, and to minimize computational operations involving derivatives and multiplications, integral and product operational matrices were derived. By introducing suitable polynomial approximations and employing the tau spectral method, the original system of FIDE was transformed into an algebraic system. Solving this algebraic system provided approximate solutions to the main system. Error bounds were computed in the Gegenbauer- weighted Sobolev space. The proposed algorithm was implemented and tested on two systems of integro-fractional differential equations to demonstrate its efficiency and simplicity. By varying the parameter a in the Gegenbauer polynomials, the impact of this variation on the approximate solutions can be observed. A comparison with another method utilizing the block-by-block approach was also presented. [ABSTRACT FROM AUTHOR]