Dynamical behaviours and stability analysis of a generalized fractional model with a real case study.

INTRODUCTION: Mathematical modelling is a rapidly expanding field that offers new and interesting opportunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly. OBJECTIVES: The main objective of this research is to explore an efficient mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework. METHODS: The new model in this paper is formulated in the Caputo sense, employs a nonlinear time-varying transmission rate, and consists of ten population classes including susceptible, infected, diagnosed, ailing, recognized, infected real, threatened, diagnosed recovered, healed, and extinct people. The existence of a unique solution is explored for the new model, and the associated dynamical behaviours are discussed in terms of equilibrium points, invariant region, local and global stability, and basic reproduction number. To implement the proposed model numerically, an efficient approximation scheme is employed by the combination of Laplace transform and a successive substitution approach; besides, the corresponding convergence analysis is also investigated. RESULTS: Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic in Italy. By using these comparisons between the simulated and measured data, we find the best value of the fractional order with minimum absolute and relative errors. Also, the impact of different parameters on the spread of viral infection is analyzed and studied. CONCLUSION: According to the comparative results with real data, we justify the use of fractional concepts in the mathematical modelling, for the new non-integer formalism simulates the reality more precisely than the classical framework.

Odd-order differential equations with deviating arguments: asymptomatic behavior and oscillation.

Despite the growing interest in studying the oscillatory behavior of delay differential equations of even-order, odd-order equations have received less attention. In this work, we are interested in studying the oscillatory behavior of two classes of odd-order equations with deviating arguments. We get more than one criterion to check the oscillation in different methods. Our results are an extension and complement to some results published in the literature.

Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior.

In this work, by using both the comparison technique with first-order differential inequalities and the Riccati transformation, we extend this development to a class of third-order neutral differential equations of the mixed type. We present new criteria for oscillation of all solutions, which improve and extend some existing ones in the literature. In addition, we provide an example to illustrate our results.

Mathematical analysis for the effect of voluntary vaccination on the propagation of Corona virus pandemic.

In this manuscript, a new nonlinear model for the rapidly spreading Corona virus disease (COVID-19) is developed. We incorporate an additional class of vaccinated humans which ascertains the impact of vaccination strategy for susceptible humans. A complete mathematical analysis of this model is conducted to predict the dynamics of Corona virus in the population. The analysis proves the effectiveness of vaccination strategy employed and helps public health services to control or to reduce the burden of corona virus pandemic. We first prove the existence and uniqueness and then boundedness and positivity of solutions. Threshold parameter for the vaccination model is computed analytically. Stability of the proposed model at fixed points is investigated analytically with the help of threshold parameter to examine epidemiological relevance of the pandemic. We apply LaSalle's invariance principle from the theory of Lyapunov function to prove the global stability of both the equilibria. Two well known numerical techniques namely Runge-Kutta method of order 4 (RK4), and the Non-Standard Finite Difference (NSFD) method are employed to solve the system of ODE's and to validate our obtained theoretical results. For different coverage levels of voluntary vaccination, we explored a complete quantitative analysis of the model. To draw our conclusions, the effect of proposed vaccination on threshold parameter is studied numerically. It is claimed that Corona virus disease could be eradicated faster if a human community selfishly adopts mandatory vaccination measures at various coverage levels with proper awareness. Finally, we have executed the joint variability of all classes to understand the effect of vaccination strategy on a disease dynamics.

A hybrid stochastic fractional order Coronavirus (2019-nCov) mathematical model.

In this paper, a new stochastic fractional Coronavirus (2019-nCov) model with modified parameters is presented. The proposed stochastic COVID-19 model describes well the real data of daily confirmed cases in Wuhan. Moreover, a novel fractional order operator is introduced, it is a linear combination of Caputo's fractional derivative and Riemann-Liouville integral. Milstein's higher order method is constructed with the new fractional order operator to study the model problem. The mean square stability of Milstein algorithm is proved. Numerical results and comparative studies are introduced.

A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model.

Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists.

On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach.

In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald-Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.

Nonstandard finite difference method for solving complex-order fractional Burgers' equations.

The aim of this work is to present numerical treatments to a complex order fractional nonlinear one-dimensional problem of Burgers' equations. A new parameter σ t is presented in order to be consistent with the physical model problem. This parameter characterizes the existence of fractional structures in the equations. A relation between the parameter σ t and the time derivative complex order is derived. An unconditionally stable numerical scheme using a kind of weighted average nonstandard finite-difference discretization is presented. Stability analysis of this method is studied. Numerical simulations are given to confirm the reliability of the proposed method.

A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator.

In this paper, we present a new fractional-order mathematical model for a tumor-immune surveillance mechanism. We analyze the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs). An efficient numerical procedure is suggested to solve these FDEs by considering singular and nonsingular derivative operators. An optimal control strategy for investigating the effect of chemotherapy treatment on the proposed fractional model is also provided. Simulation results show that the new presented model based on the fractional operator with Mittag-Leffler kernel represents various asymptomatic behaviors that tracks the real data more accurately than the other fractional- and integer-order models. Numerical simulations also verify the efficiency of the proposed optimal control strategy and show that the growth of the naive tumor cell population is successfully declined.

Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative.

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains.

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.

Certain fractional integral formulas involving the product of generalized Bessel functions.

We apply generalized operators of fractional integration involving Appell's function F 3(·) due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

[Diagnostics and therapy for normal tension glaucoma]. / Diagnostik und Therapie des Normaldruckglaukoms.

Pooled data from "The Baltimore Eye Survey", "The Blue Mountains Eye Study", "The Beaver Dam Eye Study", "The Rotterdam Study" and "The Melbourne VIP" showed a strong age-dependent prevalence of open angle glaucoma. Patients younger than 60 years have a prevalence of open angle glaucoma of less than 1 %. Beginning with the age of 60 the prevalence increases exponentially. The prevalence of open angle glaucoma in persons in the 8(th) age decade reaches up to 5 %. Among these patients, 30-50 % have normal intraocular pressure. The diagnostics and therapy for open angle glaucoma with normal intraocular pressure (IOP), also called normal tension glaucoma, is a complex and often interdisciplinary challenge. Established causative factors for developing a glaucomatous optic nerve atrophy in normal tension glaucoma are: relatively increased IOP, older age, non-dippers (0-10 %) or extreme dippers (> 20 %) concerning nocturnal arterial blood pressure drop, small vessel disease with cardiovascular disease and cerebral microgliosis (white matter lesions), decreased blood flow in the optic nerve head, extreme dip of the optic nerve head blood flow in the morning, cerebral blood flow dysregulation and the epsilon4-allele polymorphism of the apolipoprotein E-gene. Clinical pathways are presented for the diagnostics and therapy for normal tension glaucoma.

[Telemedical-supported screening of retinal vessels ("talking eyes")]. / Telemedizinisch unterstütztes Screening der retinalen Gefässe ("Talking Eyes").

BACKGROUND: Cerebral and retinal vessels behave similarly under the influence of vascular risk factors. Several groups have shown that retinal microvascular abnormalities represent an independent risk factor with regard to strokes and heart attacks. AIM OF THE STUDY: The aim of this study was to perform a prospective screening examination with regard to retinal microvascular abnormalities as well as an extended vessel diagnosis in a subgroup of patients with lower arteriovenous risk values. METHODS: In the course of a prospective cross-sectional study ("Talking Eyes") between 1.9.2001 and 1.8.2002 a telemedical-supported screening of the retina (study I) was carried out in 7,163 subjects. The patients were selected without any inclusion or exclusion criteria. The mean age was 48.2 +/- 8 years (18 - 83 years) with a sex distribution of 39.2 % females to 60.8 % males. Digital fundus photos of the right and left eyes were taken for all patients. The pictures were taken without pupil dilation using a CANON-NM camera. The pictures and case histories were stored in a central server using web-based software (MedStage, Siemens). In a central reading centre, the arteriovenous ratio of both eyes was determined telemedically using the Parr-Hubbard formula and the retinas subjected to a standardised examination by an ophthalmologist. The retinal risk factor was calculated on the basis of the arteriovenous ration, the presence of microvascular abnormalities and the case history. The reproducibility of measurement of the arteriovenous ratio (Kronbach alpha coefficient) was evaluated by double measurements on 1,332 images. In a subgroup of study I with arteriovenous ratio values < 0.76 (N = 107), an extended vessel diagnosis with measurement of 24-h blood pressure and vessel-relevant blood values (homocysteine, cholesterol, LDL, HDL, CRP, TG, HbA1c) was carried out (study II). RESULTS: Study I: The Kronbach alpha coefficient as a measure of reproducibility amounted to 0.77. The mean arteriovenous ratio of the retinal vessels was 0.83 +/- 0.09 and showed a pronounced age dependence (R = 0.9, p < 0.0001). On multivariate testing the arteriovenous ratio correlated significantly (R = 0.33, p < 0.001) with the factors age, systolic blood pressure, diastolic blood pressure and body mass index. Diastolic blood pressure followed by age had the largest influence. The prevalence of microvascular abnormalities in the right (RE) and left (LE) eyes, respectively were: cotton wool foci RE 0.0015 %, LE 0.003 %, retinal haemorrhage RE 0.1 %, LE 0.1 %, focal stenoses RE 3.4 %, LE 3.4 %, tortuositas vasorum RE 4.1 %, LE 4.0 %, arteriovenous crossing signs RE 11.2 %, LE 11.2 %. On multivariate testing the occurrence of microvascular abnormalities correlated significantly (R = 0.38, p < 0.001) with the factors high blood pressure known from case history, body mass index, and gender. Arterial hypertension had the strongest influence followed by diastolic blood pressure. The calculated retinal risk factor correlated with the prevalence of angina pectoris. Study II: 2/3 of the subjects with arteriovenous risk factor values < 0.76 exhibited pathologically high 24-h blood pressure values. For these patients there were significant correlations between the arteriovenous ratio and the low-density lipoprotein concentration as well as the Framingham risk score. CONCLUSION: In the course of a prospective, telemedical-supported screening examination of the retinal vessels of more than 7,000 subjects the arteriovenous ratio exhibited a strong dependence on age and blood pressure. Among the subjects with lowered arteriovenous ratio values, 2/3 exhibited arterial hypertension in the 24-h blood pressure determination.

Comparative application of wavelet approaches to absorption and ratio spectra for the simultaneous determination of diminazene aceturate and phenazone in veterinary granules for injection.

A comparison of two wavelet approaches, Daubechies and reverse Biorthogonal, is described for the quantitative resolution of a binary mixture of diminazene aceturate (DIMA) and phenazone (PHE) in veterinary granules for injection without any chemical separation. These two approaches were specified as db4 (a = 180) and rbior3.7 (a = 125) respectively, after testing the signal analysis parameters for the overlapping absorption spectra and ratio spectra. In the first step db4 (a = 180) was applied to the original absorbance data vector of DIMA and PHE. In the second step rbio3.7 (a = 125) was applied to the ratio spectra data vectors of DIMA using the divisor PHE. The same approach was also subjected to the ratio spectra of PHE using the divisor DIMA. The db4 (a = 180) and rbior3.7 (a = 125) calibration graphs were constructed using the transformation values obtained in the wavelet domain. In the method validation, the wavelet calibration functions were tested using synthetic mixtures and the standard addition technique. The simultaneous quantitative analysis of DIMA and PHE in the commercial veterinary preparation was achieved by the elaborated methods. The assay results were compared with each other and good agreement was observed.

Continuous wavelet transformation applied to the simultaneous quantitative analysis of two-component mixtures.

In this paper we developed a graphical method based on Haar (HA) and Mexican (MEX) one-dimensional continuous wavelet transforms and we applied it to a mixture of hydrochlorothiazide (HCT) and spironolactone (SP) in the presence of strongly overlapping signals. Keeping in mind to obtain an appropriately transformed spectrum, we tested several values of the scaling parameter a and the point number of the analysed spectrum in the concentration range of 2-22 microg/ml for both active compounds. The optimal values of the scale parameters and the corresponding frequencies were found to be a = 32 and 0.031 for HA and a = 30 and 0.008 for MEX corresponding to 400 points. HA and MEX methods based on a zero crossing technique were applied to the analysed signal and their regression lines at the selected points were obtained. The validation of the above methods was carried out by analysing different synthetic mixtures containing HCT and SP. MATLAB 6.5 software was used for one-dimensional wavelet analysis and the basic concepts about wavelet method were briefly explained. The method developed in this paper is rapid, easy to apply, inexpensive and is suitable for analysing the overlapping signals of compounds in their mixtures without any chemical pre-treatment.

Two new spectrophotometric approaches to the multicomponent analysis of the acetaminophen and caffeine in tablets by classical least-squares and principal component regression techniques.

Classical least-squares (CLS) and principal component regression (PCR) techniques were proposed for the simultaneous analysis of tablets containing acetaminophen and caffeine without using a chemical separation procedure. The chemometric calibrations were prepared by measuring the absorbances values at the 15 wavelengths in the spectral region 215-285 nm and by using a training set of the mixtures of both drugs in 0.1 M HCI. The obtained chemometric calibrations were used for the estimation of acetaminophen and caffeine in samples. The numerical calculations were performed with the 'MAPLE V' software. By applying two techniques to synthetic mixtures, the mean recoveries and the relative standard deviations in the CLS and PCR techniques were found as 99.5 and 1.29, 99.7 and 1.00% for acetaminophen and 99.9 and 1.92, 100.0 and 1.178% for caffeine, respectively. Our results were compared with those obtained previously by one of us considering HPLC method as a reference method. These two methods were successfully applied to a pharmaceutical tablet formulation of two drugs.

Spectrophotometric multicomponent analysis of a mixture of metamizol, acetaminophen and caffeine in pharmaceutical formulations by two chemometric techniques.

Inverse least squares (ILS) and factor-based (principal component analysis (PCA)) techniques were proposed for the spectrophotometric multicomponent analysis of a ternary mixture consisting of metamizol, acetaminophen and caffeine, without prior separation. In these chemometric techniques, the measurements of the absorbance values were realized in the spectral range from 225 to 285 nm in the intervals of Deltalambda=5 nm at the 13 wavelengths in the zero-order spectra of the different ternary mixtures of these active ingredients in 0.1 M HCl. The prepared calibrations of both techniques using the absorbance data and concentration matrix data sets were used to predict the concentration of the unknown concentrations of metamizol acetaminophen and caffeine in their ternary mixture. The 'MAPLE V' software was used for the numerical calculations. Mean recoveries and relative standard deviations for ILS and PCA techniques were found to be 99.8 and 1.68%, 99.9 and 1.66% for caffeine, 99.8 and 1.84%, 100.4 and 2.85% for metamizol, and 99.7 and 1.04%, 99.6 and 1.34% for acetaminophen, respectively, for the first and second techniques. The techniques were successfully applied to two pharmaceutical formulations marketed in Turkey and results were compared with a new high-performance liquid chromatography method.