Generator Matrix Research Articles

DYNAMICS OF A DENGUE FEVER MODEL WITH UNREPORTED CASES AND ASYMPTOMATIC INFECTED CLASSES IN SINGAPORE, 2020

This study is devoted to consider a novel model of dengue fever with unreported cases and asymptomatic infected classes, where infected humans is admitted to general and intensive hospitals for treatment. First, the basic reproduction number is calculated by using the next generation matrix method. The disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than one, but forward bifurcation occurs at the disease-free equilibrium when the basic reproduction number equals one. Then, the endemic equilibrium is consistent persistence when the basic reproduction number is more than one. Next, the existence of the optimal control pair is analyzed, and the mathematical expression of the optimal control is given by using Pontriagin's maximum principle. Finally, based on the dengue fever data in Singapore during the 15-52 weeks of 2020, the best fitting parameters of the model are determined by using Markov Chain Monte Carlo algorithm. The basic reproduction number is 1.6015 (95%CI:(1.5425-1.6675)). Numerical simulation and sensitivity analysis of several parameters are carried out. It is suggested that patients with dengue fever should report and receive treatment in time, which is of great significance for prevention and control of dengue fever.

Analysis of tinea capitis epidemic fractional order model with optimal control theory

Tinea Capitis is a common worldwide fungal infectious disease. This paper investigates the transmission dynamics of tinea captis infection using Caputo fractional derivative approach. The qualitative analysis part of the proposed fractional order model investigates: the model solutions existence and uniqueness by using the well-known Picard–Lindelöf criteria, the basic reproduction number using the next generation matrix approach, the model equilibrium points and their stabilities. The study re-formulates the fractional order optimal control problem with the three time-dependent controlling variables such as prevention measure, acute infection treatment measure and chronic infection treatment measure and also investigates the necessary optimality conditions and the existence of optimal control strategies by applying Pontryagin's Maximum Principle. Finally, the paper examines the effectiveness of the combinations of the three controlling strategies through numerical simulations and the results indicate that the simultaneous implementation of the three time-dependent controlling strategies significantly minimizes the number of tinea capitis infected individuals in the community.

On the spectrum and ergodicity of a neutral multi-allelic Moran model

The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.

Effect of partially and fully vaccinated individuals in some regions of India: A mathematical study on COVID-19 outbreak

In this paper, we investigate the effect of partially vaccinated and fully vaccinated individuals in preventing the transmit of COVID-19, especially in the regions of Tamil Nadu, Maharashtra, West Bengal and Delhi. Here we construct an SEIR model and analyse the behaviour. We obtained R0 by using next generation matrix approach. Also, our system shows two types of equilibria, namely disease free and endemic equilibrium. For both disease free and endemic equilibrium, local and global stability is obtained here. Our disease-free equilibrium is locally asymptotically stable whenever R0 is less than one, whereas the endemic equilibrium is locally asymptotically stable whenever R0 is greater than one. Furthermore, the global stability of disease-free equilibrium has been proven by using Lyapunov function and the global stability of endemic equilibrium has been obtained by using Poincare Bendixson technique. Also, we enhance our analytic results by numerical simulation. At the end we have attempted to fit our proposed model with the real-world data.

Dynamics of an epidemic model with general incidence rate dependent on a class of disease-related contact functions.

Starting from the idea of constructing the standard incidence rate, we take the effective contact times of individuals in the population per unit time as a contact function, $ T(\cdot) $, which depends on the population size. Considering the influence of disease on the contact function, the influence intensity factor of the disease affected by the infected person is integrated into the nonlinear incidence rate. We propose an epidemic model with a class of disease-related contact functions. Then, we analyze the well-posedness of the solutions of the model. By using the next generation matrix method, we get the basic reproduction number $ \mathcal{R}_0 $. We find that the existence and stability of the equilibria are not only related to $ \mathcal{R}_0 $, but also to the intensity of the disease affected for the infected person, $ \eta $, and the contact function, $ T(\cdot) $. We obtain some stability results under different assumptions about the contact function. Finally, we use MATLAB to simulate the system for several different contact functions. The numerical simulation results agree with our qualitative study. At the same time, we also prove that the system may have a Hopf bifurcation when the contact function $ T(\cdot) $ satisfies some corresponding conditions.

Dynamic analysis of SEIR model for Covid-19 spread in Medan

In this study, a mathematical model was studied on the population of the spread of Covid-19 in Medan which the model use an epidemic mathematical model, SEIR (Susceptible, Exposed, Infected, and Recovered). Next, we determine the basic reproduction number R0 using the next generation matrix and the equilibrium point which is analyzed using the Routh Hurwitz criteria. The disease-free equilibrium point is said to be locally asymptotically stable if R01 and the endemic equilibrium point is said to be locally asymptotically stable if R01. Numerical simulation of the model was carried out using real data on the number of Covid-19 cases in Medan and with the help of Maple software. Through the data obtained, the value of R01 indicates that Covid-19 at the time of the study was still contagious to other individuals. Furthermore, based on the simulation formed from the SEIR model with the given initial and parameters, it was found that the greater the contact rate or the transmission rate, the more spread the disease would be and the smaller the cure rate, the more the disease would spread.

Dynamics of a predator-prey model incorporating infectious disease and quarantine on prey

In this article, the dynamics of a predator-prey model incorporating infectious disease and quarantine on prey population is discussed. We first analyze the existence conditions of all positive equilibrium points. Next, we investigate the local stability properties of the proposed model using the linearization method. We also determine the basic reproduction number using the next generation matrix. Finally, some numerical simulations are performed to validate the stability of each equilibrium point.

Mathematical Analysis of SEIR Model to Prevent COVID-19 Pandemic

This paper is to analyze Susceptible-Exposed-Infectious-Recovered (SEIR) COVID-19 pandemic model. In this article, a modified SEIR model is constructed and discusses various aspects of it with mathematical analysis to study the dynamic behavior of this model. Spread of this disease through immigration can be represented by the SEIR model. COVID-19 is a highly infectious disease that spread through talking, sneezing, coughing, and touching. In this model, there is an incubation period during the spread of the disease. During the gestation period, a patient is attacked by SARS-CoV-2 coronavirus and shows symptoms of COVID-19, but does not spread of the disease. The horizontal transmission of COVID-19 worldwide can be represented and explained by SEIR model. Maximal control of the pandemic disease COVID-19 can be possible by the optimum vaccination policies. The study also investigates the equilibrium of the disease. In the study a Lyapunov function is created to analyze the global stability of the disease-free equilibrium. The generation matrix method is analyzed to obtain the basic reproduction number and has discussed the global stability for COVID-19 spreading.

A deterministic model of the spread of scam rumor and its numerical simulations

In an attempt to illustrate the dynamics of the spread of scam rumors and its impact in a heterogeneous population or social network we adopt the epidemiological model approach. In this work the SpEIsScSt (Susceptible, Exposed, Ignorant, Scammers and Stiflers) model was developed. A diagrammatic representation of the flow of individuals from one class/group to another is given from which a system of five differential equations are obtained. The Basic reproduction number, the equilibrium point of scam rumor-free and the endemic equilibrium state were obtained and discussed. For the local stability of the scam rumor-free and endemic equilibrium state the Next Generation Matrix was used. We were able to show the conditions for the existence of global stability by defining a Lyapunov function with respect to the state variables. Numerical simulation of the system was carried out in order to get a clear picture of the dynamics of the spread of scam rumor in a population or social networks, its impact in a population, and the efficiency of government/admin policies in curbing the spread of scam rumor.

Some notes on the dynamic behavior of COVID-19 epidemic disease governed by modified SEIR epidemic model with saturated treatment function

In this work, based on theory of differential systems and stability theory, we are in the first stage to study some applications of differential systems in real-world modelling problems. Here, we propose a mathematical epidemic model, namely SEIR epidemic model with saturated treatment to describe the COVID-19 infection. Based on the next–generation matrix method and the theory of Lyapunov stability, we evaluate the basic reproduction number and investigate the asymptotic behavior of disease-free equilibrium. Additionally, in the case , we also prove the existence of a unique endemic equilibrium. Finally, in order to dicuss the effect of isolation control strategies, we study the optimal control problem for this epidemic model.

Optimal quaternary $$(r,\delta )$$-locally recoverable codes: their structures and complete classification

Locally recoverable codes (LRCs) have been introduced as a family of erasure codes that support the repair of a failed storage node by contacting a small number of other nodes in the cluster. Boosted by their applications in distributed storage, LRCs have attracted a lot of attention in recent literature since the concept of codes with locality r was introduced by Gopalan et al. in 2012. Aiming to recover the data from several concurrent node failures, the concept of r-locality was later generalized as $$(r, \delta )$$ -locality by Prakash et al. An $$(r, \delta )$$ -LRCs in which every code symbol has $$(r, \delta )$$ -locality is said to be optimal if it achieves the Singleton-like bound with equality. In present paper, we are interested in optimal $$(r, \delta )$$ -LRCs over small fields, more precisely, over quaternary field. We study their parity-check matrices or generator matrices, using the properties of projective space. The classification of optimal quaternary $$(r,\delta )$$ -LRCs and their explicit code constructions are proposed by examining all possible parameters.

Universal Address Sequence Generator for Memory Built-in Self-test*

This paper presents the universal address sequence generator (UASG) for memory built-in-self-test. The studies are based on the proposed universal method for generating address sequences with the desired properties for multirun march memory tests. As a mathematical model, a modification of the recursive relation for quasi-random sequence generation is used. For this model, a structural diagram of the hardware implementation is given, of which the basis is a storage device for storing so-called direction numbers of the generation matrix. The form of the generation matrix determines the basic properties of the generated address sequences. The proposed UASG generates a wide spectrum of different address sequences, including the standard ones, such as linear, address complement, gray code, worst-case gate delay, 2i, next address, and pseudorandom. Examples of the use of the proposed methods are considered. The result of the practical implementation of the UASG is presented, and the main characteristics are evaluated.

Coinfection Dynamics of HBV-HIV/AIDS with Mother-to-Child Transmission and Medical Interventions

In this study, we analyzed the effect of mother-to-child transmission (MTCT) of hepatitis B virus (HBV) and human immunodeficiency virus (HIV) on the transmission dynamics of their coinfection to make a recommendation based on reasons to public health sector, policy makers, and programme implementers. We proved that the solutions of the sub and full models are positive and bounded. The effective reproduction numbers of the models are derived using the next generation matrix method. The disease-free and endemic equilibria of the submodels and the coinfection model are computed, and the stability of those equilibria is analyzed using Routh-Hurwitz criteria after computing the associated effective reproduction numbers. We performed a sensitivity analysis to show the influence of different parameters on the effective reproduction number of HBV-HIV/AIDS coinfection model, and we identified the most sensitive parameters are τ 2 and α 1 , which are the rate of MTCT of HIV and treatment rate for HBV infected class, respectively. The numerical simulation of the model is done using MATLAB and the findings from the simulations are discussed. From the results of numerical simulations, we observed that an increase in the rates of MTCT of HBV and HIV exacerbated HBV-HIV/AIDS coinfection, while a decrease in the rates of MTCT of these infections would decline the number of cases, minimize the spread, and help to eliminate HBV-HIV/AIDS coinfection from the society gradually.

A fractional SVIR-B epidemic model for Cholera with imperfect vaccination and saturated treatment.

Given the limited medical resources in most cholera endemic countries, in this paper, a nonlinear fractional SVIR-B (Susceptible-Vaccinated-Infected-Recovered, Bacterial) cholera epidemic model with imperfect vaccination and saturated treatment is proposed and investigated. The model performs well-posed in both epidemiologically and mathematically, especially, we demonstrate the positivity and boundedness of all solutions using the generalized mean value theorem. The control reproduction number is derived using the next generation matrix method and both local and global stability analyses for the disease-free equilibrium is performed by analyzing the characteristic equation and using Lyapunov functional. Then, the existence and stability of endemic equilibria are further addressed. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also derived. Subsequently, an optimal control problem with vaccination, media coverage, treatment, and sanitation as control strategies is proposed and analyzed, providing a rationale for cholera control and prevention. In addition, several numerical examples are employed to illustrate our theoretical results.

SYNTHESIS OF THE SYMBOLOGIES OF MULTICOLOR INTERFERENCE-RESISTANT BAR CODES ON THE BASE OF MULTI-VALUED BCH CODES

Context. The problem of constructing a set of barcode patterns for multicolor barcodes that are resistant to distortions of one or two elements within each pattern is considered.
 Objective. The goal of the work is ensuring the reliability of the reading of multi-color barcode images.
 Method. A multicolor barcode pattern has the property of interference immunity if its digital equivalent (vector) is a codeword of a multi-valued (non-binary) correcting code capable to correct errors (distortions of the pattern elements). It is shown that the construction of barcode patterns should be performed on the basis of a multi-valued correcting BCH code capable to correct two errors. A method is proposed for constructing a set of interference-resistant barcode patterns of a given capacity, which ensure reliable reproduction of data when they are read from a carrier. A procedure for encoding data with a multi-valued BCH code based on the generator matrix of the code using operations by the modulo of a prime number has been developed. A new method of constructing the check matrix of the multivalued BCH code based on the vector representation of the elements of the finite field is proposed. A generalized algorithm for generating symbologies of a multi-color barcode with the possibility of correcting double errors in barcode patterns has been developed. The method also makes it possible to build symbology of a given capacity based on shortened BCH codes. A method of reducing the generator and check matrices of a multi-valued full BCH code to obtain a shortened code of a given length is proposed. It is shown that, in addition to correction double errors, multi-valued BCH codes also make it possible to detect errors of higher multiplicity – this property is enhanced when using shortened BCH codes. The method provides for the construction of a family of multicolor noise-immune barcodes.
 Results. On the basis of the developed software tools, statistical data were obtained that characterize the ability of multi-valued BCH codes to detect and correct errors, and on their basis to design multi-color interference-resistant bar codes.
 Conclusions. The conducted experiments have confirmed the operability of the proposed algorithmic tools and allow to recommend it for use in practice for developing interference-resistant multi-color barcodes in automatic identification systems.

Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment

Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, Rph, that measures the potential spread of TB is presented by employing the next generation matrix approach. We investigate local and global stability of the TB-free equilibrium point, endemic equilibrium point, and sensitivity analysis. The analyses of the proposed model show that the model undergoes the phenomenon of backward bifurcation when the effective reproduction number (Rph) is less than one, where two stable equilibria, namely, the DFE and an EEP coexist. Further, we compute the sensitivity of the impact of each parameter on the effective reproductive number of the model by employing a normalized sensitivity index formula. Numerical simulation of the proposed model was conducted using Maple 2016 and MatLab R2020b software and compared with the theoretical results for illustration purposes. The investigation results can be useful in providing information to policy makers and public health authorities in mitigating the spread of TB infection by public health education and hospital treatment.

Vaccination effect on the dynamics of dengue disease transmission models in Nepal: A fractional derivative approach

Dengue is a vector-borne disease which is spreading rapidly around the world. It is one of the fastly growing public health problems in Nepal. Since 2004, dengue cases have been recorded in both tropical and subtropical regions of Nepal. There is no specific treatment for the dengue disease, which is brought on by the dengue virus. This study attempts the use of the Caputo fractional-order derivative to suggest SVEIRP-SEI epidemic model that integrates vaccination and hospitalization in order to precisely analyze the transmission of dengue infection phenomena. The existence and uniqueness are discussed for the model solutions. Basic reproduction number R0 with vaccination is formulated using next generation matrix. Both the local and global stability of disease free and endemic equilibrium points are explored. The parameters are estimated from the actual outbreak of infected cases in Nepal. Sensitivity analysis is then presented graphically with appropriate memory index ψ. The findings of the mathematical and simulation results demonstrate that vaccination is one of the effective strategies that lowers the prevalence of the disease significantly.

A novel method for identifying recursive systematic convolutional encoders based on the cuckoo search algorithm

The existing methods for identifying recursive systematic convolutional encoders with high robustness require to test all the candidate generator matrixes in the search space exhaustively. With the increase of the codeword length and constraint length, the search space expands exponentially, and thus it limits the application of these methods in practice. To overcome the limitation, a novel identification method, which gets rid of exhaustive test, is proposed based on the cuckoo search algorithm by using soft-decision data. Firstly, by using soft-decision data, the probability that a parity check equation holds is derived. Thus, solving the parity check equations is converted to maximize the joint probability that parity check equations hold. Secondly, based on the standard cuckoo search algorithm, the established cost function is optimized. According to the final solution of the optimization problem, the generator matrix of recursive systematic convolutional code is estimated. Compared with the existing methods, our proposed method does not need to search for the generator matrix exhaustively and has high robustness. Additionally, it does not require the prior knowledge of the constraint length and is applicable in any modulation type.

A Compartmental Model for Assessing Effects of COVID-19 Vaccination in Thailand

A dynamical model for COVID-19 spread relating to non-pharmaceutical interventions and vaccination is mathematically generated by adding a gradual vaccination compartment for the susceptible population and considering only a symptomatic infectious stage. In our model, there are seven compartments dividing a given population into susceptible <img src=image/17628733_01.gif>, vaccinated <img src=image/17628733_02.gif>, exposed <img src=image/17628733_03.gif>, infected <img src=image/17628733_04.gif>, quarantined <img src=image/17628733_05.gif>, recovered <img src=image/17628733_06.gif> and death <img src=image/17628733_07.gif> groups, respectively. Then, theoretically analysis is given by investigating the COVID-19 free and endemic equilibrium points, and computing the vaccination reproduction number of this model denoted as <img src=image/17628733_08.gif> using the next generation matrix. If <img src=image/17628733_09.gif>, then the COVID-19 transmission increases exponentially and depends on vaccine efficacy. On the other hand, if <img src=image/17628733_10.gif>, then there occurs the COVID-19 disease eradication. The risk from infection can be importantly reduced whenever the intake of COVID-19 vaccines exceeds one dose. The numerical results reveal that the nonpharmaceutical ways and the administered COVID-19 vaccines can be effective against the current variants of COVID-19, and the additional efforts such as a third vaccine booster shot should be considered and implemented to greatly mitigate the risks of emerging variants of the COVID-19 virus. Moreover, combining different types of COVID-19 vaccines can be appeared as a possible way to give better protection against COVID-19 as well.

Numerical Treatment of MHD Rotating Flow of Nano-Micropolar Fluid with Impact of Temperature-Dependent Heat Generation and Variable Porous Matrix

Numerical Treatment of MHD Rotating Flow of Nano-Micropolar Fluid with Impact of Temperature-Dependent Heat Generation and Variable Porous Matrix