Generator Matrix Research Articles

Identifying the generator matrix of a stationary Markov chain using partially observable data.

Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.

A study of a deterministic model for meningitis epidemic

Abstract Objectives A compartmental deterministic model that allows immunity from two stages of infection and carriage, and disease induced death, is used in studying the dynamics of meningitis epidemic process, in a closed population. It allows for difference in the transmission rate of infection to a susceptible by a carrier and an infective. It is generalized to allow a proportion of those susceptible infected to progress directly to infectives in stage I. Methods Two models are used in this study; the generalized model when these proportions are greater than zero (called model 1) and a special case when the proportion is zero (called model 2). The threshold conditions for the spread of carrier and infectives in stage I are derived for both models. Sensitivity analysis is performed on the reproductive number derived from the next generation matrix. The case-carrier ratio profile for various parameters and threshold values are shown. So also, are the graphs of the total number ever infected as influenced by the odds in favor of a carrier over an infective, in transmitting an infection to a susceptible, and the proportion of those susceptible that can progress directly to the infected stage. Results The infection transmission rate, these odds in favor of a carrier, and the carrier conversion rate to an infective in stage I, are identified as key parameters that should be subject of attention for any control intervention strategy. The case-carrier ratio profiles provide evidence of a critical case-carrier ratio attained before the number of reported cases grow to an epidemic level. They also provide visual evidence of epidemiological context, in this case, epidemic incidence (in later part of dry season) and endemic incidence (during rainy season). Conclusions Results from the total proportion ever infected suggest that the special case (model 2) can adequately represent, in essence, the generalized model (model 1) for this study.

Gaussian generators for the Yangian associated with the Lie superalgebra [formula omitted]

We give a new presentation of the Yangian for the orthosymplectic Lie superalgebra osp1|2m. It relies on the Gauss decomposition of the generator matrix in the R-matrix presentation. The defining relations between the Gaussian generators are derived from a new version of the Drinfeld-type presentation of the Yangian for osp1|2 and some additional relations in the Yangian for osp1|4 by an application of the embedding theorem for the super-Yangians.

Analysis of optimal control strategies on the fungal Tinea capitis infection fractional order model with cost-effective analysis

In this study, we have formulated and analyzed the Tinea capitis infection Caputo fractional order model by implementing three time-dependent control measures. In the qualitative analysis part, we investigated the following: by using the well-known Picard–Lindelöf criteria we have proved the model solutions' existence and uniqueness, using the next generation matrix approach we calculated the model basic reproduction number, we computed the model equilibrium points and investigated their stabilities, using the three time-dependent control variables (prevention measure, non-inflammatory infection treatment measure, and inflammatory infection treatment measure) and from the formulated fractional order model we re-formulated the fractional order optimal control problem. The necessary optimality conditions for the Tinea capitis fractional order optimal control problem and the existence of optimal control strategies are derived and presented by using Pontryagin’s Maximum Principle. Also, the study carried out the sensitivity and numerical analysis to investigate the most sensitive parameters and to verify the qualitative analysis results. Finally, we performed the cost-effective analysis to investigate the most cost-effective measures from the possible proposed control measures, and from the findings we can suggest that implementing prevention measures only is the most cost-effective control measure that stakeholders should consider.

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the derivative's extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a "blessing of dimensionality" result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology.

Colombia towards an electricity generation matrix using renewable energies

The purpose of this paper is present a review about to the currently legislation in Colombia with respect to renewable energies. The mainly goals are: (i) to explain the energy policy regulation related to renewables energy; and (ii) to characterize using descriptive statistics renewables projects based on wind, small hydro and photovoltaic. In addition, the analyses allow to understand the existing potential in the different regions of the country. This information was collected through of the procedures developed by UPME and CREG agencies in Colombia.

Who acquires infection from whom? A sensitivity analysis of transmission dynamics during the early phase of the COVID-19 pandemic in Belgium

Age-related heterogeneity in a host population, whether due to how individuals mix and contact each other, the nature of host–pathogen interactions defining epidemiological parameters, or demographics, is crucial in studying infectious disease dynamics. Compartmental models represent a popular approach to address the problem, dividing the population of interest into a discrete and finite number of states depending on, for example, individuals’ age and stage of infection. We study the corresponding linearised system whose operator, in the context of a discrete-time model, equates to a square matrix known as the next generation matrix. Performing formal perturbation analysis of the entries of the aforementioned matrix, we derive indices to quantify the age-specific variation of its dominant eigenvalue (i.e., the reproduction number) and explore the relevant epidemiological information we can derive from the eigenstructure of the matrix. The resulting method enables the assessment of the impact of age-related population heterogeneity on virus transmission. In particular, starting from an age-structured SEIR model, we demonstrate the use of this approach for COVID-19 dynamics in Belgium. We analyse the early stages of the SARS-CoV-2 spread, with particular attention to the pre-pandemic framework and the lockdown lifting phase initiated as of May 2020. Our results, influenced by our assumption on age-specific susceptibility and infectiousness, support the hypothesis that transmission was only influenced to a small extent by children in the age group [0,18) and adults over 60 years of age during the early phases of the pandemic and up to the end of July 2020.

Mathscr {S}mathscr {E}mathscr {I}mathscr {A}mathscr {R}mathscr {S} model for analyzing mathscr {C}mathscr {O}mathscr {V}mathscr {I}mathscr {D}-19 pandemic process via uppsi -Caputo fractional derivative and numerical simulation

The objective of this study is to develop the mathscr {S}mathscr {E}mathscr {I}mathscr {A}mathscr {R}mathscr {S} epidemic model for mathscr {C}mathscr {O}mathscr {V}mathscr {I}mathscr {D}-{textbf {19}} utilizing the uppsi -Caputo fractional derivative. The reproduction number (breve{mathscr {R}}_0) is calculated utilizing the next generation matrix method. The equilibrium points of the model are computed, and both the local and global stability of the disease-free equilibrium point are demonstrated. Sensitivity analysis is discussed to describe the importance of the parameters and to demonstrate the existence of a unique solution for the model by applying a fixed point theorem. Utilizing the fractional Euler procedure, an approximate solution to the model is obtained. To study the transmission dynamics of infection, numerical simulations are conducted by using MatLab. Both numerical methods and simulations can provide valuable insights into the behavior of the system and help in understanding the existence and properties of solutions. By placing the values texttt{t}, ln (texttt{t}) and sqrt{texttt{t}} instead of uppsi , the derivatives of the Caputo and Caputo–Hadamard and Katugampola appear, respectively, to compare the results of each with real data. Besides, these simulations specifically with different fractional orders to examine the transmission dynamics. At the end, we come to the conclusion that the simulation utilizing Caputo derivative with the order of 0.95 shows the prevalence of the disease better. Our results are new which provide a good contribution to the current research on this field of research.

Mathematical Model of Celebrity Worship Tendency Among K-Pop Fans in South Sulawesi

Celebrity worship is a form of almost obsessional participation in which people idolize and even start to “worship” their favorite celebrities. Twitter reveals that in 2021, Indonesia was ranked fourth as the country with the most number of Korean Pop (K-Pop) fans. In this study, we constructed a SEIRS mathematical model of celebrity worship tendency among K-Pop fans (CWKF) because there have not been specific mathematical models built on the problem of celebrity worship tendency among K-Pop fans. The CWKF free equilibrium point (E0), endemic equilibrium point (E1), and using the Next Generation Matrix for the Basic Reproduction Number (R0) are calculated through analysis. Using the Routh-Hurwitz criteria for equilibrium analysis, we found that E0 is locally asymptotically stable if R0 < 1, and E1 is locally asymptotically stable if R0 > 1. Secondly, we conduct numerical simulations using Maple software, and research data consisting of 204 individuals from various regions in South Sulawesi serves as the basis for this mathematical analysis. The numerical simulation obtained supports the dynamic analysis results that the number of individuals who are K-Pop fans will decrease in the population and disappear within a certain period, even without any special treatment.

Influence of seasonality on <i>Zika virus</i> transmission

<abstract><p>In order to study the impact of seasonality on <italic>Zika virus</italic> dynamics, we analyzed a non-autonomous mathematical model for the <italic>Zika virus</italic> (<italic>ZIKV</italic>) transmission where we considered time-dependent parameters. We proved that the system admitted a unique bounded positive solution and a global attractor set. The basic reproduction number, $ \mathcal{R}_0 $, was defined using the next generation matrix method for the case of fixed environment and as the spectral radius of a linear integral operator for the case of seasonal environment. We proved that if $ \mathcal{R}_0 $ was smaller than the unity, then a disease-free periodic solution was globally asymptotically stable, while if $ \mathcal{R}_0 $ was greater than the unity, then the disease persisted. We validated the theoretical findings using several numerical examples.</p></abstract>

Development on Mathematical Models of Type 2 Diabetes Mellitus (DM) in Individuals with A Genetic History

Diabetes Mellitus (DM) type 2 is one type of DM that is suffered by many DM sufferers. Type 2 DM can pass on to its offspring so it is necessary to prevent the emergence of type 2 DM. Prevention can be done such as exercise, weight loss, and dietary regulation. This study is a development of previous research with a primary focus on individuals who have a genetic history of having suffered from type 2 DM and it is recommended to choose one of the three prevention parameters involved. The next model development is to find the free equilibrium point values of DM type 2 and DM type 2 and the model will be analyzed for stability at the equilibrium point. The analysis of the basic reproduction number (R0) using the next generation matrix yielded the following values for each assumption: 0.1599; 0.1586; 0.1590; and 0.1582. Weight loss factors were discovered to play a significant influence in preventing type 2 diabetes among the three parameters studied.

Mathematical Modelling and Analysis of Dengue Transmission Dynamics

Understanding and resolving the complexity of diseases such as dengue need the use of mathematical modelling, which enables us to forecast outbreaks, create practical solutions, and allocate resources effectively to lessen the disease's effects. This research proposes a mathematical model that examines the dynamics of dengue illness transmission. Te model accounts for a number of important factors that affect the disease's spread, including the increase of the mosquito population, how long it takes an illness to develop and the extent to which infected humans also infect mosquitoes. Further, basic reproduction number has been calculated using the next generation matrix method for the single strain dengue model and SEIR-SEI model has been analyzed via mathematical theorems showing the positivity of the solutions and boundedness of the equations governing the proposed model. Sensitivity analysis has been done for basic reproduction number R0. Te findings indicate that the basic reproduction number primarily depends on factors such as the transmission rate of disease and the rate at which mosquitoes become infected by infectious human beings and depends inversely on factors such as the recovery rate of humans and the rate at which mosquitoes transmit the virus to humans. Numerical simulations have been done on a small population to show the number of people and number of mosquitoes in different compartments at any given time t and it shows the progress of disease transmission in a population.

Dynamics in a delayed rumor propagation model with logistic growth and saturation incidence

<abstract><p>This paper studies a delayed rumor propagation model with logistic growth and saturation incidence. The next generation matrix method, some inequality techniques, the Lyapunov-LaSalle invariance principle, and the Lyapunov method are used in this paper. Our results indicate that if the basic regeneration number (which is analogous to the basic reproduction number in disease transmission models) is less than 1, the rumor-free equilibrium point (which is analogous to the disease-free equilibrium point in disease transmission models) is globally stable. If the basic regeneration number is greater than 1, then the rumor is permanent, and some sufficient conditions are obtained for local and global asymptotic stability of the rumor prevailing equilibrium point (which is analogous to the endemic equilibrium point in disease transmission models). Finally, three examples with numerical simulations are presented to illustrate the obtained theoretical results.</p></abstract>

Stability analysis for a HIV model with cell-to-cell transmission, two immune responses and induced apoptosis

<abstract><p>In this paper, a dynamic HIV model with cell-to-cell transmission, two immune responses, and induced apoptosis is proposed and studied. First, the non-negativity and boundedness of the solutions of the model are given, and then the exact expression of the basic reproduction number $ R_{0} $ is obtained by using the next generation matrix method. Second, criteria are obtained for the local stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses. Furthermore, the threshold conditions are also derived for the global asymptotic stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses by constructing the suitable Lyapunov function. Finally, some numerical simulations are conducted to verify the theoretical results; the numerical simulation results show that the increase of apoptosis rate had a positive role in the control of viral infection.</p></abstract>

Modeling, analyzing and simulating the dynamics of Tuberculosis-Covid-19 co-infection

Covid-19 and tuberculosis (TB) are two significant infectious illnesses that may pose significant threats to the public health and their co-infection aggravates the issue. Through the use of a set of nonlinear ordinary differential equations and contaminated surfaces from SARS-CoV-2 in the environment, we developed and investigated a mathematical model in this study for the dynamics of the co-infection of Covid-19 and TB transmission. The free equilibrium point of the disease was determined and its stability was investigated. The next generation matrix approach is used to determine the model’s basic reproduction number. The disease-free equilibrium point is not globally asymptotically stable, but it is locally asymptotically stable if R0 < L We found key model parameters for disease dynamics spread using normalized forward sensitivity analysis. From the numerical simulation results, we conclude that government stakeholders should work in reducing the transmission mechanisms of both diseases. In addition, they should look on mechanisms for removing SARS-CoV-2 contaminated surfaces from the community.

Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments.

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.

An application of a virus optimization algorithm to the problem of computing binary self-dual and LCD codes

In this paper, we employ a virus optimization (VO) algorithm, which is one of the metaheuristic optimization techniques, and a known construction method to compute many new binary $ [72,36,12] $ self-dual codes and optimal/near-optimal linear complementary dual (LCD) codes. In particular, we obtain 39 Type Ⅰ and 19 Type Ⅱ codes of length 72, with parameters in their weight enumerators that were not known in the literature before, and 85 new binary LCD codes that are either optimal or near-optimal. We also present the generator matrix of extended Golay code $ [24,12,8] $ by a cyclic group matrix ring element.

Modeling Antiviral Treatment and Therapeutic Interventions

The severity of the Coronavirus disease (COVID-19) is still felt in some places since its outbreak in 2020 and as a result various antiviral treatments and therapeutic interventions have been researched and approved for use to help curb the disease while some are still undergoing formulation. In this work, we examined the effectiveness of these antiviral treatment and therapeutic interventions on the coronavirus disease using a deterministic SEITR model. This consists of investigating the disease-free and endemic equilibria, basic reproduction number and stability. The local stability of the disease-free equilibrium was determined by solving the Jacobian matrix of the system of the system of differential equations while the basic reproduction number was calculated using the next generation matrix method. Numerical simulations to determine the active factor(s) in the transmission, preventive and possible elimination of the disease were carried out using a computational software called Maple. It was revealed that when people comply with these non-pharmaceutical interventions the rate of recovery increases and the spread of the disease is reduced greatly.

Strategies and Impacts of COVID-19 Vaccine Rollout: A Deterministic SEIVR Model Approach

Immunization has come to play a key role in our primary health care as it has saved millions of lives yearly since it came into existence. The Corona virus disease (COVID-19) pandemic brought about an urgent need for Covid-19 vaccines. Various researches have been done and are still ongoing to help produce vaccines to help protect people by creating an immune response without the potentially severe illness or post-COVID conditions associated with COVID-19 infection. In this work, vaccine deployment strategies and their impact using a deterministic SEIVR model was examined. This consists of investigating the disease-free and endemic equilibria, basic reproduction number and stability. The local stability of the disease-free equilibrium was determined by solving the Jacobian matrix of the system of the system of differential equations. The study calculated the basic reproduction number, R0 , using the next generation matrix method and found it to be R0=1.1251426e−10. This low value suggests that vaccination efforts can be effective in reducing the spread of COVID-19.

The Generator Matrix, the Expanded Embryo, and the Multiple Dramaturgical Threads in Pedro Almodóvar

Este artigo investiga os processos e os procedimentos de criação do cineasta espanhol Pedro Almodóvar, refletindo sobre as tessituras dramatúrgicas criadas pelo cineasta no processo de construção do seu projeto poético, observando os procedimentos que envolvem o seu pensamento e a sua prática como roteirista e diretor para, assim, refletir sobre processos criativos, de autoria e comunicação. A hipótese é que o seu projeto poético é construído através da seleção e transformação de elementos temáticos e estéticos presentes na sua própria obra e na obra de outros autores, cujos trabalhos lhe afetaram a sensibilidade, fazendo com que temas e personagens se repitam, sob diferentes perspectivas, ao longo do tempo, concebendo um universo marcado pela constante reinvenção de textos e filmes para criar seus próprios sistemas fílmicos. Este estudo foi feito utilizando entrevistas do diretor e a análise de alguns de seus filmes que mantêm uma relação dramatúrgica, que funcionam como matrizes geradoras, deflagradoras de novos processos de criação. A análise foi realizada a partir das teorias de Cecilia Almeida Salles sobre processo de criação.