Generator Matrix Research Articles

Mathematical Model on The Control of Intestinal Torsion in Pigs

An institutional based research at Federal College of Education (Technical) Ekiadolor, Benin City, Edo State, Nigeria, examined a pig farm in the Ovia North East Local Government Area, investigating a mathematical model on the control of intestinal torsion in pigs through vaccination and fibre-rich nutritional feeds. It results in the application of a four-compartment mathematical model for pig intestinal torsion, which is made up of the classes S, I, V, and R for the Susceptible, Infected, Vaccinated and Recovered classes. This model is used to solve for the reproduction number (R0) using the Next Generation Matrix (NGM), to solve the positivity model, to solve for the existence and uniqueness of the model, and to plot the model graphs using Maple.

Evaluating the risk of malaria transmission within the Central African Republic with the goal of stabilising and eliminating the infection

In this paper, we formulate a multi-compartmental mathematical model for humans and mosquitoes. We construct the system of differential equations for an SEITVR for the human compartment and an SEI for the mosquito compartment. We investigate the outbreak of malaria and its effect on the Central African Republic. The analysis of the compartmental model is carried out using stability analysis and Routh Hurwitz Criterion technique is used to indicate the major impact of the model and to improve the model through minor modifications in the transformation of disease in the population. Our model exhibits two equilibrium points, disease free equilibrium points and endemic equilibrium points. The next generation matrix is used to determine the basic reproduction number R 0. A new compartmental model was framed and estimated the malaria spread after 2023 in the Central African Republic, which is the novelty of this research. Our main motivation is to make the Central African Republic a malaria free country. A Numerical example are provided to validate our results for both the disease free state and endemic state of each model. We believe that this investigation will be more effective in reducing malaria infection and stop spreading.

Mathematical analysis of the effects of precarious health system on COVID-19 transmission dynamics: case of Burkina Faso

n the early moments of the pandemic of coronavirus disease 2019 (COVID-19), the level of effectiveness in treatment and management of patients was limited due to the lack of effective medications and adapted infrastructures in several countries especially in most developing countries. In this study, we propose an SIR-pattern model to investigate the epidemiological effect of delay in hospitalization and case managements. After computing the basic reproduction number thanks to the Next Generation Matrix method, we proved that the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is less than one, and condition which ensure the appearance of two endemic equilibria indicating the possibility of having backward bifurcation is determined. When the basic reproduction number exceeds one, Lyapunov functional technique is used to establish the global stability of the unique endemic equilibrium point. Also, we presented a local sensitivity analysis that gives better understanding of the role of each model parameter appearing in the basic reproduction number. Numerical simulation of the model is also performed to corroborate the analytical results. Finally, we fitted our model by using reported data for Burkina Faso, and the results showed that it can be used for predicting the disease outcomes.

Mathematical Analysis for Honeybee Dynamics Under the Influence of Seasonality

In this paper, we studied a mathematical model for honeybee population diseases under the influence of seasonal environments on the long-term dynamics of the disease. The model describes the dynamics of two different beehives sharing a common space. We computed the basic reproduction number of the system as the spectral radius of either the next generation matrix for the autonomous system or as the spectral radius of a linear integral operator for the non-autonomous system, and we deduced that if the reproduction number is less than unity, then the disease dies out in the honeybee population. However, if the basic reproduction number is greater than unity, then the disease persists. Finally, we provide several numerical tests that confirm the theoretical findings.

Identification of CD44 as a key engager to hyaluronic acid-rich extracellular matrices for cell traction force generation and tumor invasion in 3D

Mechanical properties of the extracellular matrix (ECM) critically regulate a number of important cell functions including growth, differentiation and migration. Type I collagen and glycosaminoglycans (GAGs) are two primary components of ECMs that contribute to mammalian tissue mechanics, with the collagen fiber network sustaining tension, and GAGs withstanding compression. The architecture and stiffness of the collagen network are known to be important for cell-ECM mechanical interactions via integrin cell surface adhesion receptors. In contrast, studies of GAGs in modulating cell-ECM interactions are limited. Here, we present experimental studies on the roles of hyaluronic acid (HA) in single tumor cell traction force generation using a recently developed 3D cell traction force microscopy method. Our work reveals that CD44, a cell surface receptor to HA, is engaged in cell traction force generation in conjunction with β1-integrin. We find that HA significantly modifies the architecture and mechanics of the collagen fiber network, decreasing tumor cells’ propensity to remodel the collagen network, attenuating traction force generation, transmission distance, and tumor invasion. Our findings point to a novel role for CD44 in traction force generation, which can be a potential therapeutic target for diseases involving HA rich ECMs such as breast cancer and glioblastoma.

Fractional-Order Modeling of COVID-19 Transmission Dynamics: A Study on Vaccine Immunization Failure

COVID-19 is an enveloped virus with a single-stranded RNA genome. The surface of the virus contains spike proteins, which enable the virus to attach to host cells and enter the interior of the cells. After entering the cell, the virus exploits the host cell’s mechanisms for replication and dissemination. Since the end of 2019, COVID-19 has spread rapidly around the world, leading to a large-scale epidemic. In response to the COVID-19 pandemic, the global scientific community quickly launched vaccine research and development. Vaccination is regarded as a crucial strategy for controlling viral transmission and mitigating severe cases. In this paper, we propose a novel mathematical model for COVID-19 infection incorporating vaccine-induced immunization failure. As a cornerstone of infectious disease prevention measures, vaccination stands as the most effective and efficient strategy for curtailing disease transmission. Nevertheless, even with vaccination, the occurrence of vaccine immunization failure is not uncommon. This necessitates a comprehensive understanding and consideration of vaccine effectiveness in epidemiological models and public health strategies. In this paper, the basic regeneration number is calculated by the next generation matrix method, and the local and global asymptotic stability of disease-free equilibrium point and endemic equilibrium point are proven by methods such as the Routh–Hurwitz criterion and Lyapunov functions. Additionally, we conduct fractional-order numerical simulations to verify that order 0.86 provides the best fit with COVID-19 data. This study sheds light on the roles of immunization failure and fractional-order control.

Developing an mpox risk assessment matrix for social listening and infodemic insights generation

Abstract Issue Any effective response to an infodemic, requires prioritizing public health response on identified narrative themes that pose highest risk to health and well-being of populations. High-risk narratives often require more swift and far-reaching action and low-risk narratives require little or no response. Description of the problem Different countries are assigned different levels of risk to specific narratives or concerns, recognizing that for some issues there may be a large gap between the true public health threat and what is generating lots of conversations, concerns and press coverage that indicate a high degree of risk perception that may be unwarranted. WHO developed a risk matrix to apply to global analysis of questions, concerns and narratives about mpox, which resulted in biweekly infodemic intelligence reports that risk-assessed identified infodemic themes and corresponding recommendations for action. Results Risk assessment criteria were defined in the context of WHO’s global infodemic insights analysis for low, medium and high risk and positive sentiment in mpox global outbreak context before conducting the integrated analysis so that insights were reproducible and infodemic response actions were prioritized. This implemented the WHO/UNICEF training manual on 6 steps to produce an infodemic insights report, which includes tools on how to develop and apply a risk matrix for a specific public health question of concern and apply it in analysis. Lessons The approach resulted in rapid, systematic, reproducible and evidence-driven decision-making to respond to the infodemic accompanying the mpox global outbreak. Risk matrices must be adapted to the context of the public health event and the mandate of the health authority taking action. An mpox risk matrix for narratives included adaptation to reflect high-risk from stigmatizing narratives, as well as narratives circulating around or within LGBTQI+ communities. Key messages • A risk matrix must be developed and adapted to the context of each health emergency and level of health authority action. • Prioritization for rapid infodemic response based on a risk assessment is that it facilitates more effective response in a health emergency.

Lifted Codes with Construction of Echelon-Ferrers for Constant Dimension Codes

Finding the highest possible cardinality, Aq(n,d;k), of the set of k-dimensional subspaces in Fqn, also known as codewords, is a fundamental problem in constant dimension codes (CDCs). CDCs find applications in a number of domains, including distributed storage, cryptography, and random linear network coding. The goal of recent research papers has been to establish Aq(n,d;k). We further improved the echelon-Ferrers construction with an algorithm, and enhanced the inserting construction by swapping specified columns of the generator matrix to obtain new lower bounds.

TRANSMISSION AND CONTROL DYNAMICS OF ROTAVIRUS DIARRHEA MODEL WITH DOUBLE DOSE VACCINATION

This study introduces a six-compartmental mathematical model (S, , , E, I, R) to examine the impact of administering a double dose vaccine on the dynamic spread of diarrhea within a community. The mathematical analysis shows the existence of equilibrium points for both disease-free and endemic states in the model. The basic reproduction number was determined using the Next Generation Matrix. Analysis has shown that the basic reproduction number which indicates the disease-free equilibrium point is locally asymptotically stable. Also, using a suitable Lyapunov functional for the model system expressed in state variables and parameters defining the dynamic characteristics of spread and control strategies of the rotavirus diarrhea to obtain the global stability of disease-free equilibrium point over time. A numerical simulation was carried out by Wolfram Mathematica to show the effect of a second-dose vaccine. The inclusion of a double-dose vaccine has been found to have a significant effect on completely eliminating the outbreak of diarrhea. This is evidenced by the local and global stability results, which indicate that effective measures have been taken to prevent the reintroduction or transmission of the disease, and if there may be a risk of outbreaks or reemergence of the disease, very little continuous monitoring and intervention strategies are required to maintain control as this should be taken seriously by medical practitioners or policy health makers.

Stability Analysis and Optimal Control for Spreading Typhoid Fever Model with Direct and Indirect Transmissions

This article discusses a model of spreading typhoid fever with direct and indirect transmissions. There are five compartments included in the model, namely susceptible individuals, infected individuals, chronic carrier individuals, recovered individuals, and salmonella typhi bacteria in the environment. As an effort to control and minimize the numbers of infected individuals and chronic carrier individuals, we introduce simultaneously campaign for the susceptible individuals and treatment for the infected individuals and for chronic carrier individuals. The resulting model is a system of non-linear differential equations and quite challenging to analysis globally. Existence of both disease-free equilibrium point and endemic equilibrium point are analysed analytically. Stability of the equilibrium point is analysed locally by determining the eigenvalues of the associated Jacobian matrix, Routh-Hurwitz stability criteria, and basic reproduction number () via next generation matrix. The minimum Pontryagin principle and Hamiltonian equation are referred to minimize the numbers of infected and chronic carrier individuals. Simulation is carried out using suitable values of parameters. We found that the eigenvalues for endemic equilibrium point are all negative real numbers and . Optimal paths for state and constate variables are plotted using fourth order forward-backward Runge-Kutta method. In case there are no campaign and treatments, we found and endemic occurs. When campaign and treatments are included together in the model, we found optimal paths for all compartments that minimize the number of infected and chronic carrier individuals. Giving campaign and treatments increase significantly the susceptible and recovered individuals. At the same time, it reduces significantly the infected, chronic carrier individuals, and salmonella typhi bacteria in the environment. Involving campaign and treatment in the model of spreading typhoid fever can be considered as an effective strategy to minimize the numbers of infected and chronic carrier individuals.

On Linear Codes over Local Rings of Order p4

Suppose R is a local ring with invariants p,n,r,m,k and mr=4, that is R of order p4. Then, R=R0+uR0+vR0+wR0 has maximal ideal J=uR0+vR0+wR0 of order p(m−1)r and a residue field F of order pr, where R0=GR(pn,r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p,n,r,m,k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.

Modeling the Two-Strain Dynamics of COVID-19 in Ghana Using a Logistic Growth Model

Through mutation, viruses constantly change, bringing into existence new variants; SARS-CoV-2 is no different. In December 2020, variants with different characteristics that could affect transmissibility and death emerged around the world of which Ghana is not an exception. To address this new phenomenon, a two-strain mathematical model of SARS-CoV-2 was formulatedto analyzed thetransmission dynamicsin Ghana. Thedisease-free equilibriumwas calculated. The basicreproduction number, <I>R</I><sub>0</sub>= max{<I>R</I><sub>0A</sub>, <I>R</I><sub>0B</sub>} = max(0.9957945674, 1.109170840), associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both <I>R</I><sub>0<I>A</I></sub>, <I>R</I><sub>0<I>B</sub> <sub></I></sub>< 1, but unstable otherwise. In addition to the disease-free, the boundary equilibrium for strain <I>A</I> and strain <I>B</I> was also calculated. Using the Gershgorin’s circle theorem, it was shown that the boundary equilibrium is locally asymptotically stable when both <I>R</I><sub>0<I>A</I></sub>, <I>R</I><sub>0<I>B</sub> <sub></I></sub>> 1, but unstable when otherwise. Simulations of the model were carried out. Results indicate that the government should intensify its efforts to vaccinate a larger proportion of the population and also recommends implementing comprehensive control measures, such as the use of face masks, social distancing, and contact tracing, to mitigate the spread of the disease.

A Stochastic Approach in the Analysis of Compartmental Models in Epidemiology

The spread of infectious diseases is a significant public health issue. Mathematical modeling is being used to find appropriate answers to this question. In this paper we propose a stochastic approach to compartmental models. A new model allows to rely on the fokker-planck equation with a SEIR model, with constant rates, to take into account the random environment in the propagation of a disease, markov and kolmogorov properties have been used in the modeling of the process. We have also determined the basic reproduction rate R0 of our new stochastic model using the new generation matrix.

Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Let Fq be a finite field with an odd characteristic. In this paper, we present a new result about double cyclic codes over a finite non-chain ring. Specifically, we study the double cyclic code over Fq+vFq+v2Fq with v3=v, which is isomorphic to Fq×Fq×Fq. This study mainly involves generator polynomials and generator matrices. The generating polynomial of the dual code is also obtained. We show the relationship between the generating polynomials of the double cyclic codes and those of their dual codes. Finally, as an application of these results, we construct some optimal codes over F3.

Mathematical modelling and transmission dynamic of SEIRVQD model for epidemic diseases

In this paper, we construct and formulate a new mathematical model for the spread of epidemic diseases with vaccination. The proposed model is composed of eight compartments, namely susceptible (S), exposed (E), infected (I), recovered (R), vaccinated (V ), quarantined (Q) and died (D). We prove the existence of a unique solution. Additionally, the region of the feasibility of the system and equilibrium points are presented. In order to indicate the level of viral invasion and demonstrate the ability of infection transmission, we calculate the basic reproduction number ( R 0 ) by the next generation matrix method. Furthermore, the sensitivity of the basic reproduction number ( R 0 ) concerning the fluctuations of the biological parameters of the presented model has been investigated. Finally, Numerical simulations are performed using MATLAB to illustrate and validate the efficiency of the proposed model and its accordance using real data for Chinese measles and Portugal's COVID-19 outbreaks.

A Modified SIVR Mathematical Model of Covid-19 Post-vaccination Stability Analysis in Kenya

In this paper, the study focused on carrying out a stability analysis on a modified SIR model for the COVID-19 pandemic for the period after vaccination in Kenya. The purpose of the work was to show that whereas the rate and the extent of disease spread amongst the Kenyan people was not as wide spread as happened in other parts of the world, it was necessary for government and policy makers to roll out a robust civic education to convince the majority of the Kenyan population to embrace vaccination as a major containment measure in curbing the spread of COVID-19 and other infectious diseases since the study revealed that the infections drastically reduced to near zero after vaccination except for a few isolated cases that were and still continue to exhibit mild symptoms to none at all. This was attributed to the development of the vaccine which upon a massive campaign by the Kenyan government, led into a significant portion of the population being vaccinated. It is believed this vaccination drive enhanced herd immunity amongst the population. This development has had a significant effect in the control of more recent COVID-19 variants like JN-1 that have remained largely mild and undetected in the country. Both local and global post vaccination stabilities were analyzed for the system using the Lyapunov function. The main objective of the study was therefore to carry out a post vaccination stability analysis by estimating key parameters such as the basic reproduction number, ( R0) and herd immunity threshold (HIT) for infectious diseases by fitting data into the model to enable predictions of future dynamics of the disease. The study used the Next Generation matrix and the least square method besides the Python software to solve generated differential equations of the model for and HIT parameters. Results obtained showed that, there was a significant reduction in infections due to enhanced herd immunity attributed largely to the roll out of vaccination.

Mathematical Modeling of the Dynamics of Psychoactive Drug Abuse with Intervention of Control Agencies

Drug abuse continues to pose a significant threat to both developed and developing countries. This paper examines the dynamics of psychoactive drug abuse with intervention of control agencies. The study established that the drug abuse model remains positively invariant over time, as indicated by the positivity of the solutions to the model equations. The study identified both drug abuse endemic and drug abuse free equilibrium states. Basic reproduction number was obtained using the method of next generation matrix. The local stability of the drug abuse free equilibrium state was analyzed using Jacobian method and the analysis showed that the drug abuse free equilibrium state is locally asymptotically stable since all the Eigen values are negative. Similarly, the global stability analysis of the drug abuse free equilibrium state confirmed that the system is globally asymptotically stable. The sensitivity index was computed. The results indicate that parameters with positive sensitivity indices increase the prevalence, while those with negative sensitivity indices decrease it. This suggests that these parameters have the ability to either increase or decrease the persistence of drug abuse in the population. The numerical simulation of the model was performed using MATLAB R2015a software and results show that enlightenment and law enforcement agencies play a vital role in reducing the spread of drug abuse. This research recommends intensification of early enlightenment to curtail the spread of drug abuse. Also making more policies, enactment and enforcement of more laws that will check the use of hard drugs

Dynamical and optimal control analysis of lymphatic filariasis and buruli ulcer co-infection

This study delves into the dynamics of lymphatic filariasis and buruli ulcer coinfection, two overlooked yet impactful tropical diseases. With lymphatic filariasis, commonly referred to as elephantiasis, and buruli ulcer, a chronic affliction caused by mycobacterium ulcerans, both posing significant health challenges, understanding their interaction is crucial. Utilizing a mathematical model, this research aims to analyze the dynamics of this coinfection, elucidating its complexities. The study establishes the local asymptotic stability of the disease-free equilibrium and calculates the basic reproduction number using the next generation matrix. It uncovers transcritical and backward bifurcation phenomena within the model. Additionally, the integration of time-dependent controls enables the exploration of optimal disease management strategies. Numerical simulations highlight the efficacy of employing a comprehensive approach, utilizing all available controls simultaneously, as the most effective strategy for disease control. These findings underscore the importance of integrated interventions in combating lymphatic filariasis and buruli ulcer coinfection, offering valuable insights for public health policymakers and practitioners.

Global stability and bifurcations in a mathematical model for the waste plastic management in the ocean

The use of plastic is very widespread in the world and the spread of plastic waste has also reached the oceans. Observing marine debris is a serious threat to the management system of this pollution. Because it takes years to recycle the current wastes, while their amount increases every day. The importance of mathematical models for plastic waste management is that it provides a framework for understanding the dynamics of this waste in the ocean and helps to identify effective strategies for its management. A mathematical model consisting of three compartments plastic waste, marine debris, and recycle is studied in the form of a system of ordinary differential equations. After describing the formulation of the model, some properties of the model are given. Then the equilibria of the model and the basic reproduction number are obtained by the next generation matrix method. In addition, the global stability of the model are proved at the equilibria. The bifurcations of the model and sensitivity analysis are also used for better understanding of the dynamics of the model. Finally, the numerical simulations of discussed models are given and the model is examined in several aspects. It is proven that the solutions of the system are positive if initial values are positive. It is shown that there are two equilibria E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E^0$$\\end{document} and E∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E^*$$\\end{document} and if BR<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {B}}}{{\\mathcal {R}}}<1$$\\end{document}, it is proven that E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E^0$$\\end{document} is globally stable, while when BR>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {B}}}{{\\mathcal {R}}}>1$$\\end{document}, the equilibrium E∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E^*$$\\end{document} exists and it is globally stable. Also, at BR=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {B}}}{{\\mathcal {R}}}=1$$\\end{document} the model exhibits a forward bifurcation. The sensitivity analysis of BR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {B}}}{{\\mathcal {R}}}$$\\end{document} concludes that the rates of waste to marine, new waste, and the recycle rate have most effect on the amount of marine debris.

Modeling the transmission dynamics of banana bunch top disease in banana plants

Banana Bunch Top Disease (BBTD) is a severe viral infection that poses a significant threat to banana production in tropical regions. Caused by the Banana Bunchy Top Virus (BBTV) and transmitted primarily by the banana aphid (Pentalonia nigronervosa), BBTD results in stunted growth and poor fruit production, leading to substantial economic losses. This paper presents a mathematical model to analyze the transmission dynamics of BBTD in banana plants and evaluate various control strategies using the Next Generation Matrix approach to compute the basic reproduction number R0. The study reveals that effective control of aphid populations, timely removal of infected plants, and the use of virus-free planting material significantly reduce the spread of BBTV and improve plant health. Increasing the removal rate γ and managing the transmission rate β are crucial strategies for controlling the disease, while elevated aphid infection rates λ and progression rates σ exacerbate its spread. Numerical simulations highlight the effectiveness of targeted interventions, including the application of pesticides and the introduction of resistant banana varieties. The study recommends that, in addition to current control measures, more focus should be placed on comprehensive monitoring programs, early detection of symptoms, and community education on BBTD management strategies. Enhanced efforts in these areas are crucial to reducing the incidence of BBTD and improving the resilience of banana crops.