Articles

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Open Access
Age-Structured Transmission Dynamics Model for Vertical and Horizontal Transmission of HIV/AIDS
(Department of Mathematics, Nasarawa State University Keffi, 2008-01-06) Umar, Mallam Abdulkarim; Olowofeso, E.O.; Ademiluyi, R.A.
Transmission dynamics model for -HlV/AlDS, along the line of Mckendrick-Forester age-structured model is proposed with the natural mortality rate and the fertility functions assumed to bE age depended, similar, to Doma, Gurlin-MacCnmy definitions. The solution^ to the governing equations are obtained and the steady stales are examined for their local stability. The model is further extended to study the case of constant mortality . rate and an exponential type of interaction function. It is observed that the endemic steady'.exist, and. .. asymptotically stable.
Open Access
Age-Structures Epidemic Model for Transmission Dynamics of Tuberculosis
(Department of Mathematics, Nasarawa State University Keffi, 2007-02-02) Umar, Mallam Abdulkarim
Open Access
A Class of Block Multi-Step Methods for the Solutions of Ordinary Differential Equation (Ode)
(Department of Mathematical Science, Nasarawa State University Keffi., 2016-04-13) Nweze, N.O.; Chaku, Shammah Emmanuel; Offiong, M.N.; Maijamaa, Bilkisu
In this research, an attempt is made to derive a self starting block procedure for some K-step linear multi-step methods (for K=1, 2 and 3), using Chebyshev polynomial as the basis function. The continuous interpolant were derived and collocated at grid and off-grid points to give the discrete methods used in block and applied simultaneously for the solution of non stiff initial value problem.The regions of absolute stability of the methods are plotted and are shown to be A (α) stable. The methods for K=2 and K=3 were experimented on initial value problems and the results reveal that the newly constructed block methods have good error stability and are efficient.
Open Access
A Comparative Study of Box-Jenkins, Arima, Holt-Winter and Exponential Smoothing Models as Applied to Exchange Rate on Naira to Dollar
(Department of Mathematics, Nasarawa State University Keffi, 2005-01-06) Umar, Mallam Abdulkarim
Open Access
A Continuous Time Mathematical Model for Retiring Lecturers in the Nigerian Universities: A Case Study, of Nasarawa State University, Keffi, Nigeria
(Department of Mathematics, Nasarawa State University Keffi, 2012-01-03) Ndakwo, Husseini S.; Muhammad, A.Y.; Umar, M.A.
In this .paper, an attempt vas made to formulate a continuous tunc Mathematical model dial will solve the problems'confronting Lecturers towards and during re them end Onnh, E.S. and Mser, M.D.(2003) Mathematical Association of Nigeria worked on the general Mathematical Model for retiring workers. 'The income of a Professor and. a. Graduate Assistant in a Nigerian University was used in-the numerical example.
Open Access
The Dynamics of Natural Mortality, Life Expectancy and TB in an Structured Population.
(Department of Mathematics, Nasarawa State University Keffi, 2008-01-09) Umar, Mallam Abdulkarim
We have extended live classical Mckcndriok-Focrstcr age structured population model to study the olicet ol population area size on the transmission dynamics of tuberculosis in an ngo*otniciurccl population, in wliicli infective recovers, nfior irontmontond bocomo suscoptiblo again when in contact with Die disease. The natural mortality rate J3 assumed density mid nren size depended, wiUi the population assumed lo be in propoi tionm.0 mixing. Wo defined n niodol in which tlie area sizes and the total population influences the rale at. which death by nature in 1110 population. This model is extended to study Iho influence the area size of the population has on the natural mortality rate, life expectancy of the population and the transmission dynamics oT the infection. The analytical solutions of the governing equations arc obtained. The existence of the non trivial a toady .states in oxnminod with their local and global .stability.
Open Access
Effect of Vaccination, Treatment and Population Area Size on the Transmission Dynamics of Bird-Flu Epidemics in a Proportiona,
(Department of Mathematics, Nasarawa State University Keffi, 2007-01-06) Umar, Mallam Abdulkarim
We have examined the effect of vaccination, nnrl population nren size on the transmission dynamics oT bird flu epidemics in a poultry farm, using two class of models, an age -structured epidemic model and n homogeneous epidemics model in which the transmission rate in the ease of homogeneous model, is assumed not age-depended, while it is assumed age-dependent in the agc-struclurcd model. The per capital contact rate and the vital rates arc assumed not age depended. It is observed in both eases, that llic area size occupied by the population influences the rate of transmission of the virus. Wild birds, which arc normally on free range, arc observed to be responsible for the spread of the virus to poultry farms; however the transmission rate may be less than in poultry birds, due to their free range nature. This is not investigated this work. This is assumed to has helped to control the concentration of the wild birds population density in n particular location, where they arc found, and less per cnpilnl contact rale. Thus, the disease spread is minimal. Threshold parameters for both models arc examined to determine the stability of the non -trivial steady stales.
Open Access
Effect of Vaccination, Treatment and Population Area Size on the Transmission Dynamics of Tuberculosis in a Proportionate Mixing Population
(Department of Mathematics, Nasarawa State University Keffi, 2007-01-06) Umar, Mallam Abdulkarim
Open Access
Effects of minimum epidemic and population sizes on a global epidemic in simulations of final size data
(Department of Mathematics, Nasarawa State University Keffi, 2020-08-08) Umar, Mallam Abdulkarim; Ndakwo, Husseini S.
The stochastic SIR household epidemic model is well discussed in [2], [3] and [4]. The work of [1] • also proposed maximum likelihood based algori hm for its inference by assuming independence of epidemic in each household, contrary to the dependency assn uption in [4]. 8 ° * Using simulations, we examined the ne id for an appropriate choice of cut-off between small and large epidemics often referred to as minimum epidemic size, using rejection sampling, for a global infection to occur and then compared the estimates of the model parameters over a range of theoretical parameters, XL and Xc with corresponding z 6 [0,1]. We found that with large population size, appropriate choice of the minimum epidemic size and Xc =£ 0 facilitate the occurrence of a global epidemic. Thus, given these scenarios, the adequacy of the model fitness to the final size epidemic data is then realised.
Open Access
A Global Stability Analysis of a Susceptible-Infected-Removed-Prevented-Controlled Epidemic Model
(Department of Mathematics, Nasarawa State University Keffi, 2014-03-06) Muhammad, Yau A.; Ndakwo, Husseini S.; Umar, Mallam Abdulkarim
A mathematical model off HIV transmission dynamics Is proposed and analysed. The population is partitioned into five compartments of susceptible 5(C),Infected I{£), Removed rtp), .Prevented* t/(C) and the Controlled W(t). Each of the compartments comprises of cohort of individuals. Five sys tems of nonlinear equations are derived to represent each of the compartments. The general bility of the disease free equilibrium (DFE) and the endemic equilibrium states of the linearized model are established using the linear stability analysis (Routh-Hurwitz) method which is found to be locally asymptotically stable when the Infected individuals receive ART and use the condom. The reproduction number is also derived using the idea of Diekmann and is found to be strictly less than one. This means that the epidemic will die out.
Open Access
HARNESSING VIRTUAL CLASSROOM SYSTEM AS AN ALTERNATIVE FOR PART-TIME STUDIES IN NIGERIA
(Department Of Mathematics, Nasarawa State University Keffi., 2011-09-26) Chaku, Shammah Emmanuel; Anyachebelu, Tochukwu Kene; Offiong, N.M.
The clamour for higher education is on the increase globally, precisely, there is great need for higher education in Nigeria. For this mere fact, there was great need for part-time studies in the Nigerian universities. This need was short lived when the governing body of the Nigerian Universities; Nigerian University Commission (NUC) put a stop to this viable means of acquiring knowledge. The hope of Nigerians was salvaged with the advancement in technology, especially in the area of Information and Communication Technology (ICT). With these inventions, we have what is known as the virtual classroom (VC), a medium that works like the traditional classroom system. The virtual class room is a teaching and learning environment located within a computer-mediated communication system which supports collaborative learning. The virtual classroom has three prominent components: A Virtual Instructor, a Server and Virtual Students. These three components interact together to form the Virtual Classroom System (VCS). This paper shows how virtual classroom can be harnessed as a replacement for part-time studies in Nigeria with the aid of synchronous and asynchronous tools. The virtual instructor sends lecture notes and materials through the server (internet) and the virtual student receives the note and sends feed back through the server. The server here serves as the point of contact between the virtual lecturer and the virtual student just like what is obtainable in a traditional classroom.
Open Access
Inference of the Stochastic SIR Household Epidemic Model with Misspecification and Misclassification
(Department of Mathematics, Nasarawa State University Keffi, 2021-08-08)
Model estimate and their functions are affected by wrong choice of the infectious period distribution*, *T/ when the actual one is unknown. This is a misspecification problem which is often accompanied with biased and ijmprecise estimates as discussed in [16],. which may be taking for misclassification of the final size epidemic data . This work, examined these problems using simulations by assuming constant infectious period, Tj =4.1 and then estimated with Tj infectious period distribution, assumed; as r(2,2.05) for the household .epidemic and vice under. 'This is extended to cases wheri the final size data is misclassified, as studied in [16]. The maximum, likelihood estimates ancl the model fitness to the final size data are examined and compared under these conditions. We found that, in the two cases, the estimates biased and imprecise. Thus, model that fit poorly to the final size data is often obtained. However, the choice of appropriate model that fit better to the final size data, given these scenarios are suggested.
Open Access
Influence of Environmental Pollution on the Transmission Dynamics of Infections
(Department of Mathematics, Nasarawa State University Keffi, 2014-01-06) Umar, Mallam Abdulkarim
Open Access
Mathematical Model for Vertical Transmission of HIV/AIDS, in a Homogeneous Mixing Population
(Department of Mathematics, Nasarawa State University Keffi, 2007-03-03) Umar, Mallam Abdulkarim; Ndakwo, Husseini S.
Open Access
Mathematical prototype for the control of malaria by interrupting the life cycle of the Anopheles mosquito through the use of biological enemies in the larva, pupa and adult stages
(Department of Mathematics, Nasarawa State University Keffi, 2023-01-19) Emmanuel, Atanyi Yusuf; Oduwole, H.K.; Umar, Mallam Abdulkarim
Mathematical prototype to fight malaria by interrupting the life cycle of the Anopheles mosquito through the use of biological enemies in the larval, pupal and adult stages has been derived to eradicate larvae, pupae and adult Anopheles mosquitoes using natural predators. The new model is a control flowchart of the predator-prey interaction model in the mosquito life cycle, considering an open population of mosquitoes and predators. These models provide a osolid understanding of malaria control in our environment, especially when models are based on vector population ecology; and a solid understanding of transmission-relevant parameters and variables Model equations were derived using parameters and variables from the model Stability analysis of free equilibrium states was analyzed simultaneously using equilibrium point, Maple software, elimination and substitution methods. The number of larvae that pupate is almost zero, and the number ofp upae that turn into adults is minimal, and the number of adults that'escape to the vector stage is negligible, which means that thd life cycle could be disrupted at larval, pupal and adult stages with the introduction of natural enemies, with the natural implication 'there will be no adult Anopheles mosquito for transmission of the malaria. The contribution of this research to knowledge is to produce the model or the mathematical formula and the biologically sound methods that will contribute to the eradication of the adult Anopheles mosquito, which will also lead to° the eradication of malaria in our society.
Open Access
Modeling Horizontal Transmission of HIV/AIDS in an Age-Structured Population
(Department of Mathematics, Nasarawa State University Keffi, 2008-01-08) Umar, M.A.; Olowofeso, E.O.; Ademiluyi, R.A.
Open Access
MODELING THE DYNAMICS OFT4-CELLS COUNTS IN AN HIV-INFECTED INDIVIDUAL
(Department of Mathematics, Nasarawa State University Keffi, 2011-11-11) Umar, Mallam Abdulkarim; Ndakwo, Husseini S.
In this work, w e examined, the stages of HIV progression to AI DS and proposed .a stochastic model of • the number of T 4 - cells counts in an HIV infected person. Tlie mean number of 14 - cells in each disease, phase is obtained and the conditions for a stable level of CD +4 — lymphoc^'te cells in an infected host are suggested. TIic need for antiretroviral therapies to sustain this level is emphasized.
Open Access
Modeling Vaccination and Treatment of HTV/AIDS Epidemics in an Age-Structured Population A
(Department of Mathematics, Nasarawa State University Keffi, 2011-10-10) Umar, Mallam Abdulkarim
We proposed a deterministic age- structured vaccination and treatment model of HIV and AIDS, along the line oflvlcCamy - Foester age-structured population model, with structured population. compartments and-class dependent Activity level (interaction functions), per-capital force of infection, natural mortality rate and disease • induced death rate . The models equations Are reduced to ordinary differential equations. Their, steady state solutions are obtained and examined for local stability. The, disease-free state is found to be stable if vaccination and removal rates are simultaneously maximized. This will lend to a corresponding decrease in the size of HIV infected individuals and the number of infected cases progressing to AIDS.
Open Access
Modelling the Effects of Three Natural Predators on the Aquactic and Adult Stages of Anopheles Mosquitoes in the Control of Malaria Transmission
(Department of Mathematics, Nasarawa State University Keffi, 2023-01-16) Emmanuel, Atanyi Yusuf; Oduwole, H.K.; Umar, Mallam Abdulkarim
Modelling the effects of three natural predators on the aquatic and adult anopheles’ mosquitoes in the control of malaria transmission was derived aimed’at "eradicating anopheles’ larva, pupa and adult anopheles' mosquito by introduction of natural predators ucopepods, tadpoles and purple martins” so that there should not be anopheles ’ adult mosquito for malaria transmission in our society. The new model is a control flow diagram of predator-prey interaction model in mosquito life-cycle. The population is sub-divided based on mosquito life- cycle and natural predators. Under a mosquito life-cycle, the population is divided into four compartments', * Egg compartment E(t), Larva compartment L(t), Pupa compartment P(t), and Adult compartment A(t), and natural predators, it is divided into three compartments’, namely; Copepods CP(t), Tadpoles TP(t) and Purple martins PM (t). From the stability analysis of steady state we observed that the model free equilibrium state is stable, implies that the equilibrium point or steady state is stable and the stability of the model (1) - (10) means, there will not be anopheles adult mosquito ip our society for malaria transmission and from the idea of Beltrami’s conditions and Diekmann condition, we observed that the Determinant of the Jacobian matrix is greater than zero {Det(J) > 0}, Trace of the Jacobian matrix is less than zero {Tr(j) < 0} and R0 < 1 which implies that the model disease free equilibrium state is stable. Hence the number of larva that transform to pupa ♦ is almost zero and the number of pupa that develop to adult is minimal and number of adult that escape to vector stage are inconsequential and microscopic and that means the life-cycle could be broken at the larva, pupa, and adult stages with the introduction of natural predators, with the natural implication there will not be anopheles adult mosquito for malaria transmission and we also use maple for symbolical and numerical solution and presented the results graphically.
Open Access
On Cardinality in Finite Semigroup of Full Order-Preserving ContractionsOn Cardinality in Finite Semigroup of Full Order-Preserving Contractions
(Department of Mathematics, Nasarawa State University Keffi, 2022-04-04) Habibu, Shamsuddeen; Umar, Mallam Abdulkarim; Oduwole, H.K.; Ibrahim, Yusuf Kakangi; Braimoh, Jaiyeola Olorunshola
In this work, we considered the semigroup OCTn consisting of all mappings of a finite set Xn = {1, 2, 3,---- , n} which are both order - preserving and contraction, that is mapping a •• Xn -» Xn such that, for all x ,y E Xnfx- x a