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Item Open Access On Number of Idempotent Elements in Finite Semigroup of Full Order -Preserving Contractions(Department of Mathematics, Nasarawa State University Keffi, 2021-10-28) Amadu, Kabiru; Umar, Mallam Abdulkarim; Habibu, Shamsuddeen; Bello, Asma'uIn this paper, we considered the semigroupOCTn consisting of all mappings of a-finite set Xn - (1, 2, 3, - - - , n} which are both order — preserving and contraction, that is mappinga : Xn -» Xn ^ ich that, for all ,y e Xn, x => x a £ y a,anc | x a — *y a <1 x—y| . In particular, we proposed a closed form formula for the number of idempotent elements in OCTn, (that is elements oc satisfying oc2=oc).Item 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 OlorunsholaIn 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 aItem Open Access An Optimal Control Analysis of Malaria by Coinfection Model(Department of Mathematics, Nasarawa State University Keffi, 2022-07-03) Ashafa, S.U.; Oduwole, H.K.; Umar, Mallam AbdulkarimMalaria and Hepatitis B Virus (HB V) are diseases that poses serious challenges health wise in the world especially in countries that are developing:-Both diseases belong to t the most widespread diseases, and therefore, a major public health concerns in tropical developing countries. In this research, a mathematical model showing dynamics of Co infection of Malaria and HBV diseases was developed using ordinary differential equations which consists of 9 compartments. The study covers the model*s futdre solution positivity, model invariant region and disease-free points. The next generation matrix method was used to compute the basic reproduction number, R0, for the coinfection model using and the disease free equilibrium point and was shown to be Locally Asymptotically Stable ifJl0 < 1 and unstable if > 1. Then, the coinfection model was extended to optimal control by incorporating four control interventions. The optimality System was obtained using the Pontryagin’s maximum principle. Simulation of the optimality system was done and five strategies was proposed to qheck the effect of the controls. First, prevention only for both diseases was considered, and the result shows that, applying prevention control has a great impact in bringing down the expansion of malaria, HBV infection, and their coinfection in the specified period of time. Other approaches are prevention effort for malaria and treatment effort for HBV infection, prevention effort for HBV infection and treatment effort for malaria, treatment effort for both diseases, and using all interventions. We obtained that the listed strategies were effective in ,,minimizing the expansion of Malaria HBV coinfeciious population in the specified period of time.Item 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 AbdulkarimModelling 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.Item Open Access The Pictorial Integral of Malaria Control with Maple(Department of Mathematics, Nasarawa State University Keffi, 2023-01-03) Emmanuel, Atanyi Yusuf; Oduwole, H.K.; Umar, Mallam AbdulkarimThe pictorial integral of malaria control with maple has bepn derived to eradicate larvae, pupae and adult Anopheles mosquitoes using natural predators to eradicate "copepods, tadpoles and purple swallows" ( an organism that eats mosquitoes). 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 solid 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. Therefore, the number of larvae that pupate is almost zero, and the number of pupae that turn into adults is minimal, and the number of adults that escape to the vector stage is negligible, this means that the life cycle could be disrupted at the end of the larval, pupal and adult stages with the introduction of natural enemies, with the natural implication that there will be no adult Anopheles mosquito for transmission of the malaria, and we also use maple for the symbolic and numerical solution and graphically presented the results. . The contribution of this research to knowledge is to produce 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. ,Item 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 AbdulkarimMathematical 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.Item Open Access Influence of Environmental Pollution on the Transmission Dynamics of Infections(Department of Mathematics, Nasarawa State University Keffi, 2014-01-06) Umar, Mallam AbdulkarimItem 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, BilkisuIn 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.Item 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 AbdulkarimItem 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 AbdulkarimA 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.Item 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 AbdulkarimWe 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.Item Open Access Two Sex Model for Horizontal Transmission of STD in a Constants Homogeneous mixing population(Department of Mathematics, Nasarawa State University Keffi, 2007-01-06) Umar, Mallam AbdulkarimIn this paper, we examined llic two sex model ofl-lsich4 and Kimbir cl a/..', and extended these models to study transmission dynamics of socially ti'nnsmitlcd diseased STD ) in a proportional mixing population with vital dynamics, and no disease induced death. Our major focus is to reduce the cost of disease intervention towards eradication of curable STD. like gonorrhea and others, which is heller achieved through preventive measure rather than Irentmenl. a concept dial is cheaper and affordable, in developing economies like ours. We have proposed a variant of the Kimbir el a/..5 model of prevention of 111 V / AIDS in a varying population to study the transmission dynamics of STD. Stability of the equilibriums slates lire discussed via the reproductive ratio of the infection and the Routh -1 Iunviiz.conditions on the model parameters. Threshold conditions for eradication and persistent of the disease arc derived nod interpre tation are discussed. 'Item 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 AbdulkarimItem Open Access Age-Structures Epidemic Model for Transmission Dynamics of Tuberculosis(Department of Mathematics, Nasarawa State University Keffi, 2007-02-02) Umar, Mallam AbdulkarimItem 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 AbdulkarimWe 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.Item 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.Item 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.Item 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.Item 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.Item Open Access A RESPONSE SURFACE OUTLOOK FOR OPTIMALITY IN RICE PRODUCTION IN THE NORTHERN NIGERIA STATE OF KANO(Department of Mathemaics, Nasarawa State University, Keffi., 2014-01-01) Adehi, Mary Unekwu; Dikko, H. G.; David, I. J.With the aid o f response surface methodology a response surface model was developed for a mixed level two factors (irrigation at 3 levels and fertilizer at 4 levels) experimental design on rice yield in Kano State which is situated in the northern part o f Nigeria. A quadratic surface was fitted and a canonical analysis was carried out to determine the surface in the region of maximum. Goodness off it tests was performed in assessing the model where the R~, adjusted-R2, and residual plots were done and it was revealed that the model is adequate.