Open Access Original Research Article

Paride O. Lolika, Mlyashimbi Helikumi

Brucellosis is one of the most common zoonotic infections globally. It affects humans, domestic animals and wildlife. In this paper, we conduct an intrinsic analysis of human brucellosis dynamics in non-periodic and periodic environments. As such we propose and study two

mathematical models for human brucellosis transmission and control, in which humans acquire infection from cattle and wildlife. The first model is an autonomous dynamical system and the second is a non-autonomous dynamical system in which the seasonal transmission of brucellosis

is incorporated. Disease intervention strategies incorporated in this study are cattle vaccination, culling of infectious cattle and human treatment. For both models we conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction

numbers. Using sensitivity analysis we established that R0 is most sensitive to the rate of brucellosis transmission from buffalos to cattle, the result suggest that in order to control human brucellosis there is a need to control cattle infection. Based on our models, we also formulate

an optimal control problem with cattle vaccination and culling of infectious cattle as control functions. Using reasonable parameter values, numerical simulations of the optimal control demonstrate the possibility of reducing brucellosis incidence in humans, wildlife and cattle, within

a finite time horizon, for both periodic and non-periodic environments.

Open Access Original Research Article

Danniel Dias Augusto

In this work, we study the Magic Polygons of order 3 (P(n; 2)) and we introduce some properties that were useful to build an algorithm to find out how many Magic Polygons exists for the regular polygons up to 24 sides. The concepts of Equivalents Magic Polygons and Derivatives Magic Polygons which allowed to classify, and avoid ambiguity about the representations of such elements, are also introduced.

Open Access Original Research Article

Dereje Gutema Edossa, Alemu Geleta Wedajo, Purnachandra Rao Koya

In this study, a non-linear system of ordinary differential equation model that describes the dynamics of malaria disease transmission is formulated and analyzed. Conditions are derived from the existence of disease-free and endemic equilibria. The basic reproduction number R_{0} of the model is obtained, and we investigated that it is the threshold parameter between the extinction and persistence of the disease. If R_{0} is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable resulting in the disease removing out of the host populations. The disease can persist whenever R_{0} is greater than unity and the conditions for the existence of both forward and backward bifurcation at R_{0} is equal to unity are derived. Sensitivity analysis is also performed and the important parameter that derive the disease dynamics is identified. Furthermore, optimal combinations of time dependent control measures are incorporated to the model, and we derived the necessary conditions of optimal control using Pontryagins’s maximum principal theory. Numerical simulations were conducted using MATLAB to confirm our analytical results. Our findings were that, malaria may be controlled by reducing the requirement rate of mosquito populations and the use of a combination of vaccination, insecticide treated net ITN, indoor residual spray IRS and active treatment or strategy d can also help to reduce the number of populations with malaria symptoms to zero. We also find that the same strategy that is, strategy d proves to be efficacious and cost-effective.

Open Access Original Research Article

Szabo, Zoltan Istvan

Let and be fixed integer numbers. Assume that (a^{2}+b^{2}+c) is divisible by (ab+d) for some natural numbers a and b. Then the value of the fraction $$k ( = {(a^2+b^2+c) \over (ab+d)})$$ remains the same. Statement of this kind will be proved in pp. 1-3 and illustrated on some examples in pp. 3-10. The general method of proofs will be unified and simplified. Computing support will be provided: in pages 11-19 a simple program code is defined with the help of which one can hunt for natural numbers a, b with the same integer values of c, d and k. Here, a number of examples are given as well.

Open Access Original Research Article

Suliman Dawood, Ahmed M. Al-Audhahi, Amin Saif

Aims/ Objectives: In this paper, we study the continuity property in grill topological spaces via generalized G^{\(\omega\) }-closed sets and regular generalized G^{\(\omega\) }- closed sets. These notions are generalized G^{\(\omega\) }- continuous functions which is weak form of g^{\(\omega\) }- continuous functions, generalized G^{\(\omega\)} - irresolute functions, regular generalized G^{\(\omega\) }- continuous functions which is weak form of rg\(\omega\)continuous functions, regular generalized G^{\(\omega\) }- irresolute functions and investigates some of its properties in grill topological spaces.

Open Access Original Research Article

Manoj Kumar, Nisha Kumari

In this paper, we introduce the new definitions and fixed-point theorems for \((\hat{\alpha}-\hat{\psi})\)-Geraghty contraction with an aid of simulation function \(\zeta:[0, \infty) \times[0, \infty) \rightarrow \mathbb{R}\) in generalized metric space satisfying the following condition:

if \(\exists \hat{\beta} \in \mathcal{F}\) such that for all \(r, s \in \mathfrak{X}\), then we have

\(\zeta[\hat{\alpha}(r, s)(d(\mathcal{P} r, \mathcal{P} \mathcal{s})), \hat{\beta}(\hat{\psi}(d(r, s))) \hat{\psi}(d(r, s))] \geq 0\), where \(\hat{\psi} \in \hat{\Psi}\) and \((\mathfrak{X}, d)\) is a generalized metric space. An example is also given to support our results.

Open Access Original Research Article

Chunhua Feng, Orjul Pogue

Various enterprise interaction models with or without time delays appeared in the literature. The stability and Hopf bifurcation for one or two delays in the models were studied by many researchers. However, the periodic oscillation for a four time delays enterprise interaction model is still an open problem due to the complexity of the bifurcating equation. In this paper, by means of the mathematical analysis method, some sufficient conditions to guarantee the existence of periodic oscillatory solution for the four time delays model are obtained. An open problem is solved. Computer simulation is given to demonstrate the present results.

Open Access Original Research Article

Osakwe Charles Nnamdi, Olobayo Solomon Adelaja, Omowo Babajide Johnson

This paper presents the comparison of the two Adams methods using extrapolation for the best method suitable for the approximation of the solutions. The two methods (Adams Moulton and Adams Bashforth) of step k = 3 to k = 4 are considered and their equations derived. The extrapolation points, order, error constant, stability regions were also derived for the steps. More importantly, the consistency and zero stability are also investigated and finally, the derived equations are used to solve some non-stiff differential equations for best in efficiency and accuracy.

Open Access Original Research Article

Jones Apawu

This study which is part of a project examined the perception of preservice mathematics teachers on Technological Pedagogical Content Knowledge (TPACK) levels and the relationship among the TPACK components. The research design employed was descriptive interspersed with correlational design. The population for this study was preservice mathematics teachers at the University of Education, Winneba of Ghana. The study employed the purposive sampling technique specifically homogeneous sampling technique to select level 300 mathematics teachers from the department of mathematics education of the University of Education, Winneba. In all, the level 300 (year 3) students were 183. Raosoft sample size software tool was used to determine a sample size of 125 for the study. After the determination of the sample size, simple random sampling technique was used in selecting the respondents for the study. Questionnaire was used as instrument to collect data. Data collected were analysed quantitatively. Results showed that: (i) the perceived knowledge level of the preservice mathematics teachers on TPACK and its components were moderate and high; (ii) there were positive relationships among the components of TPACK, and all of the relationships were statistically significant. Recommendations were thereof made accordingly.

Open Access Original Research Article

Justice Kangah, Henry Otoo, Joseph Acquah

The aim of this study was to develop a hybrid optimization model for solving the routing problem identified at Zoomlion Ghana Limited in Shama district in the Western Region of Ghana. Two main optimization models were considered: Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). A hybrid algorithm was developed from the two by merging crossover and mutation operators of GA with PSO. A sample of 20 breakpoints was run through 10,000 iterations for all the models and the results of the proposed hybrid model was compared with PSO and GA separately. The optimal results of PSO, GA and the proposed models are 1160.6km, 1190.3km and 1132.3km respectively. The proposed model’s results were also compared with other hybrid models to test the robustness of the new model. This result was achieved because the new model eliminates the low convergence rate in PSO and also prevents it from easily falling into local optimum in high-dimensional space and the inclusion of crossover and mutation operators of GA improves the diversity of the iterations. After the iterations, PSO reduced a field distance of 1700 km to 1160.6 km within 780.4098 seconds. GA on the other hand reduced the same field distance of 1700 km to 1190.3km within 397.3308 seconds. The proposed hybrid model reduced the same field distance from 1700 km to 1132.3 km within 550.2527 seconds. This indicates that the proposed hybrid model performed better than PSO and GA separately. A performance test between the proposed hybrid model and other hybrid models showed that merging crossover and mutation operators into PSO gives a better optimal result. MATLAB was used for the iterations.

Open Access Original Research Article

. Deepika, Manoj Kumar

In this paper, we prove some common fixed point theorems for four weakly compatible self-maps along with (*CLR*) property in fuzzy 2- metric spaces. Our results are the improved version of the theorems proved by Shojaei et al. [1] in 2013, since our results does not require closedness of ranges of subsets of *X*.

Open Access Original Research Article

Hope Osogom Okolie, Adolphus Okechukwu Nwaoburu

This study analyses Laplace transforms method on a system of partially coupled differential equations for non-isothermal chemically reactive flow through a cylindrical channel. The dimensionless governing equation for velocity, temperature and concentration was solved using Laplace Transform. Various parameters such as Temperature boundary parameter, Concentration boundary parameter, Cooling Parameter, Grashof number, pressure gradient and Magnetic field, as well as perturbation parameter had an effect on the velocity profile as well as temperature and concentration profile. The graphs were obtained with the results showing that an increase in the temperature boundary parameter resulted to an increase in the temperature of flow, an increase in perturbation parameter resulted to an increase in temperature profile of a body and an increase in Grashof temperature number results to an increase in the velocity of the body.

Open Access Original Research Article

Alabi Oluwapelumi, Oluwagunwa Abiodun Peter, Ogundele Joshua Oluwafunminiyi, Benard O. Muse

Non response is a common problem in a survey process. Therefore, it is necessary to find a way out of handling non response whenever arises. The current study proposed difference-cum exponential ratio-type estimator for estimation of general parameters using auxiliary information which is defined in two situations of non response. A conventional estimator t^{*}_{(a,b)} is used to define population constants including population mean, standard deviation, coefficient of variation and mean square. The expression of bias and mean square error of the proposed estimators are obtained up to the first order of approximation for situation I and situation II. To compare the efficiency of the proposed estimators over the existing ones, an empirical study is carried out using real and simulated data sets. Both the theoretical and empirical study shows that the proposed class of estimators outperformed other existing estimators.

Open Access Original Research Article

M. Hassan, M. Aslam, A. Asghar, S. Ahmad, I. Rasheed, F. A. Shah, A. Qayyum

Some real life mathematical problems can be converted in the form of nonlinear equations. Solving such problems by analytical approaches is dificult in many situations. Hence numerical solution is the best way in this case. In this paper, a twelth-order iterative scheme for solving

nonlinear equations is presented and analyzed in terms of eciency. The new scheme is derived from the well-known King's method with order of convergence eight. We extend eighth-order King's method to an iterative method with memory of order 12:16 by using famous Newton's

interpolating polynomial of degree 6 to avoid the derivative used in King's method. The new derived method is a three-step and is totally derivative free with twelth order of convergence. The method requires four functional evaluations at each iteration introducing high efficiency index of (12:16) 1/4 = 1:8673: Convergence order of new method is also studied. It is achieved by using matrix method of Herzburger. Numerical results are also provided to support theoretical analysis. Comparison of the derived scheme with previously well-known iterative schemes of the same order is also presented. As diffierent schemes of same order has efficiency index of (12)1/6 = 1:5131 because they requires six functional evaluations at each iteration, hence the proposed scheme is better than other schemes.

Open Access Original Research Article

Agada Apeh Andrew, Samuel Musa, Solomon Ortwer Adee

In this research we formulated the Plants diseases model with the aim of studying the dynamics of the use of *lysobacter antibioticus *for prevention and control of rice bacterial blight. The disease free equilibrium state of the models was also obtained by equating each of the equation of the modified model to zero and simplifying. The basic reproduction number for the model was derived using the next generation matrix approach. Numerical simulation was carried out using MATLAB2018a to virtualize the dynamics of the model. Five numerical experiment was carried out and it was shown that biocontrol help to reduce the population of the pathogen as well as act as treatment for those that are already exposed or infected with the disease. It was also observed that the biocontrol agent provide immunity to rice plants against been infected with the disease. Finally, we observed from the simulation that the earlier the control is introduced the more protection plants will receive.

Open Access Original Research Article

Fulgensia Kamugisha Mbabazi, Yahaya Gavamukulya, Awichi Richard Opaka, Peter Olupot-Olupot, Samson Rwahwire, Saphina Biira, Livingstone S. Luboobi

The human{infecting corona virus disease (COVID-19) caused by the novel severe acute respiratory syndrome corona virus 2 (SARS-CoV-2) was declared a global pandemic on March11^{th}, 2020. Different countries adopted different interventions at different stages of the outbreak, with social distancing being the first option while lock down the preferred option for attening the curve at the peak of the pandemic. Lock down aimed at adherence to social distancing, preserve the health system and improve survival. We propose a Susceptible-Exposed- Infected-Expected recoveries (SEIR) mathematical model for the prevention and control of Covid-19 in Uganda. We analyze the model using available data to find the infection-free, endemic/infection steady states and the basic reproduction number. We computed the reproductive number and it worked out as *R*_{0} = 0:468. We note that *R*_{0} is less than unity, thus forecast that several strategies in combination (including travel restrictions, mass media awareness, community buy-in and medical health interventions) will eliminate the disease from the population. However, our model predicts a recurrence of the disease after one year and two months (430 days) thus the population has to be mindful and continuously practice the prevention and control measures. In addition, a sensitivity analysis done showed that the transmission rate and the rate at which persons acquire the virus, have a positive in uence on the basic reproduction number. On other hand the rate of evacuation by a rescue ambulance greatly reduces the reproduction number. The results have potential to inform the impact and effect of early strict interventions including lock down in resource limited settings and social distancing.

Open Access Original Research Article

Alabodite Meipre George, Evans Fiebibiseighe Osaisai

The behaviour of warm water discharged at 4\(^{\circ}\)C through lock-exchange in cold fresh water was investigated numerically, fixing lock volume at the centre of the domain. This investigation as presented here is practical and can also enhance policy making towards the protection of the aquatic ecosystems. Though, the aim of this study is to better fathom and as well, gain more insight into such ows. Our results have shown a speedy movement of the lock volume at the centre of the domain with a leading head at two front on the oor which resulted in a hat shape within the first few time frame. Fluid movement in the second phase is independent of the back reflected waves. We were able to identify two regimes of ow with a stepwise decreasing velocity in the second phase. Our results have shown that velocity with which the current travels with in the second regime is higher within the first time frame as compared to those with the effect of back reflected waves. One major factor that is responsible for decrease in the velocity here is mixing. Previous results have also shown that the front velocities in the collapsing phase are independent of lock volume. But this seem not to be the same here because fluid movement in the first phase (regime) is not totally independent of the lock volume and its position here, where density difference is as a result of temperature. However, our scaling power laws here in the second phase show some variations with previous studies where we have effect of back reflected waves. But results in the collapsing phase here are in strong agreement with those in the first phase of our previous simulations with small lock volume. Generally, the spreading behaviour here is dependent on lock volume, barrier position, density difference and Reynolds number.

Open Access Original Research Article

Preeti Bhardwaj, Manoj Kumar

In this manuscript, we shall introduce the implicit relation on \(\mathbb{R}_+\)^{4}. Using this implicit relation, we shall prove fixed point and common fixed point theorems on partial metric spaces.

Open Access Original Research Article

Joseph Bonazebi Yindoula, Yanick Alain Servais Wellot, Bamogo Hamadou, Francis Bassono, Youssouf Paré

We have solved the Schrödinger equation with the HPM method and the SBA method. We have noticed that with these two methods we find the same result.

Open Access Original Research Article

Cristian A. Tobias, Crispina V. Diego

Mathematics performance has been recognized as vital in the curricula in most of the country not only for academic success but also for its efficient application in everyday life. The study aimed to determine a significant relationship between the student, teacher, environmental, and parent involvement factors and students’ mathematics performance in a Catholic university for the Academic Year 2020-2021. Also, it determined which factors predict the mathematics performance of students. Using descriptive-correlational research design, 123 Grade 8 students answered a researcher-made test questionnaire to assess students' level of mathematics performance; these were interpreted using the Department of Education’s scale of standard. Furthermore, a researcher-made survey questionnaire was used to examine the level of extent of the four factors influencing the students’ mathematics performance, the correlation between these factors and students’ mathematics performance, and predictors of students’ mathematics performance. The assembled information was dealt with measurably utilizing mean, standard deviation, Kruskal-Wallis, Pearson product-moment correlation, Spearman's rho, and multiple linear regression. Results revealed that students’ level of mathematics performance was fairly-satisfactory (M=79.77, SD=8.96). The extent of influence of the factors is moderate (M=3.28, SD=0.33). There is a significant correlation between the extent of influence of factors as a whole and Mathematics performance [r(121)=0.235, p=0.009]. The results of the regression indicated that the predictor explained 7.3% of the variance [F(1, 121)=9.540, p=0.002, R^{2}=0.073)]. The individual predictors were further examined and indicated that the teacher factor [β=6.440, t=3.089, p=0.002] significantly predicts Mathematics performance. This study provides significant insights into the vital role teachers play in finding a lasting solution to the students' perennial poor performance issues in mathematics in this new normal.

Open Access Original Research Article

Charles Kojo Assuah, Louis Osei, Gershon Kwame Mantey

Using inquiry-based learning, this study investigates high school students' achievement in plane geometry. It employed the pre-test-post-test randomized experimental design, often known as the control group design. The participants (students) were randomly assigned to one of two classes/groups and were given either an intervention (the treatment group) or no intervention (the control group). One hundred and twenty (120) third-year high school students of similar mathematical aptitude (a control group = 60 students; an experimental group = 60 students) were chosen from a high school in Ghana's central region. Shapiro-Wilk had a p-value greater than.05 (p >.05) for each statistic, indicating that both the pre-test and post-test scores were normally distributed, before and after the test. The findings of the independent samples t-test showed that there was no statistically significant difference between the experimental and control pre-test scores (t = -.48, p >.05, C. I = [-1.78, 1.08]). The one-way ANOVA after inquiry-based learning showed a significant effect on student scores, F (1, 118) = 363.41, p < .05). Furthermore, independent samples t-test findings for the post-test showed statistically significant differences between the experimental and control post-test scores (t = -22.68, p < .05, C. I = [-24.29, 20.40]). The study's implications are that students can make their own connections with the content they learn. They may also comprehend the themes rather than simply recalling rules and formulas. The study concludes that inquiry-based learning improves senior high school students’ achievement in plane geometry.

Open Access Original Research Article

Deep Bhattacharjee

Any matrix multiplication is non-commutative which has been shown here in terms of suspension^{‡}, annihilator, and factor as established over a ring following the parameter *k* over a set of elements upto n for an operator to map the ring R to its opposite R^{op }having been through a continuous representation of permutation upto n-cycles being satisfied for a factor f along with its inverse f^{-1 }over a denoted orbit γ on k-parameterized ring justified via suspension η ∈ η^{0}, η^{1 }implying the same global non-commutativity for the annihilator A. This will be used for the construction of the genus–alteration scenario where the suspension η^{0 }acting with its opponent η^{1} on any topological space J can alter the geometry making a change in the manifolds for taking over the Boolean (1,0) satisfying the concerned operations.

Open Access Original Research Article

Aimin Xu

In this paper, we establish a new asymptotic expansion related to the Wallis ratio. By using the exponential Bell polynomials, we show that the coefficients of the asymptotic expansion can be recursively determined. In addition, an explicit expression for the coefficients is given. Our results improve and generalize the existing ones [1].

Open Access Original Research Article

Peter E. Ezimadu, Sophia O. Ezimadu

It has been established that trade credit can be influenced by a lot of factors. However, no specific function has been used to neither represent these factors nor consider their effects. This paper considers a supplier-retailer Stackelberg game in which the supplier as the channel leader supplies credit goods to the retailer who in-turn sells to the consumers. It uses a credit function based on credit period, supplier’s price margin and product promotion effort to model the players’ payoffs. The work considers two game scenarios: a situation involving the provision of trade credit and a situation without trade credit. The work obtains a closed-form solution for the credit period for the credit provision scenario, and the promotion efforts and payoffs for both scenarios, and shows that credit period prolongation may not be in favour of the retailer, and that the retailer can attain a larger payoff than the supplier. It also shows that the retailer’s margin is very crucial for both channel scenarios, and observes that the players are better-off with trade credit.

Open Access Original Research Article

Deep Bhattacharjee

This paper aims to create a class [O] concerning the groups associated with Gromov hyperbolic groups over correspondence and equivalence through Fuchsian, Kleinian, and Schottky when subject to Laplace – Beltrami in the Teichmüller space where for the hyperbolic 3-manifold when the fundamental groups of Dehn extended to Gromov – any occurrence of Švarc-Milnor lemma satisfies the same class [O] for quotient space and Jørgensen inequality. Thus the relation (and class) extended to Mostow – Prasad Rigidity Theorem in a finite degree isometry concerning the structure of the commensurator in higher order generalizations suffice through CAT(*k*) space. The map of the established class [O] is shown at the end of the paper.

Open Access Original Research Article

Bationo Jérémie Yiyuréboula, Bassono Francis

In this paper, we implement Regular Perturbation Method (RPM) for the Solving fractional diffusion and diffusion-convection equations, in order to determine the analytical solutions of some linear fractional diffusion and linear fractional diffusion-convection equations. In general, the solving using this method allow to obtain exact or approximate solutions. For the case of the diffusion and diffusion-convection equations solved in this document, the solutions obtained are exact. By comparing these solutions with those obtained by other researchers using other methods for a certain value of the parameter α, we obtain the same results.

Open Access Original Research Article

Levi Otanga Olwamba

This paper is committed to the study of absolute continuity of non negative functions with respect to vector measures. Almost everywhere properties are applied to establish boundedness, measurability and convergence of sequences of measurable functions. The measurability of sets with respect to vector duality functions with values in a Hilbert space is considered.

Open Access Original Research Article

Binandam Stephen Lassong, Isaac Adu, Munkaila Dasumani, Kwesi Frempong Sarfo

Computing the inverse of 3 x 3 square matrices using known methods such as Gauss-Jordon Method needs more time. During examination, candidates need to go through long process to find the inverse of 3 x 3 square matrices. Some students also find it very difficult and confusing when using Gauss-Jordon and Adjugate methods to find the inverse of the matrix. In this note, we present a new method that is simple and easy way of finding the inverse of 3 x 3 square matrices. We further give some applications of matrices in the real world phenomenon. Many other challenging problems can be addressed surprisingly by applying this strategy.

Open Access Original Research Article

Brenda Onyango, George O. Lawi, Frankline Tireito

Discrete age structured models provide a better approach of discussing the spread of infectious diseases with infectivity, mortality and recovery being age dependent. Measures such as vaccination are carried out in discrete time or applied to individuals in certain age-groups. In the past Ebola Virus disease (EVD) outbreaks, infections in children under 5 years was associated with high mortality rates. In this work, a general discrete SVEIR epidemic model with age structure is formulated. The study establishes the existence of the endemic equilibrium and further shows that it is globally stable using the graph theoretic approach on the method of Lyapunov functions. The model is then applied to EVD dynamics to study the impact of vaccination on the age structured population. The numerical simulation of the discrete age case scenario demonstrates that vaccination of children under 5 years of age against EVD has a great impact in reducing their susceptibility because of the active immune system as compared to the older population who have a poor response to vaccine immunity. For the older population, vaccination is not very eective in reducing their susceptibility.

Open Access Review Article

Nikolaos Halidias

Aims/Objectives: In this review article we study the computation of the minimum polynomial

of a matrix A and how we can use it for the computation of the matrix An. We also describe

the form of the elements of the matrix A^{-n} and we will see that it is closely related with the

computation of the Drazin generalized inverse of A. Next we study the computation of the

exponential matrix and nally we give a simple proof of the Leverrier - Faddeev algorithm for

the computation of the characteristic polynomial.