# Department of Mathematics

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Item An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection(De Gruyter Open, 2013) Obaid, Hasim; Ouifki, Rachid; Patidar, Kailash C.Show more We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.Show more Item An assessment of the age reporting in Tanzania population census(Academic Research Publishing, 2012) Mwambene, Eric; Appunni, Sathiya Susuman; Hamisi, Hamisi F.; Lougue, Siaka; Regassa, Nigatu; Ogujiuba, KanayoShow more The objective of this paper is to provide data users with a worldwide assessment of the age reporting in the Tanzania Population Census 2012 data. Many demographic and socio-economic data are age-sex attributed. However, a variety of irregularities and misstatements are noted with respect to age-related data and sex ratio data because of its biological differences between the genders. Noting the misstatement / misreporting, inconsistence of age data regardless of its significant importance in demographic and epidemiological studies, this study assess the quality of the 2012 Tanzania Population and Housing Census data relative to age. Data were downloaded from Tanzania National Bureau of Statistics. Age heaping and digit preference were measured using summary indices viz., Whipple‟s index, Myers‟ blended index, and Age-Sex Accuracy index. The recorded Whipple‟s index for both sexes was 154.43, where males had the lower index of about 152.65 while females had the higher index of about 156.07. For Myers‟ blended index, the prefrences were at digits „0‟ and „5‟ while avoidance were at digits „1‟ and „3‟ for both sexes. Finally, the age-sex index stood at 59.8 where the sex ratio score was 5.82, and the age ratio scores were 20.89 and 21.4 for males and female respectively. The evaluation of the 2012 Population Housing Censes data using the demographic techniques has qualified the data as of poor quality as a result of systematic heaping and digit preferences/avoidances in recorded age. Thus, innovative methods in data collection along with measuring and minimizing errors using statistical techniques should be used to ensure accuracy of age data.Show more Item Automorphism groups of graph covers and uniform subset graphs(Taylor and Francis Group, 2018) Mumba, Nephtale; Mwambene, EricShow more Hofmeister considered the automorphism groups of antipodal graphs through the exploration of graph covers. In this note weextend the exploration of automorphism groups of distance preserving graph covers. We apply the technique of graph covers todetermine the automorphism groups of uniform subset graphsΓ(2k,k,k−1) andΓ(2k,k,1).The determination of automorphismgroups answers a conjecture posed by Mark Ramras and Elizabeth Donovan. They conjectured that Aut(Γ(2k,k,k−1))∼=S2k×,whereTis the complementation mapX↦→T(X)=Xc={1,2,...,2k}\X,andXis ak-subset ofΩ={1,2,...,2k}.Show more Item Binary codes and partial permutation decoding sets from the odd graphs(Walter de Gruyter, 2014) Fish, Washiela; Fray, Roland; Mwambene, EricShow more For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k} , the set of all k-subsets of Ω = {1, 2, . . . , 2k + 1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = ∅. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement O(k), is investigated.Show more Item Binary codes from m-ary n-cubes Q(n) (m)(American Institute of Mathematical Sciences, 2021) Key, Jennifer D.; Rodrigues, Bernardo G.Show more We examine the binary codes from adjacency matrices of the graph with vertices the nodes of the m-ary n-cube Qmn and with adjacency de ned by the Lee metric. For n = 2 and m odd, we obtain the parameters of the code and its dual, and show the codes to be LCD. We also nd s-PD-sets of size s + 1 for s < m1 2 for the dual codes, i.e. [m2; 2m 1;m]2 codes, when n = 2 and m 5 is odd.Show more Item Closure, interior and neighbourhood in a category(Hacettepe University, 2018) Holgate, David; Slapal, JosefShow more The natural correspondences in topology between closure, interior and neighbourhood no longer hold in an abstract categorical setting where subobject lattices are not necessarily Boolean algebras. We analyse three canonical correspondences between closure, interior and neighbourhood operators in a category endowed with a subobject structure. While these correspondences coincide in general topology, the analysis highlights subtle di erences which distinguish di erent approaches taken in the literature.Show more Item Codenseness and openness with respect to an interior operator(Springer Nature, 2021) Assfaw, F.S; Holgate, DShow more Working in an arbitrary category endowed with a fixed (E, M) -factorization system such that M is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i. Some basic properties of these morphisms are discussed. In particular, it is shown that i-codenseness is preserved under both images and dual images under morphisms in M and E, respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i. Notably, we obtain a characterization of quasi i-open morphisms in terms of i-codense subobjects. Furthermore, we prove that these morphisms are a generalization of the i-open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided.Show more Item Codes from adjacency matrices of uniform subset graphs(Springer, 2017) Fish, W.; Key, J. D.; Mwambene, EricShow more Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Γ(n,k,r)Γ(n,k,r) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from Γ(n,3,r)Γ(n,3,r) for p≥5p≥5 , thus extending earlier results for p=2,3p=2,3 .Show more Item Contour integral method for European options with jumps(Elsevier, 2013) Ngounda, Edgard; Patidar, Kailash C.; Pindza, EdsonShow more We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.Show more Item Control and elimination in an SEIR model for the disease dynamics of Covid-19 with vaccination(AIMS Press, 2023) Witbooi, Peter Joseph; Vyambwera, Sibaliwe Maku; Nsuami, Mozart UmbaShow more COVID-19 has become a serious pandemic affecting many countries around the world since it was discovered in 2019. In this research, we present a compartmental model in ordinary differential equations for COVID-19 with vaccination, inflow of infected and a generalized contact rate. Existence of a unique global positive solution of the model is proved, followed by stability analysis of the equilibrium points. A control problem is presented, with vaccination as well as reduction of the contact rate by way of education, law enforcement or lockdown. In the last section, we use numerical simulations with data applicable to South Africa, for supporting our theoretical results. The model and application illustrate the interesting manner in which a diseased population can be perturbed from within itself.Show more Item Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme(SpringerOpen, 2016) Owolabi, Kolade M.; Patidar, Kailash C.Show more In this paper, we consider the numerical simulations of an extended nonlinear form of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew (J Comput Phys 176:430-455, 2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233, 2005), for the time integration of spatially discretized partial differential equations. We demonstrate the supremacy of ETDRK4 over the existing exponential time differencing integrators that are of standard approaches and provide timings and error comparison. Numerical results obtained in this paper have granted further insight to the question "What is the minimal size of the spatial domain so that the population persists?" posed by Kierstead and Slobodkin (J Mar Res 12:141-147, 1953 ), with a conclusive remark that the popula- tion size increases with the size of the domain. In attempt to examine the biological wave phenomena of the solutions, we present the numerical results in both one- and two-dimensional space, which have interesting ecological implications. Initial data and parameter values were chosen to mimic some existing patternsShow more Item Efficient numerical method for a model arising in biological stoichiometry of tumor dynamics(American Institute of Mathematical Sciences, 2019) Kolade, Owolabi; Kailash, Patidar; Shikongo, AlbertShow more In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells are further modified by blood supply through the vascularization process, we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and we observe seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.Show more Item The fibre of a pinch map in a model category(Springer Verlag, 2013) Hardie, Keith A.; Witbooi, Peter J.Show more In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X ∪ CA → X ∪ CA/ X from the mapping cone on a cofibration A → X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.Show more Item Fifth order two-stage explicit Runge-Kutta-Nystrom method for the direct integration of second order ordinary differential equations(Academic Journals, 2012) Okunuga, S.A.; Sofoluwe, A.B.; Ehigie, J.O.; Akanbi, M.A.Show more In this paper a direct integration of second-order Ordinary Differential Equations (ODEs) of the form using the Explicit Runge-Kutta-Nystrom method with higher derivatives is presented. Various numerical schemes are derived and tested on standard problems. The higher-order explicit Runge-Kutta-Nystrom (HERKN) method given in this paper is compared with the conventional Explicit Runge Kutta (ERK) schemes. Due to the limitation of ERK schemes in handling stiff problems, the extension to higher order derivative is considered. The results obtained show an improvement on ERK schemes.Show more Item A fitted numerical method for a model arising in HIV-related cancer-immune system dynamics(Tianjin Polytechnic University, 2019) Kolade, Owolabi; Kailash, Patidar; Shikongo, AlbertShow more The effect of diseases such as cancer and HIV among our societies is evident. Thus, from the mathematical point of view many models has been developed with the aim to contribute towards understanding the dynamics of diseases. Therefore, in this paper we believe by extending a system of delay differential equations (DDEs) model of HIV related cancer-immune system to a system of delay partial differential equations (DPDEs) model of HIV related cancer-immune dynamics, we can contribute toward understanding the dynamics more clearly. Thus, we analyse the extended models and use the qualitative features of the extended model to derive, analyse and implement a fitted operator finite difference method (FOFDM) and present our results. This FOFDM is analyzed for convergence and it is seen that it has has second-order accuracy. We present some numerical results for some cases of the the model to illustrate the reliability of our numerical method.Show more Item A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer(Wiley, 2019) Munyakazi, Justin B.; Patidar, Kailash C.; Sayi, Mbani T.Show more The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above.Show more Item A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems(Spring Verlag, 2013) Munyakazi, Justin B.; Patidar, Kailash C.Show more This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.Show more Item A fitted operator method for a model arising in vascular tumor dynamics(Tianjin Polytechnic University, 2020) Kolade, Owolabi; Kailash, Patidar; Shikongo, AlbertShow more In this paper, we consider a model for the population kinetics of human tumor cells in vitro, differentiated by phases of the cell division cycle and length of time within each phase. Since it is not easy to isolate the effects of cancer treatment on the cell cycle of human cancer lines, during the process of radiotherapy or chemotherapy, therefore, we include the spatial effects of cells in each phase and analyse the extended model. The extended model is not easy to solve analytically, because perturbation by cancer therapy causes the flow cytometric profile to change in relation to one another. Hence, making it difficult for the resulting model to be solved analytically. Thus, in [16] it is reported that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection, reaction processes are highly dominant and the initial data are not smooth. As a result, we construct a fitted operator finite difference method (FOFDM) coupled with non-standard finite difference method (NSFDM) to solve the extended model. The FOFDM and NSFDM are analyzed for convergence and are seen that they are unconditionally stable and have the accuracy of O(Dt +(Dx)2), where Dt and Dx denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are presented.Show more Item A fitted operator method for tumor cells dynamics in their micro-environment(Tianjin Polytechnic University, 2019) Kolade, Owolabi; Kailash, Patidar; Shikongo, AlbertShow more In this paper, we consider a quasi non-linear reaction-diffusion model designed to mimic tumor cells’ proliferation and migration under the influence of their micro-environment in vitro. Since the model can be used to generate hypotheses regarding the development of drugs which confine tumor growth, then considering the composition of the model, we modify the model by incorporating realistic effects which we believe can shed more light into the original model. We do this by extending the quasi non-linear reaction-diffusion model to a system of discrete delay quasi non-linear reaction-diffusion model. Thus, we determine the steady states, provide the conditions for global stability of the steady states by using the method of upper and lower solutions and analyze the extended model for the existence of Hopf bifurcation and present the conditions for Hopf bifurcation to occur. Since it is not possible to solve the models analytically, we derive, analyze, implement a fitted operator method and present our results for the extended model. Our numerical method is analyzed for convergence and we find that is of second order accuracy. We present our numerical results for both of the models for comparison purposes.Show more Item Generalizing the Hilton–Mislin genus group(Elsevier, 2001) Witbooi, Peter J.Show more For any group H, let H be the set of all isomorphism classes of groups K such that K H . For a finitely generated group H having finite commu- Ž .tator subgroup H, H , we define a group structure on H in terms of embed- dings of K into H, for groups K of which the isomorphism classes belong to Ž . H . If H is nilpotent, then the group we obtain coincides with the genus group Ž .GG H defined by Hilton and Mislin. We obtain some new results on Hilton Mislin genus groups as well as generalizations of known results.Show more