Research Articles (Mathematics)

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One-sided maximal inequalities for a randomly stopped bessel process
(Taylor & Francis Group, LLC, 2023) Cloud, Makasu
We prove a one-sided maximal inequality for a randomly stopped Bessel process of dimension (Formula presented.) For the special case when α = 1, we obtain a sharp Burkholder-Gundy inequality for Brownian motion as a consequence. An application of the present results is also given.
Modeling the impact of combined use of Covid Alert SA app and vaccination to curb Covid-19 infections in South Africa
(Public Library of Science, 2023) Kinyili, Musyoka; Munyakazi, Justin B.; Mukhtar, Abdulaziz Y. A.
The unanticipated continued deep-rooted trend of the Severe Acute Respiratory Syndrome Corona-virus-2 the originator pathogen of the COVID-19 persists posing concurrent anxiety globally. More effort is affixed in the scientific arena via continuous investigations in a prolific effort to understand the transmission dynamics and control measures in eradication of the epidemic. Both pharmaceutical and non-pharmaceutical containment measure protocols have been assimilated in this effort. In this study, we develop a modified SEIR deterministic model that factors in alternative-amalgamation of use of COVID Alert SA app and vaccination against the COVID-19 to the Republic of South Africa’s general public in an endeavor to discontinue the chain of spread for the pandemic. We analyze the key properties of the model not limited to positivity, boundedness, and stability.
A NSFD method for the singularly perturbed Burgers-Huxley equation
(Frontiers Media, 2023) Derzie, Eshetu B.; Munyakazi, Justin B.; Dinka, Tekle G.
This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and e cient numerical solution for this equation via the classical numericalmethods. To overcome this challenge, a nonstandard finite di􀀀erence (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary di􀀀erential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniformconvergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme.
Binary codes from m-ary n-cubes Q(n) (m)
(American Institute of Mathematical Sciences, 2021) Key, Jennifer D.; Rodrigues, Bernardo G.
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 < m􀀀1 2 for the dual codes, i.e. [m2; 2m 􀀀 1;m]2 codes, when n = 2 and m 5 is odd.
On the exact constants in one-sided maximal inequalitiesfor Bessel processes
(Taylor and Francis Group, 2023) Makasu, Cloud
In this paper, we establish a one-sided maximal moment inequalitywith exact constants for Bessel processes. As a consequence, weobtain an exact constant in the Burkholder-Gundy inequality. Theproof of our main result is based on a pure optimal stopping prob-lem of the running maximum process for a Bessel process. The pre-sent results extend and complement a number of related resultspreviously known in the literature.
A NSFD method for the singularly perturbed Burgers-Huxley equation
(Frontiers Media, 2023) Derzie, Eshetu B.; Munyakazi, Justin B.; Dinka, Tekle G.
This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and efficient numerical solution for this equation via the classical numerical methods. To overcome this challenge, a nonstandard finite difference (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary differential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniform convergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme.
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.
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.
Some meta-cayley graphs on dihedral groups
(Springer, 2019) Allie, Imran; Mwambene, Eric
In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.
Modeling the dynamics of an epidemic under vaccination in two interacting populations
(Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, Kailash
Mathematical modeling of the numerical evolution of infectious diseases has become an important tool for disease control and eradication when possible. Much work has been done on the problem of how a given population is affected by another population when there is mutual interaction. The mere presence of migrant people poses a challenge to whatever health systems are in place in a particular region. Such epidemiological phenomena have been studied extensively, described by mathematical models with suggestions for intervention strategies. The epidemiological effect of migration within the population itself was modeled for sleeping sickness in a paper 1 by Chalvet-Monfray et al. In the case of malaria, there is for instance a study 2 by Tumwiine et al. on the effect of migrating people on a fixed population. The latter two diseases are vector borne. Diseases that propagate without a vector spread perhaps more easily when introduced into a new region. Various studies of models with immigration of infectives have been undertaken for tuberculosis, see for instance 3 by Zhou et al., or the work 4 of Jia et al., and for HIV, see the paper 5 of Naresh et al.
Generalizing the Hilton–Mislin genus group
(Elsevier, 2001) Witbooi, Peter J.
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.
Graphs, designs and codes related to the n-cube
(Elsivier, 2009) Fish, W; Key, J D; Mwambene, E
For integers n 1; k 0, and k n, the graph 􀀀 k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 􀀀 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 􀀀 2 n when n 0 .mod 4/, n 8.
Protein interaction networks as metric spaces: A novel perspective on distribution of hubs
(BMC, 2014) Fadhal, Emad; Gamieldien, Junaid; Mwambene, Eric C
In the post-genomic era, a central and overarching question in the analysis of protein-protein interaction networks continues to be whether biological characteristics and functions of proteins such as lethality, physiological malfunctions and malignancy are intimately linked to the topological role proteins play in the network as a mathematical structure. One of the key features that have implicitly been presumed is the existence of hubs, highly connected proteins considered to play a crucial role in biological networks. We explore the structure of protein interaction networks of a number of organisms as metric spaces and show that hubs are non randomly positioned and, from a distance point of view, centrally located.
Optimal strategy for controlling the spread of HIV/AIDS disease: A case study of South Africa
(Taylor and Francis Group, 2012) Yusuf, Tunde T.; Benyah, Francis
HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested infighting it. There is a need to integrate the series of control measures available to ensure a consistentreduction in the incidence of the disease pending the discovery of its cure. We present a deterministic modelfor controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapyas control measures. We formulate a fixed time optimal control problem subject to the model dynamicswith the goal of finding the optimal combination of the two control measures that will minimize the costof the control efforts as well as the incidence of the disease. We estimate the model state initial conditionsand parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin’smaximum principle to derive the optimality system and solve the system numerically. Compared withthe practice in most resource-limited settings where ARV treatment is given only to patients with full-blown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress tothe pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals’ sexualhabits reduces both the incidence and prevalence of the disease faster. In fact, the results predict thatthe implementation of the proposed strategy would drive new cases of the disease towards eradication in10 years.
Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers
(KOREAN MATHEMATICAL SOC, 2014) Munyakazi, Justin B.; Patidar, Kailash C.
Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.
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 Umba
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.
Modeling the impact of combined use of Covid Alert SA app and vaccination to curb Covid-19 infections in South Africa
(Public Library of Science, 2023) Kinyili, Musyoka; Munyakazi, Justin B.; Mukhtar, Abdulaziz Y. A.
The unanticipated continued deep-rooted trend of the Severe Acute Respiratory Syndrome Corona-virus-2 the originator pathogen of the COVID-19 persists posing concurrent anxiety globally. More effort is affixed in the scientific arena via continuous investigations in a prolific effort to understand the transmission dynamics and control measures in eradication of the epidemic. Both pharmaceutical and non-pharmaceutical containment measure protocols have been assimilated in this effort. In this study, we develop a modified SEIR deterministic model that factors in alternative-amalgamation of use of COVID Alert SA app and vaccination against the COVID-19 to the Republic of South Africa’s general public in an endeavor to discontinue the chain of spread for the pandemic. We analyze the key properties of the model not limited to positivity, boundedness, and stability. We authenticate the model by fitting it to the Republic of South Africa’s cumulative COVID-19 cases reported data utilizing the Maximum Likelihood Estimation algorithm implemented in fitR package.
Binary codes and partial permutation decoding sets from the odd graphs
(Walter de Gruyter, 2014) Fish, Washiela; Fray, Roland; Mwambene, Eric
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.
Relative homotopy in relational structures
(Cambridge University Press, 2018) Witbooi, Peter
The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs (X, A) where X is a poset and A is a subposet of X. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.
On maximal inequalities via comparison principle
(SpringerOpen, 2015) Makasu, Cloud
Under certain conditions, we prove a new class of one-sided, weighted, maximal inequalities for a standard Brownian motion. Our method of proof is mainly based on a comparison principle for solutions of a system of nonlinear first-order differential equations.
A robust spectral method for pricing of American put options on zero-coupon bonds
(Global-Science Press, 2018) Pindza, Edson; Patidar, Kailash C.
American put options on a zero-coupon bond problem is reformulated as a linear complementarity problem of the option value and approximated by a nonlinear partial differential equation. The equation is solved by an exponential time differencing method combined with a barycentric Legendre interpolation and the Krylov projection algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity (a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor at maturity. A zero-coupon bond is a bond that makes no periodic interest payments.

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