Research Articles (Mathematics)
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Item Modeling the population dynamics of HIV/AIDS with opportunistic infections at the severe stage of HIV(American Institute of Mathematical Sciences, 2025) Nsuami, Mozart Umba; Witbooi, Peter JosephIn this paper, we present a deterministic model for the population dynamics of HIV/AIDS, wherein some individuals at the severe symptomatic phase of HIV develop serious opportunistic infections (OIs) such cryptococcal, tuberculous, pneumococcal, and other bacterial meningitis due to an inappropriate treatment or lack of counseling. OIs are responsible for significant mortality and disability on individuals with HIV in many countries. Cryptococcal meningitis (CM) is among frequent OIs responsible for significant mortality and disability of individuals with HIV in limited resource settings. However, there are also cases of high mortality due to CM on HIV-uninfected individuals, but the burden of CM is more frequent in people living with HIV. We proved the global stability of the disease-free as well as the endemic equilibrium points. In addition, we performed the study of sensitivity analysis of the basic reproduction number with the parameters of the model as well as with some compartmental classes. We illustrated our theoretical results by way of numerical simulations using a projection on the HIV historical data of South Africa since 2024. Our analysis showed that a combination of ART and OI specific treatments may reduce the number of death related cases.Item A mathematical model for malaria disease dynamics with vaccination and infected immigrants(American Institute of Mathematical Sciences, 2024) Duve, Pride; Munyakazi, JustinThe world is aiming to eliminate malaria by 2030. The introduction of the pilot project on malaria vaccination for children in Kenya, Ghana, and Malawi presents a significant thrust to the elimination efforts. In this work, a susceptible, infectious and recovered (SIR) human-vector interaction mathematical model for malaria was formulated. The model was extended to include a compartment of vaccinated humans and an influx of infected immigrants. Qualitative and quantitative analysis was performed on the model. When there was no influx of infected immigrants, the model had a disease-free equilibrium point that was globally asymptotically stable when a threshold known as the basic reproductive number denoted by R0 was less than one. When there was an influx of infected immigrants, the model had endemic equilibrium points only. Parameter sensitivity analysis on R0 was performed and results showed that strategies must be implemented to reduce contact between mosquitoes and humans. Results from different vaccine coverage indicated that in the absence of an influx of infected immigrants, it is possible to achieve a malaria-free society when more children get vaccinated and the influx of infected humans is avoided. The analysis of the optimal control model showed that the combined use of vaccination, personal protective equipment, and treatment is the best way to curb malaria incidence, provided the influx of infected humans is completely stoppedItem Modeling the effect of imported malaria on the elimination programme in KwaZulu-Natal province of South Africa(African Field Epidemiology Network, 2024) Witbooi, Peter Joseph; Abiodun, Gbenga Jacob; Maharaj, RajendraIntroduction: with imported malaria cases in a given population, the question arises as to what extent the local cases are a consequence of the imports or not. We perform a modeling analysis for a specific area, in a region aspiring for malaria-free status. Methods: data on malaria cases over ten years is subjected to a compartmental model which is assumed to be operating close to the equilibrium state. Two of the parameters of the model are fitted to the decadal data. The other parameters in the model are sourced from the literature. The model is utilized to simulate the malaria prevalence with or without imported cases. Results: in any given year the annual average of 460 imported cases, resulted in an end-of-year season malaria prevalence of 257 local active infectious cases, whereas without the imports the malaria prevalence at the end of the season would have been fewer than 10 active infectious cases. We calculate the numerical value of the basic reproduction number for the model, which reveals the extent to which the disease is being eliminated from the population or not. Conclusion: without the imported cases, over the ten seasons of malaria, 2008-2018, the KwaZulu-Natal province would have been malaria-free over at least the last 7 years of the decade indicated.Item Optimal control strategies applied to a mathematical model of meningitis(New York Business Global, 2025) Patidar, Kailash; Mohamed, Sahar; Obaid, HasimIn this paper, we deal with the problem of optimal control for the transmission dynamics of the meningococcal meningitis. The problem is a mathematical model described by a system of nonlinear differential equations. Based on this, two controls are formulated and the resulting system is solved as an optimal control problem. Aiming to minimize the number of illnesses or deaths in the population, we used a control representing a vaccination and another one representing a treatment strategy. We prove that these controls are capable of reducing the number of carriers and infectious individuals. Numerical simulations are carried out to show how to perform the two strategies.Item On automorphism groups of the conjugacy class type Cayley graphs on the symmetric and alternating groups(Taylor and Francis Ltd., 2025) Habineza, Olivier; Mwambene, EricThe automorphism groups of Cayley graphs on symmetric groups, Cay(G, S), where S is a complete set of transpositions have been determined. In a similar spirit, automorphism groups of Cayley graphs Cay(An, S) on alternating groups An, where S is a set of all 3-cycles have also been determined. It has, in addition, been shown that these graphs are not normal. In all these Cayley graphs, one observes that their corresponding Cayley sets are a union of conjugacy classes. In this paper, we determine in their generality, the automorphism groups of Cay(G, S), where G ∈ {An, Sn} and S is a conjugacy class type Cayley set. Further, we show that the family of these graphs form a Boolean algebra. It is first shown that Aut(Cay(G, S)), S ∉ {∅, G \ {e}}, is primitive if and only if G = An. Using one of the results obtained by Praeger in 1990, we exploit further the other cases, thereby proving that, for n > 4 and n ≠ 6, Aut(Cay(An, S)) ≅ Hol(An ) ⋊ 2, with Hol(G) ∼= G ⋊ Aut(G), provided that S is preserved by the outer automorphism defined by the conjugation by an odd permutation. Finally, in the remaining case G = Sn, n > 4 and n ≠ 6, we show that Aut(Cay(Sn, S) ≅ (Hol(An) ⋊ 2) ≀ S2 for S ⊂ An \ {e}, and that Aut(Cay(Sn , S)) ≅ Hol(Sn) ⋊ 2 otherwise; provided that S does not contain Sn \ An or S ≠ An \ {e}, S ∉ {∅, Sn \ {e}}.Item Overview of Császár orders and quasi-uniformities on complete lattices(Elsevier B.V., 2025) Bakulikira C, Iragi.In this paper we introduce a theory of quasi-uniformities and syntopogenous structures on complete lattices, extending the results from [7] and [5] in a general categorical context. In particular, we show that topogenous orders on a complete lattice encompass both closure and interior operations, and that a syntopogenous structure is a base of a quasi-uniformity. In fact, not only does this paper extend various structures from category theory, but it also demonstrates that these constructs can be studied without relying on (E,M)-factorizations, which have historically been necessary for their study in a categorical setting. In closing, we show that any ⋀-structure of a complete lattice can be used to construct a base of transitive quasi-uniformity on the lattice. © 2025 Elsevier B.V.Item Generalizing β- and λ-maps(Elsevier B.V., 2025) Avilez, Ana BelénWe generalize the notions of β- and λ-maps in terms of selections of sublocales, obtaining different classes of localic maps. These new classes of maps are used to characterize almost normal, extremally disconnected, F- and Oz-locales, among other types of locales, in a manner akin to the characterization of normal locales via β-maps. As a byproduct we obtain a characterization of localic maps that preserve the completely below relation (that is, the right adjoints of assertive frame homomorphisms).Item A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equations(Tamkang University, 2025) Munyakazi, Justin; Mohye, Mekashaw Ali; Dinka, Tekle GemechuThe objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolson discretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions.. Model examples are provided to validate the practicality of the proposed numerical scheme.Item A characterization of procyclic groups via complete exterior degree(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Rodrigues, Bernardo; Russo, Francesco GWe describe the nonabelian exterior square (Formula presented.) of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of (Formula presented.) and new structural results for (Formula presented.). Then, we investigate a generalization of the probability that two randomly chosen elements of G commute: this notion is known as the “complete exterior degree” of a pro-p-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degreeItem An efficient numerical scheme for a time-fractional black–scholes partial differential equation derived from the fractal market hypothesis(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Nuugulu, Samuel M.; Patidar, Kailash C.; Gideon, FrednardSince the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in realtime market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.Item Ground state representations of topological groups(Springer Science and Business Media Deutschland GmbH, 2024) Neeb, Karl-Hermann; Russo, Francesco GLet α : R → Aut(G) define a continuous R-action on the topological group G. A unitary representation (π , H) of the extended group G := G α R is called a ground state representation if the unitary one-parameter group π (e, t) = eitH has a nonnegative generator H ≥ 0 and the subspace H0 := ker H of ground states generates H under G. In this paper, we introduce the class of strict ground state representations, where (π , H) and the representation of the subgroup G0 := Fix(α) on H0 have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of G0. This is particularly effective if the occurring representations of G0 can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representationsItem One-sided maximal inequalities for a randomly stopped bessel process(Taylor & Francis Group, LLC, 2023) Cloud, MakasuWe 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.Item 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.Item 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 dierence (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 dierential 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.Item 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 < m1 2 for the dual codes, i.e. [m2; 2m 1;m]2 codes, when n = 2 and m 5 is odd.Item On the exact constants in one-sided maximal inequalitiesfor Bessel processes(Taylor and Francis Group, 2023) Makasu, CloudIn 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.Item 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.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.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.Item Some meta-cayley graphs on dihedral groups(Springer, 2019) Allie, Imran; Mwambene, EricIn 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.Item Modeling the dynamics of an epidemic under vaccination in two interacting populations(Hindawi, 2012) Ahmed, Ibrahim H. I.; Witbooi, Peter J.; Patidar, KailashMathematical 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.