Research Articles (Mathematics and Applied Mathematics)

Permanent URI for this collectionhttp://hdl.handle.net/2263/1978

A collection containing some of the full text peer-reviewed/ refereed articles published by researchers from the Department of Mathematics and Applied Mathematics

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Now showing 1 - 20 of 727

On the impact of re-mating and residual fertility on the sterile insect technique efficacy : case study with the medfly, Ceratitis capitata
(Public Library of Science, 2024-05-06) Dumont, Yves; Oliva, Clelia F.
The sterile insect technique (SIT) can be an efficient solution for reducing or eliminating certain insect pest populations. It is widely used in agriculture against fruit flies, including the Mediterranean fruit fly (medfly), Ceratitis capitata. The re-mating tendency of medfly females and the fact that the released sterile males may have some residual fertility could be a challenge for the successful implementation of the SIT. Obtaining the right balance between sterility level and sterile male quality (competitiveness, longevity, etc) is the key to a cost-efficient program. Since field experimental approaches can be impacted by many environmental variables, it is difficult to get a clear understanding on how specific parameters, alone or in combination, may affect the SIT efficiency. The use of models not only helps to gather knowledge, but it allows the simulation of a wide range of scenarios and can be easily adapted to local populations and sterile male production. In this study, we consider single- and double-mated females. We first show that SIT can be successful only if the residual fertility is less than a threshold value that depends on the basic offspring number of the targeted pest population, the re-mating rates, and the parameters of double-mated females. Then, we show how the sterile male release rate is affected by the parameters of double-mated females and the male residual fertility. Different scenarios are explored with continuous and periodic sterile male releases, with and without ginger aromatherapy, which is known to enhance sterile male competitiveness, and also taking into account some biological parameters related to females that have been mated twice, either first by a wild (sterile) male and then a sterile (wild) male, or by two wild males only. Parameter values were chosen for peach as host fruit to reflect what could be expected in the Corsican context, where SIT against the medfly is under consideration. Our results suggest that ginger aromatherapy can be a decisive factor determining the success of SIT against medfly. We also emphasize the importance of estimating the duration of the refractory period between matings depending on whether a wild female has mated with a wild or sterile male. Further, we show the importance of parameters, like the (hatched) eggs deposit rate and the death-rate related to all fertile double-mated females. In general, re-mating is considered to be detrimental to SIT programs. However, our results show that, depending on the parameter values of double-mated females, re-mating may also be beneficial for SIT. Our model can be easily adapted to different contexts and species, for a broader understanding of release strategies and management options.
Data-driven cold starting of good reservoirs
(Elsevier, 2024-12) Grigoryeva, Lyudmila; Grigoryeva, Lyudmila; Kemeth, Felix P.; Kevrekidis, Yannis; Manjunath, Gandhi; Ortega, Juan-Pablo; Steynberg, Matthys J.; manjunath.gandhi@up.ac.za
Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described. This strategy is called cold-starting, and it is based on a map called the starting map, which is determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space. The time series generated by the reservoir system using that initial state can be used to run the system in autonomous mode in order to produce accurate forecasts of the time series under consideration immediately. By utilizing this map, the lengthy ‘‘washouts’’ that are necessary to initialize reservoir systems can be eliminated, enabling the generation of forecasts using any selection of appropriately short histories of the observations.
Bounds for extreme zeros of Meixner–Pollaczek polynomials
(Elsevier, 2025-05) Jooste, Alta; Jordaan, Kerstin Heidrun; alta.jooste@up.ac.za
Please read abstract in the article.
Parameter spaces for cross-diffusive-driven instability in a reaction-diffusion system on an annular domain
(World Scientific Publishing, 2025-04) Yigit, Gulsemay; Sarfaraz, Wakil; Barreira, Raquel; Madzvamuse, Anotida
In this work, the influences of geometry and domain size on spatiotemporal pattern formation are investigated to establish the parameter spaces for a cross-diffusive reaction–diffusion model on an annulus. By applying the linear stability theory, we derive the conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether the selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on nonconvex geometries, e.g. annulus. Hence, the key research methodology and outcomes of our studies include a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction–diffusion system; a linear stability analysis to characterize the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through the lower and upper bounds of the domain size; the full classification of the model parameters; and a demonstration of how cross-diffusion relaxes the general conditions for the reaction–diffusion system to exhibit pattern formation. To validate the theoretical findings and predictions, we employ the finite element method to reveal the spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction–diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumors. Specifically, the cross-section of an actively invasive region in a hypoxic tumor can be effectively approximated by an annulus.
Static hedging of vanilla and exotic options in a South African context
(Operations Research Society of South Africa, 2024-07) Levendis, Alexis; Mare, Eben
In this paper, we test the performance of a static hedging strategy for a long-dated European call option and European spread call option in South Africa. The stochastic volatility double jump (SVJJ) model is calibrated to historical FTSE/JSE Top40 returns to generate real-world FTSE/JSE Top40 prices at future dates. The SVJJ model is also calibrated to the FTSE/JSE (Top40) implied volatility surface in order to value the options under the risk-neutral measure. Two static hedging programs are then implemented to test their effectiveness when replicating a long-dated European call option and European spread call option. Our results indicate that static hedging is a simple, yet effective, solution when hedging non-exchange-traded options with vanilla exchange-traded options.
Homogenization of partial differential equations with Preisach operators
(Tamkang University, 2025-02) Pokam Kakeu, Achille Landri; al.pokamkakeu@up.ac.za
The current work deals with initial boundary value parabolic problems with Preisach hysteresis whose the density functions are allowed to depend on the variable of space. The model contains nonlinear monotone operators in the diffusion term, arising from an energy. Thanks to the properties of Preisach hysteresis operators and to the sigma-convergence method, we obtain the convergence of the microscopic solutions to the solution of the homogenized problem. The effective operator is obtained in terms of a solution of a nonlinear corrector equation addressed in the usual sense of distributions, leading in an approximate scheme for the homogenized coefficient which is an important step towards the numerical implementation of the results from the homogenization theory beyond the periodic setting.
Optimal monetary and fiscal policies to maximise non-parallel risk premia in sovereign bond markets
(MDPI, 2024-11) Hariparsad, Sanveer; Mare, Eben; eben.mare@up.ac.za
In this paper, we analysed several emerging market (EM) and developed market (DM) sovereign yield curves to identify the proportion of parallel and non-parallel shifts over time. We found that non-parallel shifts are more prevalent in EM due to higher political and economic risks. Key drivers include systemic risk events like wars, debt distress, and pandemics. By backtesting a long butterfly strategy to extract non-parallel risk premia from June 2007 to March 2024, we observed that steeper slopes and greater curvature result in higher returns. We also quantified monetary and fiscal regimes to determine what types of policies are required to extract non-parallel risk premia from these sovereign yield curves. Our research suggests that countries with opposing monetary and fiscal policies possess higher return opportunities whilst countries with complementing policies require tactical butterfly strategies to optimise returns.
Mathematical assessment of control strategies against the spread of MERS-CoV in humans and camels in Saudi Arabia
(AIMS Press, 2024-07) Alatawi, Adel; Gumel, Abba B.
A new mathematical model for the transmission dynamics and control of the Middle Eastern respiratory syndrome (MERS), a respiratory virus caused by MERS-CoV coronavirus (and primarily spread to humans by dromedary camels) that first emerged out of the Kingdom of Saudi Arabia (KSA) in 2012, was designed and used to study the transmission dynamics of the disease in a human-camel population within the KSA. Rigorous analysis of the model, which was fitted and cross-validated using the observed MERS-CoV data for the KSA, showed that its disease-free equilibrium was locally asymptotically stable whenever its reproduction number (denoted by R0M) was less than unity. Using the fixed and estimated parameters of the model, the value of R0M for the KSA was estimated to be 0.84, suggesting that the prospects for MERS-CoV elimination are highly promising. The model was extended to allow for the assessment of public health intervention strategies, notably the potential use of vaccines for both humans and camels and the use of face masks by humans in public or when in close proximity with camels. Simulations of the extended model showed that the use of the face mask by humans who come in close proximity with camels, as a sole public health intervention strategy, significantly reduced human-to-camel and camel-to-human transmission of the disease, and this reduction depends on the efficacy and coverage of the mask type used in the community. For instance, if surgical masks are prioritized, the disease can be eliminated in both the human and camel population if at least 45% of individuals who have close contact with camels wear them consistently. The simulations further showed that while vaccinating humans as a sole intervention strategy only had marginal impact in reducing the disease burden in the human population, an intervention strategy based on vaccinating camels only resulted in a significant reduction in the disease burden in camels (and, consequently, in humans as well). Thus, this study suggests that attention should be focused on effectively combating the disease in the camel population, rather than in the human population. Furthermore, the extended model was used to simulate a hybrid strategy, which combined vaccination of both humans and camels as well as the use of face masks by humans. This simulation showed a marked reduction of the disease burden in both humans and camels, with an increasing effectiveness level of this intervention, in comparison to the baseline scenario or any of the aforementioned sole vaccination scenarios. In summary, this study showed that the prospect of the elimination of MERS-CoV-2 in the Kingdom of Saudi Arabia is promising using pharmaceutical (vaccination) and nonpharmaceutical (mask) intervention strategies, implemented in isolation or (preferably) in combination, that are focused on reducing the disease burden in the camel population.
Measuring the mathematical maturity of students in an academic development programme
(Springer, 2024-08) Simelane, Bridgette Makhosazana; Engelbrecht, Johann; johann.engelbrecht@up.ac.za
This study focuses on students who are registered for the University of Pretoria’s academic development programme, the Four-year Programme (FYP). The programme was introduced as a gateway for students who are under-prepared but have the potential to succeed. This programme helps them to then continue their studies in mainstream science programmes. Our research focuses on measuring the change in the academic maturity of these students. In the theoretical framework that we developed, academic maturity is subdivided into two components, namely, non-subject based maturity, and subject based maturity (mathematical maturity). A mathematics test was also administered twice (at the beginning of the year and after the first semester) and was used to measure the subject based maturity of students. The results of the pre- and post-tests were compared to measure the improvement in students’ mathematical skills. The results showed that in all of the topics and constructs, there was an improvement in students’ mathematical abilities. The study also shows that students still struggle with the fundamentals of some mathematics topics, even after a semester of tuition in the FYP.
Perturbed weighted trapezoid inequalities for convex functions with applications
(Universidad de la Frontera, 2024-12) Dragomir, Sever S.; Kikianty, Eder; eder.kikianty@up.ac.za
Please read abstract in the article.
Mathematical assessment of wastewater-based epidemiology to predict SARS-CoV-2 cases and hospitalizations in Miami-Dade county
(Springer, 2025-03) Pant, Binod; Safdar, Salman; Ngonghala, Calistus N.; Gumel, Abba B.
This study presents a wastewater-based mathematical model for assessing the transmission dynamics of the SARS-CoV-2 pandemic in Miami-Dade County, Florida. The model, which takes the form of a deterministic system of nonlinear differential equations, monitors the temporal dynamics of the disease, as well as changes in viral RNA concentration in the county’s wastewater system (which consists of three sewage treatment plants). The model was calibrated using the wastewater data during the third wave of the SARS-CoV-2 pandemic in Miami-Dade (specifically, the time period from July 3, 2021 to October 9, 2021). The calibrated model was used to predict SARS-CoV-2 case and hospitalization trends in the county during the aforementioned time period, showing a strong correlation between the observed (detected) weekly case data and the corresponding weekly data predicted by the calibrated model. The model’s prediction of the week when maximum number of SARS-CoV-2 cases will be recorded in the county during the simulation period precisely matches the time when the maximum observed/reported cases were recorded (which was August 14, 2021). Furthermore, the model’s projection of the maximum number of cases for the week of August 14, 2021 is about 15 times higher than the maximum observed weekly case count for the county on that day (i.e., the maximum case count estimated by the model was 15 times higher than the actual/observed count for confirmed cases). This result is consistent with the result of numerous SARS-CoV-2 modeling studies (including other wastewater-based modeling, as well as statistical models) in the literature. Furthermore, the model accurately predicts a one-week lag between the peak in weekly COVID-19 case and hospitalization data during the time period of the study in Miami-Dade, with the model-predicted hospitalizations peaking on August 21, 2021. Detailed time-varying global sensitivity analysis was carried out to determine the parameters (wastewater-based, epidemiological and biological) that have the most influence on the chosen response function—the cumulative viral load in the wastewater. This analysis revealed that the transmission rate of infectious individuals, shedding rate of infectious individuals, recovery rate of infectious individuals, average fecal load per person per unit time and the proportion of shed viral RNA that is not lost in sewage before measurement at the wastewater treatment plant were most influential to the response function during the entire time period of the study. This study shows, conclusively, that wastewater surveillance data can be a very powerful indicator for measuring (i.e., providing early-warning signal and current burden) and predicting the future trajectory and burden (e.g., number of cases and hospitalizations) of emerging and re-emerging infectious diseases, such as SARS-CoV-2, in a community.
VEMcomp : a virtual elements MATLAB package for bulk-surface PDEs in 2D and 3D
(Springer, 2024-08-31) Frittelli, Massimo; Madzvamuse, Anotida; Sgura, Ivonne
We present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulksurface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order k = 1 on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order k = 1 as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.
Existence and convergence of stochastic processes underlying a thin layer approximation of a coupled bulk-surface PDE
(Elsevier, 2025-05) Bobrowski, Adam; Madzvamuse, Anotida; Ratajczyk, Elżbieta
We study a system of coupled bulk-surface partial differential equations (BS-PDEs), describing changes in concentration of certain proteins (Rho GTPases) in a living cell. These proteins, when activated, are bound to the plasma membrane where they diffuse and react with the inactive species; inactivated species diffuse inside the cell cortex; these react with the activated species when they are close to the plasma membrane. For our case study, we model the cell cortex as an annulus, and the plasma membrane as its outer circle. Mathematically, the aim of the paper is twofold: Firstly, we show the master equation for the changes in concentration of Rho GTPases is the Kolmogorov forward differential equation for an underlying Feller stochastic process, and, in particular, the related Cauchy problem is well-posed. Secondly, since the cell cortex is typically a rather thin domain, we study the situation where the thickness of the annulus modeling the cortex converges to 0. To this end, we note that letting the thickness of the annulus to 0 is equivalent to keeping it constant while increasing the rate of radial diffusion. As a result, in the limit, solutions to the master equation lose dependence on the radial coordinate and can be thought of as functions on the circle. Furthermore, the limit master equation can be seen as describing diffusion on two copies of the circle with jumps from one copy to the other.
A mathematical model on the impact of awareness and traditional medicine in the control of Ebola : case study of the 2014-2016 outbreaks in Sierra Leone and Liberia
(Oxford University Press, 2024-08) Tassé, Arsène Jaures Ouemba; Tsanou, Berge; Kum, Cletus K.; Lubuma, Jean
Please read abstract in article.
Weighted Fejer, Hermite–Hadamard, and trapezium-type inequalities for (h1,h2–Godunova–Levin Preinvex function with applications and two open problems
(MDPI, 2024-02) Ahmadini, Abdullah Ali H.; Afzal, Waqar; Abbas, Mujahid; Aly, Elkhateeb S.
This note introduces a new class of preinvexity called (h1, h2)-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature.
A characterization of procyclic groups via complete exterior degree
(MDPI, 2024-04) Rodrigues, Bernardo G.; Russo, Francesco G.; bernardo.rodrigues@up.ac.za
Please read abstract in the article.
A metapopulation model with exit screening measure for the 2014–2016 West Africa Ebola virus outbreak
(MDPI, 2024-12) Tasse, Arsene Jaures Ouemba; Tsanou, Berge; Woukeng, Jean Louis; Lubuma, Jean M.-S.
We construct a new metapopulation model for the transmission dynamics and control of the Ebola Virus Disease (EVD) in an environment characterized by considerable migrations and travels of people. It is an extended SEIR model modified by the addition of Quarantine and Isolated compartments to account for travelers who undergo the exit screening. The model is well-fitted by using the reported cases from the neighboring countries Guinea, Liberia and Sierra Leone where the 2014–2016 Ebola outbreak simultaneously arose. We show that the unique disease-free equilibrium (DFE) of the model is unstable or locally asymptotically stable (LAS) depending on whether the control reproduction number is larger or less than unity. In the latter case, we prove that the DFE is globally asymptotically stable (GAS) provided that the exit screening is 100% negative. We also prove the GAS of the DFE by introducing more explicit thresholds, thanks to which the existence of at least one boundary equilibrium is established. We design two new nonstandard finite difference (NSFD) schemes, which preserve the dynamics of the continuous model. Numerical simulations that support the theory highlight that exit screening is useful to mitigate the infection. They also suggest that the disease is controlled or the explicit threshold is less than unity provided that the migration and the exit screening parameters are above a critical value.
Hyers–Ulam stability of 2D-convex mappings and some related new Hermite–Hadamard, Pachpatte, and Fejér Type integral inequalities using novel fractional integral operators via totally interval-order relations with open problem
(MDPI, 2024-04-19) Afzal, Waqar; Breaz, Daniel; Abbas, Mujahid; Cotirla, Luminita-Ioana; Khan, Zareen A.; Rapeanu, Eleonora; mujahid.abbas@up.ac.za
The aim of this paper is to introduce a new type of two-dimensional convexity by using totalorder relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.
Optimal control approach for implementation of sterile insect techniques
(Springer, 2024-03) Bliman, P.-A.; Cardona-Salgado, D.; Dumont, Yves; Vasilieva, O.
The vector or pest control is essential to reduce the risk of vector-borne diseases or crop losses. Among the available biological control tools, the sterile insect technique (SIT) is one of the most promising. However, SIT-control campaigns must be carefully planned in advance in order to render desirable outcomes. In this paper, we propose SIT-control intervention programs that can avoid the real-time monitoring of the wild population and require to mass-rear a minimal overall number of sterile insects, in order to induce a local elimination of the wild population in the shortest time. Continuous-time release programs are obtained by applying an optimal control approach, and then laying the groundwork of more practical SIT-control programs consisting of periodic impulsive releases.
A climate-based metapopulation malaria model with human travel and treatment
(Springer, 2025-03) Danquah, Baaba A.; Chirove, Faraimunashe; Banasiak, Jacek
A climate-based metapopulation malaria model is formulated by incorporating human travel between zones with varying climatic factors, effective and counterfeit drug treatments, and time-periodic parameters for the mosquito population to understand the effect of human travel on malaria transmission. We study the existence, uniqueness, and stability of positive periodic solutions in the model and carry out numerical simulations for three climatic zones of Ghana. The study shows that the climate effects introduce fluctuations in the solutions, while human travel between zones affects the disease prevalence in each zone and the local transmission dynamics of malaria. We observed different outcomes depending on various restrictions imposed on human travels. The study also suggests that it is essential to ban the sale, importation or manufacture of counterfeit drugs and punish the offenders to ensure the effective use of high-quality drugs in the population.

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