Applied Mathematics Publications

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Applied Mathematics Publications

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    Bifurcation Analysis of a 5D Nutrient, Plankton, Limnothrissa miodon Model with Hydrocynus vittatus Predation
    (Journal of Applied Mathematics, 2022-07-20) Mutasa, F.K.; Jones, B.; Tendaupenyu, I.H.; Nhiwatiwa, T.; Ndebele-Murisa, M.R.
    In this paper, we construct and analyze a theoretical, deterministic 5D mathematical model of Limnothrissa miodon with nutrients, phytoplankton, zooplankton, and Hydrocynus vittatus predation. Local stability analysis results agree with the numerical simulations in that the coexistence equilibrium is locally stable provided that certain conditions are satisfied. The coexistence equilibrium is globally stable if certain conditions are met. Existence, stability, and direction of Hopf bifurcations are derived for some parameters. Bifurcation analysis shows that the model undergoes Hopf bifurcation at the coexistence point for the zooplankton growth rate with periodic doubling leading to chaos.
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    Malunguza, N. J. 2016. MATHEMATICAL MODELLING AND ANALYSIS OF EXISTING AND EMERGING PROBLEMS INFLUENCING THE TRANSMISSION DYNAMICS OF HIV/AIDS IN AFRICAN HETEROSEXUAL SETTINGS. PhD Thesis.
    (2016) Malunguza, N. J.
    The human immunodeficiency virus (HIV) is spread through exchange of body fluids and affects specific cells of the immune system, called CD4 + T cells. Progressive depletion of these cells over time leads to a condition called the acquired immuno deficiency syndrome (AIDS). In Africa, transmission in the general population is predominantly heterosexual. Confluence of various complex social, cultural and behavioural tendencies influence the incidence and distribution of the virus in the region and there are sub-populations that are particularly at heightened risk of acquiring or passing on the virus. In this thesis, HIV transmission models are formulated to obtain qualitative and quantitative insights into the transmission dynamics of the disease. The models are designed to capture the effect of differential infectivity and susceptibility through stratification of the population into sub-groups according to some risk factors peculiar to the group. Alcohol consumption in sexual settings reflective of gender inequities and same sex interaction between men and men in heterosexual settings are studied as inherently problematic pre-existing risks. Super-infection by different HIV subtypes, contraceptive use and heterosexual anal intercourse are also considered as emerging challenges in disease control. The models are rigorously analysed using analytical and computational techniques to determine solutions and their dynamical properties and transmission characteristics. Results show that superinfection by two non-competing HIV strains increases disease burden and must be factored in the HIV policing framework. Partnership formation rates between low and high risk groups are shown to be pivotal in the spread of HIV across heterogeneous populations so that constraining formation of partnerships between the general population and high risk groups such as alcohol consumers and men who have sex with men will be beneficial in the fight against HIV. Whereas literature on the acquisition and transmission risk posed by hormonal contraceptive use remains inconclusive, model results indicate the potential for accelerated spread and the bridging effect of men who may be the link between the high risk hormonal contraceptive users and the low risk general population. The model fit to data for Zimbabwe supports heterosexual anal intercourse (AI) only by a small percentage (< 1%) of the population in order for parameters to remain plausible with results showing that even at that low prevalence, heterosexual AI would compound HIV transmission. Study results show that MSM increase HIV spread in heterosexual settings through their bisexual activities and that curtailing bisexuality would overally have a negative impact on HIV spread. To bridge the gap between theory and real world observations, some models were fitted to sentinel data for Zimbabwe. Although data at the onset of the epidemic is sparse, an estimate of the basic reproductive number R0 quantifying the magnitude of the control effort necessary for epidemic control, was drawn. Projections of incidence and prevalence are made to allow for forward planning. Results from the assessment of each individual model collectively contribute to the general understanding of the problem surrounding the population level transmission dynamics of HIV for effective disease prevention and control.
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    Stability Analysis and Optimal Control of a Limnothrissa Miodon Model with Harvesting
    (Discrete Dynamics in Nature and Society, 2022-05-05) Mutasa, F.K.; Jones, B.; Hove-Musekwa, S.D.; Tendaupenyu, I.H.; Nhiwatiwa, T.; Ndebele-Murisa, M.R.
    We construct a theoretical, deterministic mathematical model of the dynamics of Limnothrissa miodon with nutrients, phytoplankton, and zooplankton and investigate the effect of harvesting on the population density of Limnothrissa miodon in a lake. For the autonomous model, results from local stability analysis are in agreement with numerical simulations in that the coexistence equilibrium is locally stable, provided certain conditions are satisfied. The coexistence equilibrium is globally unstable if it is feasible. Numerical results show that a stable limit cycle exists for the nonautonomous model. Optimal control results show an optimal harvesting monthly effort of 15394 boat nights which corresponds to 505 fishing units, showing that there is overcapacity in Lake Kariba. A maximum sustainable annual catch of 34669 tonnes is obtained and simulation results show that Limnothrissa miodon abundance is more closely related to nutrient inflow than to harvesting.
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    A mathematical model for bioconvection flow with activation energy for chemical reaction and microbial activity
    (Indian Academy of Sciences, 2022-01-23) Dhlamini, M.; Mondal, H.; Sibanda, P.; Mosta, S.S.; Shaw, S.
    In most of the industrial processes, it is of paramount importance to control the heat and mass transfer rates to ensure high-quality products. Using nanofluids instead of ordinary fluids and using motile micro-organisms are some of the techniques to control heat and mass transfer rates. In some recent studies of bioconvection flow, activation energy, Brownian motion and thermophoretic effects are considered only for the solute and not for the microbes. Our current study incorporates these effects for the motile micro-organisms too. Few, if any results of this nature exist in literature. A system of partial differential equations is formulated to incorporate the effects of these parameters. The system of equations are solved numerically using the spectral quasi-linearisation method to gain an insight into the influence of key parameters on the fluid and flow properties. The thermophoretic force, the Brownian motion and activation energy are significant contributors in the microbes’ dynamics. The concentration of microbes decreases with an increase in the thermophoretic force and increases with increasing microbe’s Brownian motion parameter. Based on our results, we conclude that increasing activation energy leads to a decrease in microbes’ velocity. The inclusion of the microbes’ Brownian motion proved to be significant as this was shown to have an impact on the temperature, solute concentration and microbes’ concentration in the boundary layer.
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    Numerical Analysis of Couple Stress Nanofluid in Temperature Dependent Viscosity and Thermal Conductivity
    (Springer Nature, 2021-02-24) Dhlamini, M.; Mondal, H.; Sibanda, P.; Motsa, S.
    This communication reports on an innovative study of two-dimensional couple stress fluid 3 with effect of viscosity and conductivity. We proposed a new model based on temperature dependent variable thermal conductivity on kinetic theory. Our model assumes that thermal conductivity is a decreasing function of temperature rather than an increasing function. The effect of the three key parameters, viscosity, thermal conductivity and couple stress parameter are analyzed. The coupled non-linear system is further validated numerically using the spectral quasilinearization method. The method is found to be accurate and convergent. Increasing the temperature dependent parameter for viscosity is shown to reduce the heat mass transfer rates at the surface. Increasing thermal conductivity and the couple stress parameter increased the heat mass transfer rates on the boundary surface.