Browsing by Author "De Silva, T. M. M."

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Hopf bifurcation in a periodic toxin producing phytoplankton model
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Dilshani, P. P.; De Silva, T. M. M.
Harmful algal blooms (HABs) caused by toxin-producing phytoplankton (TPP) have become increasingly common worldwide. Understanding the complex interactions between TPP and other organisms in the ecosystem is crucial. This study focuses on the Hopf bifurcation analysis of plankton interactions between TPP and zooplankton, with uptake function and a periodic toxin production. The maximum toxin liberation rate is considered as a bifurcation parameter. The aim is to determine how the toxin liberation rate affects the system. One of the proposed models assumes constant toxin production by TPP, resulting in an autonomous system of ordinary differential equations. To incorporate natural day and night, tidal, or seasonal cycles, the model is extended to a periodic system. The study examines the existence of steady states and trivial periodic solutions and analyses the stability of both models. Moreover, using the concept of uniform persistence, we derive sufficient conditions for the coexistence of the periodic system based on the model parameters. Due to instability of equilibria, we observe Hopf bifurcations in the constant toxin-producing model, providing insights into the system's dynamic behaviour. Numerical simulations are performed to validate the analytical findings of the proposed models and their implications.
Mathematical modeling of diabetes mellitus
(Faculty of Science, University of Kelaniya Sri Lanka, 2024) Subawickrama, H. D. K. M.; Munasinghe, J.; De Silva, T. M. M.
Diabetes is becoming a silent epidemic that is endangering public health worldwide. For instance, the majority of the 422 million diabetics globally reside in low- and middle-income nations, and the illness directly causes 1.5 million fatalities per year. In recent decades, the prevalence and complexity of the chronic medical condition diabetes have raised serious concerns about global health. As a result, studying and developing mathematical models becomes increasingly important since they provide an advanced perspective for realizing the complex nature of the disease and creating realistic care with preventive measures. Furthermore, it is a useful tool for tracking the rising incidence of the disease and creating affordable control measures for both its incidence and consequences. Thus, the main focus of this study is to propose a mathematical model that is employed to forecast changes in the prevalence of diabetes using a saturated incidence rate, which reflects a diminishing rate of new infections as the number of affected individuals increases, by extending the Diabetes Complication (DC) and Susceptible Diabetes Complication (SDC) models. It has been established which factors lead to both endemic and disease-free equilibriums. The conditions that contribute to the equilibrium between endemic and disease-free states are identified. Based on the eigenvalues of the Jacobian matrix, the local stability of the equilibrium points has been determined. Since diabetes is not a transmissible disease, there is no disease-free equilibrium and when the constant term 𝑅0 < 1, the endemic equilibrium point is locally asymptotically stable. Additionally, the Lyapunov function theory has been utilized to study global stability. We observe the effect of the term saturated incidence rate under some parameter conditions. Based on data gathered from annual mortality reports and health bulletins that have been published by the Ministry of Health, Sri Lanka and the United Nations’ World Population Prospects the parameters for complications related mortality rate, the natural mortality rate, and the birth rate are estimated. Numerical simulations using MATLAB’s ODE 45 technique are conducted to validate the analytical findings of the proposed model and assess its approach. This model also monitors the number of susceptible people as well as the population with and without diabetes complications. We predict that the 2.29 million will be the approximate Sri Lankan total prevalence of diabetes in 2024 and additionally, it notes an average annual increase of about 0.2199% in diabetes prevalence. The model’s accuracy is highlighted by its minimal average relative error, making it effective for forecasting and valuable for informing disease management strategies.
Qualitative and quantitative analysis of the dengue fever model with reference to the data obtained from Sri Lanka
(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Jayasooriya, J. A. D. L. K.; Munasinghe, J.; De Silva, T. M. M.
Dengue is a rapidly emerging pandemic disease in many parts of the world, especially in tropical and non-tropical areas. The dengue outbreak has a multisectoral impact on the medical, societal, economic, and political sectors. The main economic impact of dengue is due to production costs. Lower-income groups may be more vulnerable to dengue and may also bear a higher financial cost as a result of it. Moreover, once a family’s primary wage worker contracts the sickness, the lost productivity that results from the illness puts a financial strain on the family. This affects the household’s ability to pay for treatment. Resource-poor countries are particularly hard hit because of inadequate public health infrastructure, lack of resources to combat the vector, and limited health care services to manage cases. Dengue incidence has increased in Sri Lanka over the past 20 years, with deaths and illnesses increasing disproportionately among adults compared to children. Dengue fever is caused by Dengue virus, first recorded in the 1960s in Sri Lanka. Aedes aegypti and Aedes albopictus are both mosquito species native to Sri Lanka. In this study, a SIR model for the human population and SI model for the vector (mosquito) population with a constant treatment function is considered to describe dengue transmission. The equilibrium points and the basis reproduction number (𝑅0) are computed. It is emphasized that reproduction number affects the asymptotic stability for both endemic and disease-free equilibrium points. The conditions leading to the disease-free and endemic equilibrium are determined. The eigenvalues of the Jacobian matrix corresponding to the reduced system are used to demonstrate the local stability for the equilibrium points, and the Lyapunov function theory is used to assess the global stability. When 𝑅0 ≤ 1, the disease-free equilibrium point exhibits global asymptotic stability. But as 𝑅0 > 1, the endemic equilibrium point becomes globally asymptotically stable. Based on actual data gathered from the Institute of Epidemiology Unit Ministry of Health in Sri Lanka, the parameters for infection and disease-related death rates are estimated. The numerical simulation is used to validate the findings of the analytical results. It is important to determine a suitable capacity for treating a disease. We have observed that the treatment function affects the infected compartment. That is, the increased rate of treatment function reduces the infection. This shows that to eliminate the disease adequate treatment facilities must be provided.
Stochastic modelling of Lotka-Volterra competition
(Faculty of Science, University of Kelaniya Sri Lanka, 2022) Ayesha, K. M. S.; De Silva, T. M. M.
The concept of stochasticity, which is based on probability theory, has played a vital role in describing the population fluctuations in most species. Demographic and environmental stochasticity are the main branches of stochasticity and occur due to the random nature of events and irregular or noisy dynamics, respectively. In particular, this study focuses on the concept of demographic stochasticity for studying the distribution of two competing populations. In the literature, the deterministic models of two competing populations have been studied, including the Lotka-Volterra competition model. Unlike prior work, we analyse the stochastic modelling of two competing populations where one population is subject to the Allee effect and understocking. The deterministic model of two competing populations, which is based upon the classical Lotka-Volterra competition model, is used to construct the corresponding continuous-time Markov chain (CTMC) and Ito stochastic differential equations (SDEs). Moreover, in the construction of CTMC and SDEs, demographic variability due to random birth and death have been applied to the populations, which is absent in the deterministic setting. In addition, the moments of the random variables in the populations based on the moment-generating functions of the transition probabilities are derived theoretically in such a way that the transition probabilities satisfy the forward Kolmogorov differential equations. Also, there is an infinite number of SDE models that correspond to the same ordinary differential equation system. In this study, we formulate two SDE models considering two different birth and death rates to see the variability in population interactions. The parameter values are taken from existing literature to justify the analytical results. The Euler-Maruyama numerical method is applied to simulate the numerical solutions of the Ito stochastic differential equations for comparing both types of stochastic models with the deterministic system numerically. From the numerical simulation, we have observed that the sample paths of the SDEs are closer to the solution of the deterministic model. Moreover, variabilities of the population interactions are highly correlated with the birth and death rates. In addition, for the chosen parameter values, though the populations coexist in the deterministic setting, we capture sudden population extinction in the stochastic setting. The study concludes that the theoretical results established in the deterministic setting may not be valid in the stochastic models due to random effects of the birth and death process embedded in the populations. Therefore, stochastic modelling with the Allee effect and stocking can significantly affect the competition outcomes and population interactions.