Communication in Biomathematical Sciences

ANALYSIS OF DYNAMICS AND BIFURCATIONS OF A DISCRETE HOST-PARASITOID SYSTEM WITH HOST REFUGE MECHANISM

Abadi Abadi

Nicholson–Bailey host–parasitoid dynamics are studied in a modified model that includes a host refuge
mechanism to reduce the instability of the classical formulation. The impacts of the host refuge fraction and
the host reproduction rate on fixed-point stability and on qualitative changes in system dynamics are examined.
Local stability is obtained from Jacobian linearization, while bifurcation types are determined using a geometric
approach in the trace-determinant plane and supported by numerical simulations. Two fixed points are present:
extinction and coexistence. The extinction fixed point loses stability through a transcritical bifurcation when
the host reproduction rate exceeds a critical threshold. At the coexistence fixed point, the host refuge fraction
becomes an important stability parameter; decreasing the refuge level beyond a critical value leads to a loss of
stability via a Neimark–Sacker bifurcation, producing bounded oscillations on an invariant closed curve. The
main outcome is a two-parameter stability map that clearly separates regions of extinction, stable coexistence,
instability with bounded oscillations, and solution divergence caused by the assumption of unbounded growth.
This map summarizes how combinations of host intrinsic reproduction rate and host refuge fraction level affect
stability and the qualitative behavior of the model across parameter space.

Bifurcation Analysis of Top-down Effects and Phosphorus Loading in Lake Eutrophication

Md. Abdul Aziz

Lake eutrophication is a complex process that can be influenced by both bottom-up (such as nutrient loading) and top-down (such as grazing and predation by herbivores) factors. Lake studies traditionally focus on nutrient management and the roles of top-down factor has been overlooked. To address this issue, this
paper aims (i) to formulate a mathematical model capturing the dynamic interactions among phosphorus, algae, and zooplankton, (ii) to assess the top-down effects in lake eutrophication and (iii) to provide limnological implications based on the bifurcation results. The formulated model is applied to Tasik Harapan, a shallow tropical eutrophic lake in Universiti Sains Malaysia. Then, we conduct both co-dimension one and co-dimension two bifurcation analyses by using MatCont to identify critical thresholds in the lake dynamics. The external phosphorus loading rate, zooplankton grazing rate, zooplankton mortality rate and fish predation rate are chosen as the bifurcation parameters. In codimension one bifurcation analysis, both transcritical and Hopf bifurcations are detected. Further simulations in co-dimension two bifurcation analysis identify different dynamic regions, including (i) stable equilibrium (co-existence of algal, zooplankton and phosphorus), (ii) stable equilibrium (coexistence of algal and phosphorus only, extinction of zooplankton) and (iii) stable limit cycle. The simulation results reveal that zooplankton grazing, zooplankton mortality and fish predation rates will affect the lake eutrophication. Our study suggests that top-down control can be used as a complementary strategy to nutrient reduction in lake eutrophication management.

Spatial Fractional Reaction Diffusion Systems for Pattern Formation by Cancer and Immune Cell Dynamics

E.A.S. Navodya

This research investigates reaction-diffusion systems, including their fractional counterparts, to model the spatial and temporal dynamics of cancer cell proliferation within tissue environments. The study examines the interactions between cancer and immune cells, highlighting how these dynamics, coupled with diffusion processes, give rise to spatial patterns. Both the classical reaction-diffusion model, which employs local diffusion, and fractional models, characterized by non-local diffusion, are analyzed to simulate cancer cell spread. The mathematical models are solved using a semi-implicit finite difference method for the classical
model due to its stability and computational efficiency, and the Fourier spectral method for the fractional model to accurately capture the non-local diffusion behavior. A stability analysis identifies conditions for Turing instabilities, which are essential for understanding the mechanisms underlying pattern formation in reaction-diffusion systems. Key findings demonstrate that when the fractional order μ decreases, the generated patterns are increasingly irregular reminiscent of chaotic and invasive growth observed in realistic cancer dynamics in 2D. Extending this work to 3D simulations has provided deeper insights into the emergence of more complex and fractal like spatial patterns and improved the biological realism of the models. These results underscore the critical role of reaction, diffusion kinetics in understanding cancer cell behavior and offer promising avenues for developing predictive tools to inform future cancer treatment strategies. Future research will focus on incorporating additional biological complexities which generate non symmetric patterns, such as heterogeneous tissue environments and adaptive immune responses, to further enhance the models’ applicability.

Sensitivity Analysis and Optimal Control for Nipah Virus Outbreak

Binti Mualifatul Rosydah

Nipah virus (NiV) is a zoonotic virus that potentially causes outbreaks with high mortality rates, namely about 40-75%. The spread of the virus occurs through direct contact between humans and infected animals, consumption of contaminated food, or human-to-human transmission. As there is no vaccine or specific treatment for Nipah virus, a public health policy-based control is required. In this study, we developed a mathematical model to study the dynamics of Nipah virus spread. Furthermore, calculating the basic reproduction number R0 as an indicator of the spread of the disease, conducting sensitivity analysis of the main parameters of the model, and evaluating the effectiveness of three control strategies, namely health campaigns on exposed individuals, quarantine on infected individuals, and treatment to increase the recovery rate. The purpose of this study is to minimize the number of exposed and infected individuals and the control implementation cost. The mathematical model used in this study is a compartmental model that divides the human population into five subpopulations: susceptible (S), exposed (E), infected (I), recovered (R), and deceased (D). The Pontryagin Maximum Principle (PMP) is applied to solve the problem. The Forward-Backward Sweep method is used to solve optimization problem numerically. The results of the R0 calculation show that the effective contact rate, the duration of the incubation period, and the treatment rate are the most influence parameters in determining the spread of the disease. Sensitivity analysis confirmed that reducing R0 is most effective by improving the efficiency of health campaigns and treatment. Numerical simulation results show that the optimal combination of the three control strategies significantly reduces the number of exposed and infected individuals compared to the application of one control strategy separately. With an optimally designed combination of control strategies, the implemented policy can achieve a balance between reducing virus spread and cost efficiency.