Communication in Biomathematical Sciences

Comparative Analysis of Integer and Fractional Order Pharmacokinetic Models for Metformin

SANTOSHI PANIGRAHI, SUNITA CHAND

In this work, a fractional-order compartmental model for the pharmacokinetics of metformin is proposed by extending the classical integer-order system in the Caputo sense. The model describes the transport of the drug between the physiological compartments and incorporates memory effects in the inter-compartmental transfer. The equilibrium point and stability of the system are analysed. Numerical simulations are carried out using the predictor--corrector method for the fractional derivative. A comparison between the integer-order and fractional-order models is performed for the kidney drug concentration, which represents the measurable pharmacokinetic quantity. The fitting performance is evaluated through the root mean square error, which shows that the fractional-order model provides a closer agreement with the concentration--time data. The influence of the fractional order on the drug transport process is examined and it is observed that the fractional model leads to a slower drug depletion and a prolonged elimination phase. A sensitivity analysis is also presented to identify the parameters that significantly affect the drug concentration. The results indicate that the dissolution rate and the renal clearance rate play an important role in the pharmacokinetic behaviour of metformin. The proposed model provides a more flexible and physiologically meaningful framework compared to the classical integer-order formulation.

Fractional-order ratio-dependent May-Holling-Tanner model with an alternative food source for the predator

Marco, Santiago, Bertha

This work introduces and examines a predator-prey model with a mixed functional response (ratio-dependent Holling type II), alternative food for the predator, and logistic growth in both populations, using the ABC (Atangana--Baleanu--Caputo) operator to reflect the effects of population memory. A positively invariant region is established, strictly demonstrating the uniqueness and existence of solutions. Through a Volterra-type Lyapunov function, sufficient conditions are obtained for the local asymptotic stability of critical points, and global stability of interior critical points. Two numerical scenarios where the fractional order $\alpha$ regulates memory intensity are used to confirm the theoretical results. It is found that population memory has a significant impact under unstable circumstances: low values of $\alpha$ (strong memory) cause continuous oscillations and can result in the extinction or recovery of species; conversely, high values of $\alpha$ (weak memory or classical system) lead to rapid convergence to the critical point. The numerical approximations obtained through the predictor-corrector method are reliable, thanks to Ulam-Hyers stability. By including the system's history, this study offers a more realistic interpretation of predator-prey dynamics.

A Physics-Informed Neural Network for the Two-Dimensional Spatial SIR Model

Aditya Firman Ihsan

Spatially-explicit epidemic models describe how an outbreak propagates through a geographic region, capturing phenomena---infection fronts, spatial heterogeneity, travelling waves---that the classical spatially-homogeneous SIR model cannot represent. Such models take the form of reaction-diffusion partial differential equations (PDEs), whose numerical solution by classical mesh-based methods can be costly in two or more spatial dimensions. In this work we develop a physics-informed neural network (PINN) for the forward problem of the two-dimensional spatial SIR model, in which the infected compartment diffuses over the unit square subject to zero-flux (Neumann) boundary conditions. We present a formulation in which all derivatives appearing in the PDE residual, including the Laplacian $\nabla^2 I$, are computed by exact analytic differentiation through the network via recursive chain-rule propagation of first- and second-order derivative tensors, achieving machine-precision accuracy ($<10^{-11}$ relative error) without any automatic-differentiation library. We introduce an adaptive collocation strategy that concentrates residual evaluation near the infection front. Using a finite-difference solution as ground truth
($\beta=0.50$, $\gamma=0.10$, $D_I=0.01$, basic reproduction number $\Rzero=5$), the proposed network reproduces the susceptible field to a relative $L^2$ error of $0.74\%$ and the infected field to an absolute root-mean-square error of $6.6\times 10^{-3}$. A systematic ablation study over collocation strategy, network architecture, loss-component weighting, and activation function reveals two findings of practical importance for biomathematical modellers: (i) network depth scaling is non-monotone, the moderate-depth network outperforming both shallower and deeper alternatives on the infected field; and (ii) over-weighting the initial-condition loss relative to the PDE residual prevents convergence of the time-evolution entirely. The framework provides a mesh-free route to spatial epidemic simulation and a foundation for future data-driven parameter inference.

A Compartmental Model for Tuberculosis Transmission Dynamics with Control Measures

MORGAN WALICHO

Tuberculosis is a global endemic that claims millions of lives every year. Therefore,there is need for continuous research in order to understand its dynamics for effective prevention and control as recommended by WHO. The study developed a six compartmental $SVLATR$ mathematical model that incorporated TB transmission dynamics with control measures to assess their impact on TB spread . The threshold quantity called basic reproduction number $R_0$ ,that determines whether the disease persists and spreads or gets eliminated was computed using the Next Generation Matrix ($NGM$) and found to be $ R_0 \approx 1.73 > 1$ indicating persistence of the endemic. The model was used to predict future trends in TB and projected a decline in TB incidence,prevalence and mortality for the next 5 years but insufficient to fully eliminate TB. The stability properties of the system were analyzed using Jacobian matrix and eigenvalue analysis. The condition for stability of the Disease Free Equilibrium ($DFE$) is stable if $R_0 < 1$ . Since $R_0 > 1$ it was found to be unstable while Endemic Equilibrium ($EE$) is locally asymptotically stable since $R_0 >1$ . Bifurcation analysis indicated that the system is at the endemic state and measures to lower reproduction number below unity are needed to eliminate TB. The simulated results using the real world epidemiological data indicates persistence of TB under the current intervention measures. This modeling approach provides policymakers and public health stakeholders with evidence based recommendations for improving TB control.