Ordinary differential equations provide a common language for representing rates of change in chemical, biological, epidemiological, and financial systems. This study develops and computationally examines four model families drawn from physical and life-science applications: the dimensionless Lengyel–Epstein model for the chlorine dioxide–iodine–malonic acid reaction; a six-compartment demographic, exposure, infection, and AIDS-progression model motivated by delayed first sexual intercourse; the classical susceptible–infectious–removed epidemic model; and exponential growth and radioactive-decay models. Equilibria and local stability conditions are derived analytically, while numerical solutions are obtained with adaptive Runge–Kutta integration. For the chemical model with illustrative parameters $a=12$ and $b=0.30$, the positive equilibrium is unstable and the numerical trajectory approaches sustained oscillation. The delayed-intercourse model is locally asymptotically stable when the feedback between the sexually active and under-age compartments is weaker than total demographic removal, specifically when $(d_1+m_1)(d_2+m_2+b_2)>b_1m_1$. For the epidemic illustration, the effective transmission rate is $0.8$ per day, the recovery rate is $0.125$ per day, and $R_0=6.4$; the infectious population peaks at approximately $554$ persons near day $13$ in a population of $1,000$. Continuous $5%$ financial growth increases $20,000$ monetary units to $23,236.68$ after three years, whereas an $800,\mathrm{mg}$ bismuth-210 sample with a five-day half-life declines to $12.5,\mathrm{mg}$ after $30$ days. The results demonstrate how a shared differential-equation framework supports model formulation, stability analysis, simulation, and transparent comparison across distinct applications. All numerical outcomes are illustrative and are not fitted to clinical or laboratory observations.
Plasmodium falciparum</em>, the most virulent human malaria parasite, possesses a highly regulated genome that supports survival, adaptation, virulence, drug resistance, developmental switching, and stress responses within the human host. This study characterizes stable regulatory attractors in a malaria-parasite gene regulatory network using an asynchronous Boolean update method. The network is formulated as a Boolean dynamical system with nine biological components: \(X_1=\mathrm{PfEMP1}\), \(X_2=\mathrm{PfCRT}\), \(X_3=\mathrm{PfMDR1}\), \(X_4=\mathrm{PfDHFR}\), \(X_5=\mathrm{AP2\text{-}G}\), \(X_6=\mathrm{PfSIR2A}\), \(X_7=\mathrm{PfK13}\), \(X_8=\mathrm{HP1}\), and \(X_9=\mathrm{H3K9me3}\), representing regulators associated with virulence, drug resistance, developmental regulation, stress response, and epigenetic control. External pressure is represented by a general stress signal \(\omega\) together with selectors for chloroquine \((\lambda)\), antifolate pressure \((\alpha)\), artemisinin pressure \((\beta)\), and partner-drug pressure \((\rho)\). Fixed points are states satisfying \(F(X)=X\); such states are invariant under either synchronous or asynchronous updating. Under the all-stress-OFF condition, the model yields three fixed points, whereas the all-stress-ON condition yields four fixed points. The attractors show how stress-dependent logical regulation can shift the network between distinct stable expression patterns and provide a mathematical framework for exploring regulatory states associated with parasite adaptation and antimalarial pressure.
Nonlinear Volterra integral equations (VIEs) of the second kind arise in numerous physical contexts, including nonlinear heat conduction with memory, viscoelasticity, and population dynamics. However, when the kernel satisfies only a one-sided Lipschitz condition, classical contraction-based numerical methods fail, and efficient, provably convergent solvers are lacking. In this paper, we develop a fully discrete, computationally efficient numerical scheme for such equations. The method combines an implicit Euler time discretisation with an inexact Mann inner solver that requires no Jacobian evaluations, making it ideally suited for large-scale parallel computations. We establish global convergence of the adaptive, fully discrete scheme under realistic smoothness assumptions, and introduce a novel adaptive time-stepping strategy based on the Mann residual. The performance of the method is demonstrated on a physically motivated model of one-dimensional (1D) nonlinear heat conduction with memory. Extensive numerical comparisons against Picard iteration and Newton’s method show that the Mann-based solver is robust for large time steps, scales efficiently to high-resolution spatial discretisations, and maintains linear convergence rates as predicted by theory. Our open-source implementation provides a practical, ready-to-use tool for the computational physics community, and the framework is easily extended to partial integro-differential equations (PIDEs) and fractional-order memory kernels.
