Ordinary Differential Equation Models of Chemical Kinetics, HIV-Prevention Pathways, Epidemic Spread, and Growth–Decay Processes

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.