UC Merced

The Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation is widely used to describe thekinetics of materials phase transformations, such as solidification, precipitation, or recrystallization. Beyond materials science, the JMAK equation has been applied to many non-thermodynamic transformations in the life sciences and social sciences. Through a cross-disciplinary lens, my research has explored parallels between the JMAK model and the Susceptible-Infected-Recovered (SIR) epidemiological model, particularly in the context of modeling COVID-19 data. This effort aspires to promote communication about the spread of an epidemic via a mathematical framework that is more accessible than the SIR model to non-technical audiences. The effectiveness of the JMAK model at both retroactively describing and proactively predicting (forecasting and projecting) the rise in case numbers as a function of time is assessed. The JMAK framework is also shown to be useful in identifying possible kinetic mechanisms of spread of COVID-19. Application of the model is extended to other infectious diseases, demonstrating versatility beyond the scope of a single disease.

Furthermore, my study broadens the applicability of the JMAK formulation in materialsphase transformation by incorporating a variety of time-dependencies into the nucleation and growth rates. Many common phase transformations involve kinetics in which the growth rate varies in proportion to time raised to a power that can range from -0.5 (diffusion control; Fickian diffusion; Case I diffusion), through values that represent anomalous diffusion, to zero (interface control; Case II diffusion) and beyond (Super Case II diffusion). However, classic derivation of the JMAK equation primarily assumes that growth rate is constant, and that nucleation is either instantaneous or sporadic at a constant rate; consideration of a possible time dependence of these rates is limited to the time dependence being linear, which is uncommon in practice. My study extends the classic derivation by generalizing the formulation of the growth and nucleation rates, and introduces some additional refinements to represent mechanistic processes. The reaction equations derived in this process explicitly demonstrate how the time dependence of nucleation rate and growth rate affects the constants in the JMAK equation, allowing the possibility of values beyond their classical range. The study also identifies several circumstances under which the two principal parameters contained in the JMAK model are explicitly linked.