Abstract. We present a mathematical model of the asexual blood stage of a malaria infection from Plasmodium falciparum, including time, maturation of infection, and phase of infection. The phases of infection include the merozoite phase, immature trophozoite phase, trophozoite phase, schizont phase, as well as the production of gametocytes. Our model shows oscillatory behavior of the parasite between the phases with a period that strongly depends on the generation reproduction number of the parasite. We separate parasite evolution into different growth regimes classifying final sizes when the parasite population does not change and final proportions when it grows exponentially. Without fitting parameters, we compare our model to data for parasitized red blood cells showing excellent agreement. The phase structure of our model allows us to also gain insight into sequestered parasitized red blood cells which are unmeasured. We demonstrate how parameters within the phase structure can impact parasite growth and maturation with potential implications for antimalarial drugs and other therapies.
Abstract. In the context of PDE-constrained optimization theory, source identification problems traditionally entail particles emerging from an unknown source distribution inside a domain, moving according to a prescribed stochastic process, e.g., Brownian motion, and then exiting through the boundary of a compact domain. Given information about the flux of particles through the boundary of the domain, the challenge is to infer as much as possible about the source. In the PDE setting, it is usually assumed that the flux can be observed without error and at all points on the boundary. Here we consider a different, more statistical presentation of the problem, in which the data has the form of discrete counts of particles arriving at a set of disjoint detectors whose union is a strict subset of the boundary. In keeping with the primacy of the stochastic processes in the generation of the model, we present a stochastic gradient descent algorithm in which exit rates and parameter sensitivities are computed by simulations of particle paths. We present examples for both Itô diffusion and piecewise-deterministic Markov processes, noting that the form of the sensitivities depends only on the parameterization of the source distribution and is universal among a large class of Markov processes.
Abstract. We adapt the article of Forien, Pang, Pardoux, and Zotsa [ Ann. Appl. Probab., 35 (2025), pp. 2175–2216 ] on epidemic models with varying infectivity and waning immunity to incorporate the memory of the last infection. To this end, we introduce a parametric approach and consider a piecewise deterministic Markov process modeling both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly chosen for the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. The local stability of these equilibria is also analyzed. The endemicity threshold is computed for several examples, including models that incorporate a vaccination policy, and a local stability result is obtained for a memory-free SIS-type model.
Abstract. This paper examines the Maxwell system of electrodynamics within the framework of distributions. A primary objective is to establish general boundary conditions for fields at interfaces when the charge and current densities are measures localized on the interface. From this analysis, the paper presents a derivation of the generalized Snell’s law, along with formulas for the amplitudes of the reflected and transmitted waves in terms of the incident amplitude.
Abstract. Control of chemical release from soluble, or partially soluble, porous systems is important in a variety of applications, including drug release from pharmaceutical tablets. The underlying process involves complex interactions between dissolvable microparticles in the structure and a solvent, governed by factors like solute diffusion, dissolution reaction kinetics, and matrix porosity. We use asymptotic homogenization to develop a tractable model for chemical release from a three-dimensional porous domain consisting of soluble spherical microparticles. The effect of the evolving microstructure on release is captured in systematically-derived macroscale bulk diffusion and reaction terms. The model is analyzed and applied to a radially-symmetric spherical tablet. When drug particles near the tablet boundary dissolve completely, the tablet shrinks, initiating a moving boundary at the macroscale. A differential equation is derived to track the evolution of this boundary over time. We investigate how the physicochemical properties of a tablet affect its internal drug transport dynamics, mass release through its surface, and the time required to achieve complete dissolution. This study lays the groundwork to model more complex microstructural geometries and partially soluble systems, offering a promising framework to optimize release profiles from real drug delivery devices.
