Alzheimer's disease is the most common neurodegenerative disorder. Its pathological development is connected with the misfolding and accumulation of two toxic proteins: amyloid-beta and tau proteins. Mathematical models provide a valuable quantitative tool for monitoring disease progression. Here, we propose and compare a novel framework where the spatio-temporal dynamics of amyloid-beta and tau proteins is modeled based on employing either three-dimensional patient-specific geometries or through reduced network-based models defined on the brain connectome. More specifically, a high-fidelity biophysical model is proposed on three-dimensional brain geometries reconstructed from magnetic resonance imaging, whereas a network-based reduced formulation is defined on the brain connectome. For both approaches, a suitable numerical discretisation is proposed. A sensitivity analysis is presented to quantify the influence of model parameters on protein concentration patterns as well as compare the quality of the predictions. For both approaches, the results are validated against PET-SUVR clinical data using 18FAZD4694 for amyloid-beta and 18FMK6240 for tau protein. The results indicate that the three-dimensional model provides the most accurate and biologically consistent description of the disease progression, but remains computationally demanding. On the other hand, the reduced graph-based model is cheaper, but it is not always able to achieve reliable results.

Aedes aegypti is the primary vector of several arboviruses of major public health concern such as dengue, Zika, chikungunya, and yellow fever. Wolbachia is a symbiotic bacterium that is transmitted maternally and induces cytoplasmic incompatibility. Wolbachia suppresses replication of Aedes aegypti. We present a spatiotemporal model that characterizes the spatial distribution and persistence of Aedes aegypti mosquito populations after the release of Wolbachia-infected individuals. The model is formulated as a coupled system of reaction–diffusion equations with mechanisms of imperfect transmission, spatial heterogeneity and mosquito movement. The model describes invasion thresholds, stability conditions and qualitative behavior for different parameters. Numerical simulations are performed to address the influence of release size and spatial distribution on establishment success, strain-dependent conditions that maximize long-term persistence and the interplay between environmental heterogeneity and infection dynamics over extended time horizons. Our results provide some quantitative guidance for designing sustainable biological control strategies for mosquito-borne diseases, particularly dengue.

Age-related macular degeneration (AMD) is a leading cause of severe vision loss among the elderly population worldwide. Current treatment strategies rely on repeated intravitreal injections of anti-vascular endothelial growth factor (anti-VEGF) agents such as aflibercept and ranibizumab. To better understand the transport and efficacy of these drugs, we develop a three-dimensional pharmacodynamic model describing their distribution and interaction with VEGF in the eye.

We present a three-dimensional computational pharmacodynamic model that describes the transport, distribution, and therapeutic action of intravitreally administered anti-VEGF agents. The model consists of two coupled compartments representing the vitreous body and the retina. Drug transport in the vitreous is governed by a system of coupled convection–diffusion–reaction equations accounting for aqueous humor flow, molecular diffusion, and reversible binding kinetics. In the retina, diffusion–reaction equations describe drug penetration through the inner limiting membrane and subsequent interaction with VEGF. The model incorporates experimentally determined transport parameters and realistic ocular geometry.

The resulting system of partial differential equations is discretized in time using one-step θ-schemes and in space by the finite element method. Numerical simulations are performed to compare the pharmacodynamic behavior. Quantitative analyses include retinal drug exposure, VEGF suppression dynamics, and drug-specific treatment durability. Our simulations demonstrate that only a limited fraction of the injected drug reaches the retinal tissue due to the complex ocular physiology, including additional transport pathways and boundary conditions such as aqueous humor outflow through the Schlemm’s canal. Furthermore, the model predicts substantial differences in the duration of VEGF suppression between the two therapeutic agents.