In this study, we propose a compartmental mathematical model that considers two interacting populations (citrus plants and insect vectors) and investigate the transmission dynamics of Huanglongbing in citrus crops. This disease is caused by the bacterium Candidatus Liberibacter asiaticus and is vectored by the psyllid Diaphorina citri. The disease is modeled under the following three main assumptions: there is vital dynamics with constant recruitment rates of citrus plants, the force of infection in both populations is a spatially dependent function varying with geographic location, and there is a spatial displacement of the vectors. In the main results of the paper, we formulate a coupled ordinary and partial differential equation system with initial and zero flux boundary conditions, establish the existence and uniqueness of solutions to the proposed model by applying semigroup theory, and introduce a numerical approximation of the system. Moreover, we develop a stability and persistence analysis. From the analytical point of view, we calculate the basic reproduction number $R_0$ and prove three facts: the disease-free equilibrium is globally asymptotically stable when $R_0<1$; the disease-free equilibrium is globally asymptotically stable when $R_0>1$; and the hybrid system exhibits uniform persistence of infection when $R_0>1$. In addition, we present some numerical examples.

Advancing Sustainable Development Goals 4 and 10 in higher education requires understanding students’ diverse academic backgrounds and how they relate to performance in early mathematics courses that are critical for academic progression. This study addresses this need by linking cohort-level admission profiles with individual course-specific predicted probabilities of failure. Anonymized admission data from 193 first-year engineering students at a regional university in southern Chile were analyzed using 7 academic and school-background characteristics. Gower dissimilarity and partitioning around medoids identified 3 partially overlapping profiles (average silhouette width $=0.435$) that summarized cohort heterogeneity and were associated with subsequent Algebra and Calculus trajectories (Cramér’s $V=0.285$). Separate ridge-penalized logistic models, evaluated through 100 stratified training–validation splits, achieved mean areas under the receiver operating characteristic curve (AUC) of $0.832$ and $0.735$, with Brier scores of $0.185$ and $0.221$ for Algebra and Calculus, respectively, indicating that the same admission information was more predictive of failure in Algebra. Internal validation showed that the predicted probabilities of failure spanned too narrow a range in both courses. Their use to determine which students are offered support requires validation in new cohorts. Individual probabilities nevertheless varied substantially within profiles and overlapped across them. Admission profiles can therefore inform cohort-level support planning, while course-specific probabilities provide preliminary signals for individual review without assigning fixed labels. These complementary signals may inform early, differentiated, and non-stigmatizing academic support, whose effects on equitable student progression and retention remain to be evaluated.

In this article, we are interested in studying and analyzing the heat conduction phenomenon in a multi-layered solid. We consider the physical assumptions that the dual-phase-lag model governs the heat flow on each solid layer. We introduce a one-dimensional mathematical model given by an initial interface-boundary value problem, where the unknown is the solid temperature. More precisely, the mathematical model is described by the following four features: the model equation is given by a dual-phase-lag equation at the inside each layer, an initial condition for temperature and the temporal derivative of the temperature, heat flux boundary conditions, and the interfacial condition for the temperature and heat flux conditions between the layers. We discretize the mathematical model by a finite difference scheme. The numerical approach has similar features to the continuous model: it is considered to be the accuracy of the dual-phase-lag model on the inside each layer, the initial conditions are discretized by the average of the temperature on each discrete interval, the inside of each layer approximation is extended to the interfaces by using the behavior of the continuous interface conditions, and the inside each layer approximation on the boundary layers is extended to state the numerical boundary conditions. We prove that the finite difference scheme is unconditionally stable and unconditionally convergent. In addition, we provide some numerical examples.