Cell sorting mediated by direct cell–cell contact or cellular protrusions can be described by nonlocal cell–cell adhesion models of haptotaxis type, whereas communication through diffusible chemical signals is described by Keller–Segel type chemotaxis systems. We investigate the mathematical relationship between these two descriptions. We first prove that, in the fast signal diffusion limit, subsequences of weak solutions of a Keller–Segel type parabolic–parabolic chemotaxis system with possibly degenerate nonlinear diffusion and density saturation converge to weak solutions of the corresponding parabolic–elliptic system. Eliminating the elliptic chemical fields rewrites the population equation as a nonlocal adhesion model whose interaction kernel is a finite linear combination of Green functions. We then prove, in arbitrary spatial dimensions, that the periodization of the gradient of a prescribed radially symmetric interaction potential can be approximated by finite linear combinations of gradients of Green functions. Combining these results, we prove that suitable weak solutions of the nonlocal adhesion model and a parabolic–parabolic chemotaxis system can be chosen with an arbitrarily small difference over any fixed finite time interval. Numerical simulations compare the nonlocal, parabolic–elliptic, and parabolic–parabolic models and examine how the number of kernel terms and the relaxation time affect their differences.
Human papillomavirus (HPV) is a common sexually transmitted infection whose transmission and disease progression depend strongly on age and are shaped by prevention strategies such as vaccination and screening. Temporal changes in transmission conditions, contact patterns, and healthcare conditions may induce epidemiological regimes that switch over time, making it difficult to characterize long-term transmission and invasion thresholds. We study an age-structured model for HPV transmission in a randomly fluctuating epidemiological environment, where temporal variability in epidemiological conditions is represented by a finite-state continuous-time Markov chain, while vaccination is assumed to remain fixed across regimes. Each regime is governed by a deterministic age-structured system. Using semigroup methods, we characterize the invasion threshold of the randomly switching system, relate it to the reproduction numbers of the regime-specific models, and derive fast- and slow-switching limits. We further establish exponential extinction of the infected population in mean and almost surely when the random reproduction number is below one, while above the threshold we obtain instability of the linearized first-moment dynamics. Our results provide a rigorous mathematical framework for understanding HPV dynamics under temporally varying epidemiological conditions.
In the practice of infectious disease prevention and control, the regional and cyclical interventions (e.g., seasonal sub-regional disinfection, localized lockdowns, or phased vaccination campaigns) are key factors in regulating the dynamic characteristics of disease transmission. In this paper, we investigate the dynamics of two reaction-diffusion susceptible-infected-susceptible (SIS) epidemic models incorporating spatio-temporal periodic control strategies. The models simulate the application of two strategies: one involves establishing local spatial control zones where the disease transmission rate is reduced to zero; the other entails population-wide control (e.g., large-scale vaccination, mask-wearing, or voluntary quarantine) to achieve an overall reduction in the transmission rate. By analyzing the impact of control cycles on disease spread, we find that longer durations and higher proportions of control measures enhance the likelihood of disease eradication, and further derive the critical period value that determines the outbreak and extinction of infectious diseases. In particular, we obtain the optimal and worst positions of the control zone in one-dimensional periodic space. We establish the threshold dynamics of the ordinary differential equation SIS model with multi-stage temporal control and the partial differential equation SIS model with multi-stage population-wide control. Finally, numerical simulations indicate that the optimal control zones should be prioritized for establishment in high-risk areas rather than low-risk areas. These insights may provide theoretical guidance for designing effective disease control policies.
This paper examines the competition dynamics between two mosquito species, Aedes aegypti and Aedes albopictus, incorporating seasonal effects and reproductive interference based on experimental data. We develop a hybrid model that switches between continuous growth during favorable seasons and discrete overwintering updates, capturing both the asymmetric mating interference observed in laboratory studies and the seasonal constraints in temperate regions. To analyze this complex system, we first characterize the continuous subsystem’s global dynamics, then incorporate seasonal switching to derive explicit thresholds determining long-term outcomes. The main contributions include establishing a separatrix that divides initial conditions leading to competitive exclusion, deriving six biologically interpretable thresholds that classify the system’s behavior, and proving rigorous stability criteria for single-species states, together with the necessary and sufficient conditions for the existence and uniqueness of the coexistence state. Our approach bridges experimental data and mathematical theory, providing a general framework for analyzing seasonal competition systems. Based on our results, we propose four mathematical questions on basin characterization, convergence time estimation, multistability analysis, and high-dimensional extensions. These questions contribute to advancing both the theory of hybrid dynamical systems and the mathematical modeling of ecological invasions.
Optimal control problems in mathematical biology are commonly solved by the forward-backward sweep on the Pontryagin’s Maximum Principle necessary conditions. The sweep has known stability issues at constraint-saturated optima. We frame the same problem within the universal differential equation framework. A neural network represents the control. The cost functional serves as the training loss. We prove that any stationary point of the training loss satisfies the continuous first-order necessary conditions to a discretisation-controlled accuracy. On a grid of bifurcation-parameter and initial-condition scenarios drawn from a tumor-immune dynamics model, the standard sweep fails on most scenarios in the nominal regime. Neural network training reaches a stable optimum on every scenario, with a lower final cost than the converged damped sweep iterate. The pipeline accommodates cost functional variants by changing only the loss function. A Sobol sensitivity analysis confirms robustness of the trained schedule to perturbations of the cost weights.
