Weight-loss maintenance remains a persistent public health challenge, partly because social influence and behavioral relapse can return individuals who have achieved a healthy weight to an overweight state. This study extends a previously developed social obesity model by introducing a time-dependent follow-up control κ(t) that reduces the effective relapse rate from the recovered class. An optimal control problem is formulated to minimize the prevalence of overweight and obesity together with the implementation cost of follow-up support. We prove the existence of an optimal control and use Pontryagin's Maximum Principle to derive the corresponding adjoint system and pointwise characterization of the optimal strategy. Under the baseline weighting, initial condition, and 30-year planning horizon, the optimal intervention lowers the combined overweight-and-obese prevalence at T=30 by approximately 6.9% relative to the uncontrolled trajectory and maintains a larger recovered population, although the total burden is not reversed over the intervention horizon. Over the 30-year horizon considered, the optimal strategy is front-loaded, with stronger support early and lower intensity when implementation is more costly. Post-intervention simulations show that the benefit persists for many years after support is withdrawn but gradually erodes as the system relaxes toward the uncontrolled endemic equilibrium.
Human behavior shapes epidemic dynamics, yet most models represent it by rescaling transmission, conflating behavior with biology and removing the memory carried by sustained protective behavior. We develop and analyze a susceptible-infected-recovered-aware model in which prevalence-driven awareness, eroded by fatigue, moves susceptible and infectious individuals between accessible and inaccessible states rather than modifying transmission probabilities. The basic reproduction number is R0 = β/(γ+μ), independent of behavioral parameters, and a unique endemic equilibrium exists exactly when R0 > 1 excluding backward bifurcation. Infection and awareness are uniformly persistent whenever R0 > 1. We derive an exact decomposition of the reduction in endemic burden into susceptible withdrawal and infectious sequestration, establish the effect of each behavioral parameter, and show that the endemic state depends on awareness response and fatigue only through their ratio, with prevalence vanishing linearly as fatigue vanishes. In the instantaneous-relaxation limit, the model reduces to a transmission-modulating formulation whose endemic equilibrium is locally asymptotically stable. At finite relaxation rates, however, numerical stability and bifurcation analyses reveal a Hopf transition and self-sustained epidemic cycles without imposed delay. Relative to the SIR benchmark, awareness lengthens transient inter-epidemic periods and slows damping rather than creating the underlying transient oscillations.
The 2014-2016 Ebola Virus disease (EVD) outbreak came with an unprecedented challenge. It simultaneously arose in three different countries with considerable inbound and outbound travels. This led the World Health Organization (WHO) to suggest implementing exit screening of travelers at their borders. In addition to this measure, some countries accounted for entry screening and/or vaccination to curb the disease. In this paper, we consider two optimal controls for patch models in which other interventions supplement exit screening. The first problem is defined through the exit and entry screenings of travelers. The second involves the combination of exit screening and vaccination interventions. We prove the well-posedness of both optimal control problems. For each problem, we construct a new Nonstandard Finite Difference Forward-Backward-Sweep-Method, which preserves the dynamics of the continuous model. Numerical simulations that support the theory are provided. They also suggest that vaccination alone is effective in reducing the disease burden.
We investigate the impact of the weak Allee effect on the total biomass of a single species inhabiting a two-patch environment connected by asymmetric dispersal. In the proposed model, the first patch exhibits logistic growth, while the second patch follows a weak Allee threshold. We establish the global existence, nonnegativity, and boundedness of solutions, and prove that the system admits a unique positive equilibrium that is globally asymptotically stable. Analytical results are derived for the asymptotic regime of high dispersal rates using singular perturbation theory, revealing an emergent weak Allee effect at the metapopulation scale. We conduct a complete classification of the parameter space to determine when dispersal increases, decreases, or has a unimodal effect on total equilibrium biomass, and provide geometric and monotonicity analyses of the equilibrium. Numerical simulations illustrate the theoretical trichotomy and highlight the ecological implications of dispersal management in fragmented landscapes. Our results generalize previous studies on logistic models to include the weak Allee effect, offering new insights into the interplay between dispersal intensity, asymmetry, and population persistence.
HIV/AIDS remains a major public health challenge in South Africa, where denialism continues to undermine prevention and treatment efforts. This study develops a mathematical transmission model that explicitly incorporates denial behaviours, including refusal of preventive measures, refusal to initiate antiretroviral therapy (ART), and discontinuation of ART. Analytical and simulation results show that denial substantially increases HIV incidence, accelerates progression to AIDS, and raises the likelihood of a backward bifurcation, an occurrence where HIV may persist even when the basic reproduction number falls below one. Sensitivity analysis identified denial-related transmission as a key driver of epidemic dynamics. These findings underscore the need to address denialism by strengthening HIV testing, ART uptake and retention, and community-level education to counter misinformation and stigma.
