Rumors have become a serious issue in today’s modern era, particularly in view of increased activity in social and online platforms. False information can go viral almost instantaneously through social networks, which immediately affect society and people’s minds. The form of rumor it develops within, whether fabricated intentionally or not, impacts public perspectives through manipulation of emotion and cognition. We propose and analyze a mathematical model describing how rumors can spread through an online social media (OSM) platform. Our model focuses on two coexisting rumors (two strains). The results provide some conditions under which rumors die out or become persistent, and they show the influence of delays, skepticism levels, and incidence rates on the dynamics of information spread. We combine analytical tools (Routh–Hurwitz tests and delay-induced stability switches) with MATLAB/Python simulations to validate the theoretical predictions.

This mathematical model delves into the dynamics of the immune system during Chronic Myeloid Leukemia (CML) therapy with imatinib. The focus lies in elucidating the allergic reactions induced by imatinib, specifically its impact on T helper (Th) cells and Treg cells. The model integrates cellular interactions, drug pharmacokinetics, and immune responses to unveil the mechanisms underlying the dominance of Th2 over Th1 and Treg cells, leading to allergic manifestations. Through a system of coupled delay differential equations, the interplay between healthy and leukemic cells, the influence of imatinib on T cell dynamics, and the emergence of allergic reactions during CML therapy are explored.

Fake news has become a major challenge in online social media because false information can propagate rapidly and influence public perception and collective behavior. In this paper, we propose and analyze a six-dimensional delay differential equation model for the spread of two competing fake-news strains on an open online platform. The model includes recruitment and natural exit of users, saturated incidence rates for the adoption of each strain, delayed withdrawal from the platform, and delayed strain-specific anti-spreading activity. We prove positivity and boundedness of solutions and identify the main equilibria, including the fake-news-free, leaving-only, single-strain, and coexistence equilibria. For the fake-news-free equilibrium, we derive local stability conditions and establish sufficient conditions for global asymptotic stability by means of a Lyapunov–Krasovskii functional. For the coexistence equilibrium, we study the characteristic equation and use a block-reduction approach to investigate delay-dependent stability. Numerical simulations and a Monte Carlo sensitivity analysis show that, for the chosen parameter set, the coexistence equilibrium is stable and fake news persists.

Antimicrobial Resistance (RAM) poses a significant threat to global public health, making important medicines less useful. While the medical and biological reasons behind RAM are well studied, we still don't know enough about how false health information affects people's actions, which can speed up RAM. This study presents a new mathematical model to investigate the complex interplay between the spread of misinformation and the dynamics of RAM. We adapt a multi-strain fake news model, including distinct population compartments representing individuals susceptible to, believing in, or skeptical of various ideas related to antibiotic use. The model considers multiple "strains" of misinformation, such as the wrong belief that antibiotics are effective for viral infections or not trusting medical advice regarding prudent antibiotic prescription. Time delays are integrated to reflect the latency in information processing, behavioral change, and the manifestation of resistance. Through stability analysis and numerical simulations, this research aims to identify critical factors and parameters that influence the propagation of harmful beliefs and their consequent impact on behaviors contributing to RAM. The findings could help develop public health campaigns to reduce the negative impact of misinformation on fighting antimicrobial resistance.

We consider the wave equation on an $ N $-edge star of strings with fixed outer endpoints and a Neumann-type control at the central node, imposed by $ \sum_{j = 1}^N u_{j,x}(t,0) = h(t) $. We derive a Green-type identity adapted to this central actuation and obtain the moment formulation $ {\displaystyle {\int }}_0^T \beta_n e^{i\lambda_n t}h(t)\,dt = u_{1,n}-i\lambda_n u_{0,n}, \qquad \beta_n = \theta_n(0), $ where $ (\lambda_n^2,\theta_n) $ are eigenpairs of the graph Laplacian. The spectrum is encoded by an entire generating function $ q $, whose real zeros are precisely the eigenfrequencies. From the properties of these eigenfrequencies, we place the analysis in the broader framework of nonharmonic Fourier series. This connects our moment method approach to the Ingham–Beurling type observability results with weakened gap conditions developed for networks with a common node. Using the Paley–Wiener biorthogonal family associated with $ q $, we construct controls in $ L^2(0,T) $ for every $ T>2\sum_{j = 1}^N \ell_j $. Assuming $ \mathbb{Q} $-linear independence of the lengths, which ensures that the eigenmodes have nonzero values at the central node, we obtain spectral controllability. Finally, we relate the control cost to the quantities $ \beta_n $ and $ q'(\lambda_n) $, showing that modes close to Dirichlet frequencies on individual edges can require large controls, and we confirm this behavior numerically in MATLAB.