University of Bío-Bío

We study a new three-dimensional mathematical model of cancer dynamics describing the interaction among tumor cells, effector immune cells, and the cytokine IL-2 in a patient. The model capture both the functional exhaustion or depletion of effector cells and their tumor-induced proliferation. We analyze tumor size oscillationsto explain long-term phenomena, such as tumor elimination or recurrence, under the influence of cellular immunotherapy. The model captures the nonlinear dynamics of immunogenic tumors. In particular, we show that the trajectories are bounded and defined for all positive time, invariant subsets exist, and under certain parameter conditions, the system admits at most five equilibrium points in the first octant: one tumor-free and up to four coexistence states. This improves upon the previous upper bound of six coexistence equilibria reported in the literature. Analytically, we demonstrate the existence of transcritical and pitchfork bifurcations emerging from the tumor-free equilibrium. Furthermore, we establish the existence of a Hopf bifurcation leading to a stable periodic orbit, which represents cyclic oscillations in the cell populations characterized by a strong dominance of abnormally growing tumor cells. Our analysis reveals that the model exhibits rich dynamical behavior, significantly influenced by the parameters μ1 (death rate of effector cells) and p1 (production rate of effector cells stimulated by IL-2). In particular, the inequality μ1 > p1 plays a critical role in determining whether the tumor persists or is eliminated. A comparative analysis of our main results is performed for triple-negative breast cancer (TNBC).

In this work, we present two Nitsche-based mixed finite element methods. The first one concerns the scalar Poisson problem on a domain whose boundary is partitioned into Dirichlet and Neumann (linear) conditions. For this problem, we introduce a new discrete variational formulation that enables the weak imposition of mixed boundary conditions. The proposed method is analyzed within the Babu & scaron;ka-Brezzi framework, ensuring the well-posedness of the discrete problem. We derive optimal apriori error estimates and present numerical experiments that confirm the theoretical convergence rates and demonstrate the robustness of the proposed scheme. The second method is a mixed Nitsche-type discretization for the unilateral (non-linear) Signorini conditions in the scalar case, as a simple prototype for contact and friction problems or other variational inequalities associated with boundary conditions. We present the full variational formulation. The performance of the method is assessed by numerical experiments that illustrate the accurate enforcement of the boundary conditions and the expected behavior under mesh refinement.

The incorporation of probiotic dairy and non-dairy foods is often limited by the loss of bioactivity caused by stresses to which microorganisms are subjected during processing, storage, and digestion. This frequently results in probiotic counts below the minimum level required to exert their probiotic function (10⁷ CFU/g) at the time of consumption. Maintaining cell viability remains a challenge for probiotic food manufacturers. The omics technologies have enabled the identification of stress response pathways in microorganisms; however, their application to the design of reproducible and scalable processes remains limited. This review integrates microbial stress physiology with bioprocesses and food matrix engineering to support the design of robust probiotic products. Biological mechanisms underlying increased tolerance to environmental stresses are analyzed and preadaptation strategies based on sublethal exposure to heat, acid and bile, ultrasound, pulsed electric fields, and high-pressure processing are discussed. Encapsulation systems are considered protective structures that reduce stress during food processing, storage, and digestion. To enhance the bioactivity of probiotics, a robustness-by-design approach is proposed. This begins with selecting the probiotic strain and the food. After selecting the probiotic strain, it is subjected to sublethal stress treatments to induce adaptive responses. The conditioned probiotic is incorporated into the food. The proposed scheme integrates survival kinetics and biomarkers to support process design. Biosafety aspects associated with sublethal stress are considered critical for the development of stable and safe probiotic foods on an industrial scale.

We investigate a cosmological model based on matter creation in a single-component universe, its late time behavior and observational constraints derived from observational data. Furthermore, we explore the equivalence between this framework and interacting dark sector models, which establishes a connection between the matter creation rate and cosmological interactions. We first focus on the case of a constant equation of state parameter, where both known and novel interaction terms naturally emerge from matter creation, numerous of them exhibiting a sign-changeable behavior. The analysis is then extended to a time-dependent equation of state by using dynamical systems techniques.

In this paper, we consider a coupled parabolic system with multiple Henón-type components ( ∂ _t u - Δ _𝔾u) (x,t) = H(x,t,u) for (x,t) ∈𝔾× (0,T) , where u=(u_1,⋯ ,u_m) is the unknown, 𝔾 is a homogeneous Carnot group on ℝ^N , Δ _𝔾 is the operator whose components are given by the sub-Laplacian on 𝔾 , and H(x,t ,u)=( t^s_1 |x|_𝔾^γ _1 u_2^p_1, t^s_2 |x|_𝔾^γ _2 u_3^p_2, ⋯ , t^s_m |x|_𝔾^γ _m u_1^p_m) , with p_i ≥ 1 , ∏ _i=1^m p_i>1 , γ _i≥ 0 , s_i> -1 for i =1, ⋯ , m , where |· |_𝔾 denotes a homogeneous norm on 𝔾 . We determine the Fujita-type exponent for this system, which depends on the homogeneous dimension Q of 𝔾 . In contrast to previous studies, our results are obtained by iterative methods involving the symmetric submarkovian semigroup associated with the operator Δ _𝔾 . In particular, the derivation of the blow-up result is delicate and is carried out through a novel approach based on Stirling’s asymptotic formula.