In the work, we have constructed the mathematical model of respiratory viral infection propagation in the tissue with influence of resident macrophages. The model is based on reaction–diffusion equations with integral terms. In this system, the following biologically significant characteristics are determined: 1) the basic replication number, 2) the viral load, 3) the wave speed, and 4) the immunity effectiveness number. The basic replication number shows whether the infection will propagate. The viral load, i.e., the space integral from the concentration of viral particles, allows us to estimate the infectivity of the viral disease, i.e., how fast infection transmits between individuals. The wave speed corresponds to the severity of the disease. The immunity effectiveness number helps to estimate the impact of immunity in the condition of the infection propagation. The model is analyzed by applicable methods of analysis of reaction–diffusion systems and numerically. The conditions of the infection propagation, the infectivity and the severity of the disease, and the influence of the immunity are investigated, and their quantity estimates are obtained. The results may help in planning treating strategies and identification of targets for the treatment. The work was supported by the Russian Science Foundation No. 24-11-00073, https://rscf.ru/en/project/24-11-00073/. References: M. Bouzari, L. Ait Mahiout, A. Mozokhina, V. Volpert Infection propagation in a tissue with resident macrophages. Mathematical Biosciences. 2025; 381: 109399. https://doi.org/10.1016/j.mbs.2025.109399

Cancer remains one of the leading causes of morbidity and mortality worldwide, posing a significant global health challenge despite advances in therapeutic interventions. The process of tumor growth is a complex interplay among cancer cells, the immune system, and healthy tissues, and this interplay depends on the specific therapeutic approach. Mathematical modeling can be a valuable tool for dissecting these processes and identifying thresholds for different outcomes.

This work revisits and generalizes the widely used tumor-immune system model of de Pillis and Radunskaya by developing a set of nonlinear ordinary differential equations that incorporate the impact of therapy. Drug-cell interactions are modeled using Michaelis-Menten Kinetics, while immune suppression due to tumor invasion is represented by a Hill-type functional response. In this study, we establish fundamental dynamical properties of the system, including the positivity and boundedness of solutions, determine all equilibrium points, and characterize the local stability of the tumor-free state. In addition, we introduce a tumor invasion number that represents the threshold conditions governing successful tumor establishment.

In the context of cancer immuno-editing, the elimination, equilibrium, and escape stages are considered, and the parameter ranges governing transitions between these states via transcritical bifurcations are identified. Simulations of the system are also conducted to verify the findings, investigate the effects of various treatments on tumor-immunotherapy dynamics, and perform a sensitivity analysis to identify the key biological parameters that affect outcomes. The proposed framework enhances the theoretical understanding of tumor-immunotherapy interactions and may contribute to the development of more effective treatment strategies.

We develop and analyze a reaction–diffusion prey–predator model that integrates predator-induced fear, constant prey refuge, and cooperative interactions among specialist predators. To capture realistic density regulation, intraspecific competition within the prey population is modelled using Richards-type growth, while predator harvesting is incorporated as an external ecological control mechanism.

The dynamical behaviour of the associated non-spatial system is first examined, revealing the presence of multiple equilibria, namely the trivial equilibrium, the predator-free (axial) equilibrium, and a biologically feasible coexistence equilibrium. Local stability analysis demonstrates that the persistence of the coexistence state is strongly governed by critical ecological parameters, including handling time, fear intensity, and harvesting rate. Variations in these parameters induce qualitative shifts in system dynamics, and the coexistence equilibrium is shown to undergo Hopf bifurcation, resulting in the emergence of stable and oscillatory solutions. The corresponding threshold values are determined through numerical continuation techniques implemented in MATLAB and MATCONT.

The model is subsequently extended to a spatial framework to explore diffusion-driven instability. Through combined analytical investigation and numerical simulations, we establish conditions for the onset of Turing instability and the formation of self-organized spatial patterns. The results underscore the significant influence of diffusion rates and biological interactions on species dispersal and spatial organization. In particular, the dispersion analysis indicates that Turing patterns arise when the prey diffusion rate exceeds that of the predator, highlighting the role of differential mobility in pattern formation.

Overall, this work presents a biologically consistent modelling framework that elucidates the interplay between ecological interactions and spatial processes, offering new insights into the mechanisms underlying complex spatiotemporal dynamics in prey–predator systems.

Additive manufacturing is increasingly adopted in engineering practice for producing geometrically complex structures with high precision. However, predicting the mechanical behavior of polymer-based components remains challenging due to insufficiently accurate data on their elastic moduli, particularly under dynamic loading. Classical experimental mechanics methods primarily based on quasi-static tests are often unsuitable for accurately characterizing dynamic elastic properties. This study proposes a two-stage algorithm to solve an inverse coefficient problem for determining the elastic moduli of specimens fabricated from polymeric materials via fused deposition modeling (FDM), leveraging experimental dispersion characteristics of guided elastic waves. Surface out-of-plane velocities, measured using laser Doppler vibrometry, are processed with the matrix pencil method to extract dispersion curves. In the first stage, a fast method based on the Fourier transform of the waveguide's Green's matrix provides an initial approximation of the elastic moduli. This estimate is refined in the second stage by minimizing the discrepancy between theoretical and experimental dispersion relations; the objective function selectively weights experimental points in proximity to the current theoretical prediction, which improves convergence robustness. When validated on synthetic data, the two-stage approach reduces parameter estimation error by approximately 50 % compared to single-stage optimization. Experimental validation on FDM-printed specimens shows strong correlation between the identified Young's modulus and reference values obtained from standard uniaxial tensile tests, confirming the method's applicability for non-destructive dynamic characterization of additively manufactured polymers. This research is carried out with the financial support of the Kuban Science Foundation, Ortho-Market LLC in the framework of the project Num. NTIP-25.1/8.

The objective of time scale theory is to bridge continuous and discrete models and to better capture hybrid dynamics arising in real-world applications. Within the framework of this theory, the dynamics of pancreatic cancer cells under radiotherapy will be investigated by incorporating both continuous and discrete mathematical formulations. In this study, the mathematical model describing pancreatic cancer cell behavior under radiotherapy will first be examined on an arbitrary time scale, allowing for a general representation of tumor growth and treatment effects. For this general case, explicit solutions of the proposed model will be derived, providing analytical insight into the underlying system dynamics. Subsequently, continuous and discrete models will be obtained as special cases of the general time scale formulation, corresponding to classical differential and difference equation models, respectively. A detailed comparison between these continuous and discrete models and their solutions will then be carried out in order to evaluate differences in tumor response and treatment effectiveness. This comparative analysis aims to identify the most appropriate time scale that leads to the fastest and most effective therapeutic response under radiotherapy. By capturing biological processes across different time scales in a unified manner, this approach offers a more realistic and flexible representation of pancreatic cancer dynamics. Ultimately, the proposed modeling framework contributes to a deeper understanding of cellular responses to radiotherapy and may support the development of improved treatment strategies.

This paper investigates exact analytical solutions of the Ivancevic option pricing model, a nonlinear mathematical framework designed to describe complex features of financial markets, including market volatility and nonlinear asset price dynamics. Unlike classical option pricing models, the Ivancevic model incorporates nonlinear wave behavior, making it suitable for capturing irregular and extreme market movements. To obtain closed-form solutions, the modified extended direct algebraic method is employed as an effective analytical tool. Through this approach, several families of exact wave solutions are derived, including periodic, dark-bright, mixed dark–bright, singular, and hyperbolic function solutions. These solution structures reveal a wide range of possible option price behaviors under different market conditions and parameter settings. To further understand the economic and dynamical implications of the obtained solutions, numerical simulations are carried out using carefully selected model parameters. The numerical analysis includes two- and three-dimensional wave profiles as well as contour plots, which provide clear visual insight into the evolution and interaction of pricing waves over time and space. In particular, contour plots are used to identify the extrema and concentration regions of option price fluctuations within specified temporal and spatial intervals. These visualizations help clarify the influence of model parameters on market dynamics. Overall, the results enhance the analytical understanding of the Ivancevic option pricing model and demonstrate the effectiveness of the proposed method. The findings offer a solid mathematical foundation for future research on stability analysis, parameter sensitivity, and nonlinear wave propagation in complex financial and economic systems.

Introduction: Cassava, a key carbohydrate staple in low-income countries, is severely affected by cassava mosaic disease (CMD), primarily transmitted via infected stem cuttings and whiteflies with East African cassava mosaic virus and African cassava mosaic virus (ACMV) being the most significant. ACMV affects all cassava-growing regions of sub-Saharan Africa, India and southern Asia leading to widespread infection rates of 80-100% among crops and yield losses between 20-90%, hence the need to further study the transmission dynamics and possible controls.

Objectives: The objective is to investigate the transmission dynamics and control of CMD by examining the effects of resistant efficacy on infectious cassava. Methods: A deterministic compartmental model governed by a six-dimensional system of ODE was formulated to describe the transmission dynamics and control of CMD between the interacting cassava and whiteflies populations. A piecewise-constant resistant efficacy rate was included to assess its impact on infected cassava transmission dynamics. The control reproduction number, R_e, was calculated using the Next-generation matrix method via Maple. This threshold quantity was used to study the asymptotic behaviours of the disease-free equilibrium (DFE) and endemic equilibrium (EE) of the model. Numerical simulations demonstrated how varying resistant efficacies affect disease dynamics in both susceptible and infectious cassava.

Results: It was found that the DFE (EE) is both locally and globally asymptotically stable whenever R_e<1 (R_e>1) and unstable otherwise. Sensitivity analysis revealed that resistance efficacy and roguing efforts positively influenced the dynamics of CMD.

Conclusion: The results suggest that increasing the resistance efficacy toward 1 significantly reduces new cases of CMD and increasing roguing efforts considerably result to healthier, better quality and quantity of harvested cassava.

We investigate effective quantum state-transfer dynamics on weighted path graphs by employing a fermionic formulation combined with stationary perturbation theory. This approach leverages the well-established equivalence between XX spin chains and continuous-time quantum walks on path graphs in the single-excitation sector. By restricting the dynamics to this single-excitation subspace, the full Hamiltonian of the system can be projected onto a lower-dimensional effective subspace, significantly reducing the complexity of the problem while retaining the essential physics.

In the dispersive regime, where the endpoint vertices are weakly coupled to the intermediate nodes, second-order perturbation theory is applied to systematically eliminate the intermediate vertices. This procedure yields a reduced effective Hamiltonian that directly couples the endpoints of the graph through virtual transitions, capturing the dominant contributions to the transfer dynamics. The resulting effective model allows for the derivation of closed-form analytical expressions for key quantities, including effective coupling strengths, energy shifts, and state-transfer fidelity.

Analysis of the fidelity reveals coherent oscillatory behavior, which is entirely determined by the perturbative parameters and the spectral gaps of the original graph Hamiltonian. This analytical insight provides a clear understanding of how the structure and weighting of the path graph influence the efficiency of quantum state transfer. To validate the approach, we perform numerical comparisons with exact diagonalization of finite weighted path graphs. The results demonstrate that the effective Hamiltonian reproduces the main features of the transfer dynamics and provides accurate estimates of the fidelity within its regime of applicability.

Our findings highlight a practical method for obtaining analytical expressions for quantum state-transfer fidelity in systems with weighted interactions, offering both computational efficiency and physical insight. This work provides a foundation for the design and optimization of engineered spin chains and quantum networks for high-fidelity state transfer, with potential applications in quantum information processing and quantum communication.

Fractional calculus provides useful extensions to models based on ordinary calculus, enabling the description of physical effects such as dissipation and memory. A notable application of this framework in quantum mechanics is the fractional-time Schrödinger equation (FTSE), in which the standard time derivative is replaced by a Caputo derivative carrying a power-law memory kernel. This modification, however, inherently leads to non-unitary evolution of the quantum state. In this work, we apply the FTSE within the paradigmatic Jaynes–Cummings (JC) model to study the evolution of a binomial state of light interacting with matter. The binomial distribution that characterizes the binomial states possesses both coherent and number states as special cases, while the JC model is a cornerstone for studying quantum light–matter interactions, with experimental validation in cavity quantum electrodynamics and applications in quantum information processing. To restore unitarity in our analysis, we follow a recently introduced technique based on time-dependent Dyson maps -- invertible operators $\eta(t)$ that relate the unitary $\left(\hat{u}(t)\right)$ and non-unitary $\left(\hat{U}(t)\right)$ evolution operators via $\hat{u}(t) = \eta(t)\hat{U}(t)\eta^{-1}(0)$. Three distinct binomial distributions are analyzed with population inversion as the primary figure of merit. We show how different derivative orders $\left(\alpha\right)$ distinctly influence the dynamics: a decreasing number of oscillations $\left(\alpha=0.75\right)$, periodicity $\left(\alpha=0.50\right)$, and aperiodicity $\left(\alpha=0.40\right)$.

Introduction: HIV continues to be a major global public health problem, with ongoing transmission across all countries. At the end of 2024, around 40.8 million people were living with HIV worldwide, while an estimated 630,000 people died from HIV-related causes. In Indonesia, an estimated 570,000 individuals are living with HIV. Despite the absence of a cure, effective prevention, diagnosis, treatment, and care have made HIV a manageable chronic disease, enabling people living with HIV to maintain long and healthy lives. In Indonesia, treatment coverage among people living with HIV is low, with only 31% accessing antiretroviral therapy (ART) and 14% attaining viral suppression. Addressing the treatment gap is among the most critical challenges facing the country’s HIV response. Thus, the objective of this study is to assess the optimal strategy required to contain HIV/AIDS transmission in Indonesia using optimal testing (or diagnosis) and ART.

Methods: An optimal control framework based on a non-autonomous deterministic HIV/AIDS transmission model was developed. The model incorporates time-dependent ART use and testing strategies. Pontryagin’s maximum principle was used for the qualitative analysis of the model. The associated optimality system was simulated to demonstrate the effects of separate and combined use of the optimal controls on HIV/AIDS dynamics in Indonesia.

Results: With optimal ART, more people become treated while fewer people become acute and chronic HIV-infected and full-blown AIDS patients. Activation of optimal testing only leads to fewer acute HIV-infected people, and fewer people progress to AIDS as more people are aware of their status and are receiving ART. Simultaneous implementation of the two optimal control shows fewer number of acute and chronic HIV-infected people, while fewer people progress to AIDS as more people receive ART.

Conclusion: Optimal implementation of testing (or diagnosis) and ART can significantly help in stemming the spread of HIV/AIDS in Indonesia.