This paper overviews the recent advances how spatial-temporal heterogeneity affects the spread of epidemic diseases from the perspective of basic reproduction number. First part focuses upon the theoretical studies and applications of seasonal effects on infectious invasion. Then the investigations are introduced to demonstrate the role of spatial structure and spatial contact patterns in the prevention of disease outbreaks. Suggestions are presented for potential future researches.

Mathematical oncology is a rapidly evolving interdisciplinary field that uses mathematical models to enhance our understanding of cancer dynamics, including tumor growth, metastasis, and treatment response. Tumor-immune interactions play a crucial role in cancer biology, influencing tumor progression and the effectiveness of immunotherapy and targeted treatments. However, studying tumor dynamics in isolation often fails to capture the complex interplay between cancer cells and the immune system, which is critical to disease progression and therapeutic efficacy. Mathematical models that incorporate tumor-immune interactions offer valuable insights into these processes, providing a framework for analyzing immune escape, treatment response, and resistance mechanisms. In this review, we provide an overview of mathematical models that describe tumor-immune dynamics, highlighting their applications in understanding tumor growth, evaluating treatment strategies, and predicting immune responses. We also discuss the strengths and limitations of current modeling approaches and propose future directions for the development of more comprehensive and predictive models of tumor-immune interactions. We aim to offer a comprehensive guide to the state of mathematical modeling in tumor immunology, emphasizing its potential to inform clinical decision-making and improve cancer therapies.

Biological experiments have verified that chromatin organization has importance influence on gene expression, but the conventional models of gene expression neglect this influence, in particular the effect of enhancer-promoter (E-P) communication on gene expression. Here we first review properties of the classical Rouse model that is a quite accurate description of chromatin as confirmed by microscopy experiments. Second, we extend this model to the polymer models with long-range interactions so that they include E-P communications that are typically long-range interactions. We also carry out theoretical analysis for the extended models. Third, we establish mathematical models for the whole process of gene transcription, which consider connections between upstream chromatin dynamics and downstream promoter kinetics. These connections consider two possible ways of regulation: The one via E-P encounter probability and the other via E-P spatial distance, both supported by a different experimental measurement. These models lay solid foundations not only for the deep study of gene-expression dynamics but also for the statistical inference of experimental data.

This paper studies an susceptible-infected-susceptible reaction-diffusion model in spatially heterogeneous environment proposed in [Allen et al., Discrete Contin. Dyn. Syst., 21, 2008], where the existence and uniqueness of the endemic equilibrium are established and its stability is proposed as an open problem. However, till now, there is no progress in the stability analysis except for special cases with either equal diffusion coefficients or constant endemic equilibrium. In this paper, we demonstrate the first criterion in determining the stability of the non-constant endemic equilibrium with different diffusion coefficients. Thanks to this criterion, when one of the diffusion rates is small or large, the impact of spatial heterogeneity on the stability can be characterized based on the asymptotic behavior of the endemic equilibrium.

Severe fever with thrombocytopenia syndrome (SFTS) is an emerging tickborne zoonotic disease caused by severe fever with thrombocytopenia syndrome virus. In recent years, there has been an increasing number of human SFTS cases in central and northeast China. In the study region, Dalian, the number of human cases in years between 2011 and 2019 exhibited recurrent patterns in synchrony with the seasonal temperature variation. Here, we develop a transmission dynamics model to incorporate contact characteristics of animal and human hosts from published literature, and fit the model to historical temperature and human incidence data in our study region to analyze trends in human SFTS incidence and the time trends of SFTS prevalence within the natural tick-host cycle. Our analysis highlights the contributions of the systemic, co-feeding, and transovarial transmission routes, and provides insights for cost-effective public health interventions targeted to reducing transmission in these coexisting transmission pathways.

Stem cell regeneration is crucial for development and maintaining tissue homeostasis in self-renewing tissues. The dynamics of gene regulatory networks (GRNs) play a vital role in regulating stem cell renewal and differentiation. However, integrating the quantitative dynamics of GRNs at the single-cell level with populationlevel stem cell regeneration poses significant challenges. This study presents a computational framework that links GRN dynamics to stem cell regeneration through an inheritance function. This function captures epigenetic state transitions during cell division in heterogeneous stem cell populations. Our model derives this function using a hybrid approach that integrates cross-cell-cycle GRN dynamics, effectively connecting cellular-level GRN structures with population-level regeneration processes. By incorporating GRN structure directly into stem cell regeneration dynamics, this framework simulates cross-cell-cycle gene regulation using individual-cell-based models. The scheme is adaptable to various GRNs, providing insights into the relationship between gene regulatory dynamics and stem cell regeneration. Additionally, we propose a future perspective that integrates single-cell ribonucleic acid sequencing data, GRN analysis, and cell regeneration dynamics using AI-driven tools to enhance the precision of regenerative studies.

This work introduces a repulsive chemotaxis susceptible-infected-susceptible (SIS) epidemic model with logarithmic sensitivity and with mass-action transmission mechanism, in which the logarithmic sensitivity assumes that the chemotactic migration of susceptible populations is suppressed by large density of infected individuals while the biased movement is strongly sensitive to a density variation of small infected population. Under suitable regular assumption on the initial data, we firstly assert the global existence and boundedness of smooth solutions to the corresponding no-flux initial boundary value problem in the spatially one-dimensional setting. Second, we investigate the effect of strong chemotaxis sensitivity on the dynamics of solutions through extensive numerical simulations. Our numerical study suggests that although this chemotaxis model includes an unbounded force of infection, the blow-up of solutions cannot occur in two dimensions, which remains to be analytically verified. Moreover, the numerical studies on the asymptotic profiles of the endemic equilibrium indicate that the susceptible populations move to low-risk domains whereas infected individuals become spatially homogeneous when the repulsive-taxis coefficient is large. Furthermore, simulations performed in the one- and two-dimensional cases find that rich patterns, like periodic peaks, structured holes, dots and round circles, may arise at intermediate times, but eventually are smoothed out, and that clusters of infection can emerge in a heterogeneous environment. Additionally, our numerical simulations suggest that the susceptible population with larger chemosensitivity, tends to respond better to the infected population, revealing the effect of strong chemotaxis sensitivity coefficient on the dynamics of the disease.