Deep brain stimulation (DBS) is a prominent therapy for neurodegenerative disorders, particularly advanced Parkinson’s disease (PD), offering relief from motor symptoms and lessening dependence on dopaminergic drugs. Yet, theoretically comprehending either DBS mechanisms or PD neurophysiology remains elusive, highlighting the importance of neuron modeling and control theory. Neurological disorders ensues from abnormal synchrony in neural activity, as evidenced by abnormal oscillations in the local field potential (LFP). Complex systems are typically characterized as high-dimensional, necessitating the application of dimensional reduction techniques pinpoint pivotal network tipping points, allowing mathematical models delve into DBS effectiveness in countering synchrony within neuronal ensembles. Although the traditional closed-loop control policies perform well in DBS technique research, computational and energy challenges in controlling large amounts of neurons prompt investigation into event-triggered strategies, subsequently machine learning emerges for navigating intricate neuronal dynamics. We comprehensively review mathematical foundations, dimension reduction approaches, control theory and machine learning methods in DBS, for thoroughly understanding the mechanisms of brain disease and proposing potential applications in the interdisciplinary field of clinical treatment, control, and artificial intelligence.

Human heterogeneity is a critical issue in infectious disease transmission dynamics modelling, and it has recently received much attention in COVID-19 studies. In this article, a general human heterogeneous disease model with mutation is proposed to comprehensively study the effects of human heterogeneity on basic reproduction number, final epidemic size and herd immunity. We show that human heterogeneity may increase or decrease herd immunity level, strongly depending on some convexity of the heterogeneity function, which gives new insights and extends the results in [Britton et al., Science, 369: 846–849, 2020]. Moreover, human heterogeneity may decrease the basic reproduction number but increase the level of herd immunity, implying the unreliability of the basic reproduction number in characterizing the spread and control of infectious diseases with human heterogeneity.

Cancer is a multifaceted disease caused by dynamic interaction between genetic mutations and environmental factors. Understanding the genetic mutations underlying the development and progression of cancer is the stepstone for developing effective treatments and therapies. However, these mutations occurred in only a small fraction of cancer patients and it is extremely difficult to associate with cancer. Here, we propose MutNet, a heterogeneous network embedding method which integrate biomolecular network with cancer genomics data. Using pan cancer genomic data from The Cancer Genome Atlas program and public protein-protein interaction and pathway data, MutNet identifies rarely mutated cancer genes often overlooked by conventional genetic studies. In addition, the unified vector representation of biological entities allows us to reveal the tumor type specific cancer genes, cancer gene modules, and potential relationships among different tumor types. Our heterogeneous network embedding method holds the promise for the underlying mechanisms of cancer and potential therapeutic targets.

Granger causality (GC) stands as a powerful causal inference tool in time series analysis. Typically estimated from time series data with finite sampling rate, the GC value inherently depends on the sampling interval $\tau.$ Intuitively, a higher data sampling rate leads to a time series that better approximates the real signal. However, previous studies have shown that the bivariate GC converges to zero linearly as $\tau$ approaches zero, which will lead to mis-inference of causality due to vanishing GC value even in the presence of causality. In this work, by performing mathematical analysis, we show this asymptotic behavior remains valid in the case of conditional GC when applying to a system composed of more than two variables. We validate the analytical result by computing GC value with multiple sampling rates for the simulated data of Hodgkin-Huxley neuronal networks and the experimental data of intracranial EEG signals. Our result demonstrates the hazard of GC inference with high sampling rate, and we propose an accurate inference approach by calculating the ratio of GC to $\tau$ as $\tau$ approaches zero.

How does the movement of individuals influence the persistence of a single species and the competition of multiple populations? Studies of such questions often involve the principal eigenvalues of the associated linear differential operators. We explore the significant roles of the principal eigenvalue by investigating two types of mathematical models for arbitrary but finite number of competing populations in spatially heterogeneous and temporally periodic environment. The interaction terms in these models are assumed to depend on the population sizes of all species in the whole habitat, representing some kind of nonlocal competition. For both models, the single species can persist if and only if the principal eigenvalue for the linearized operator is of negative sign, suggesting that the best strategy for the single species to invade when rare is to minimize the associated principal eigenvalue. For multiple populations, the global dynamics can also be completely characterized by the associated principal eigenvalues. Specifically, our results reveal that the species with the smallest principal eigenvalue among all competing populations, will gain a competitive advantage and competitively exclude other populations. This suggests that the movement strategies minimizing the corresponding principal eigenvalue are evolutionarily stable, echoing the persistence criteria for the single species.

In this paper, we propose a maximum entropy method for predicting disease risks. It is based on a patient’s medical history with diseases coded in International Classification of Diseases, tenth revision, which can be used in various cases. The complete algorithm with strict mathematical derivation is given. We also present experimental results on a medical dataset, demonstrating that our method performs well in predicting future disease risks and achieves an accuracy rate twice that of the traditional method. We also perform a comorbidity analysis to reveal the intrinsic relation of diseases.

In this paper, we establish the existence and nonlinear stability of a hyperbolic system of conservation laws derived from a repulsive singular chemotaxis model. By the phase plane analysis alongside Poincaré-Bendixson theorem, we first prove that this hyperbolic system admits three different types of traveling wave profiles, which are explicitly illustrated with numerical simulations. Then using a unified weighted energy estimates and technique of taking anti-derivatives, we prove that all types of traveling wave profiles, including non-monotone pulsating wave profiles, are nonlinearly and asymptotically stable if the initial data are small perturbations with zero mass from the spatially shifted traveling wave profiles.