Working memory -- the ability to store and recall precise temporal patterns of neural activity -- remains an open challenge for spiking neural networks (SNNs). We propose a recurrent SNN of $N$ neurons in which each synapse is equipped with $D = 41$ delays, modelled as a weight tensor $\mathbf{W} \in \mathbb{R}^{N \times N \times D}$ and trained end-to-end with surrogate-gradient backpropagation through time. The network stores $M$ arbitrary target spike patterns by representing each as a sequential chain of overlapping Spiking Motifs: contiguous windows of length $D$ that uniquely predict spikes at the next time step. On a synthetic benchmark of $M=16$ patterns ($N=512$ neurons, $T=1000$ steps), training achieves a mean F1 score of $1.0$, with recall emerging first near the clamped initialisation window and propagating forward in time. This result demonstrates that heterogeneous delays provide an efficient substrate for working memory in SNNs, enabling energy-efficient neuromorphic edge deployment.

Antibody lead optimization is inherently a multi-objective challenge in drug discovery. Achieving a balance between different drug-like properties is crucial for the development of viable candidates, and this search becomes exponentially challenging as desired properties grow. The ever-growing zoo of sophisticated in silico tools for predicting antibody properties calls for an efficient joint optimization procedure to overcome resource-intensive sequential filtering pipelines. We present BOAT, a versatile Bayesian optimization framework for multi-property antibody engineering. Our `plug-and-play' framework couples uncertainty-aware surrogate modeling with a genetic algorithm to jointly optimize various predicted antibody traits while enabling efficient exploration of sequence space. Through systematic benchmarking against genetic algorithms and newer generative learning approaches, we demonstrate competitive performance with state-of-the-art methods for multi-objective protein optimization. We identify clear regimes where surrogate-driven optimization outperforms expensive generative approaches and establish practical limits imposed by sequence dimensionality and oracle costs.

Gene-sharing networks provide a powerful framework to study the evolution of viruses and mobile genetic elements. These bipartite networks, which link genes to the genomes that contain them, exhibit characteristic degree distributions: a scale-free distribution for genes and an exponential-like decay for genomes. Here, we propose a mechanistic model that explains these patterns through fundamental evolutionary processes including horizontal gene transfer, capture of new genes, emergence of new genomes, and gene loss. Using a mean-field approximation, we derive analytical expressions for the asymptotic gene and genome degree distributions, recapitulating a power-law distribution for genes and an exponential distribution for genomes. Numerical simulations validate these predictions and yield parameter values that closely fit empirical data from dsDNA viruses, RNA viruses, and prokaryotic pangenomes. This simple model with only two parameters provides a generative framework for bipartite gene-sharing networks, offering qualitative and quantitative insights into the main evolutionary forces driving genome plasticity. Setting the gene loss rate to zero, the gene and genome degree distributions of the model closely fit the empirically observed distributions. Thus, evolution of viruses appears to be dominated by gene gain, in agreement with the results of independent reconstructions of viral evolution.

Assembly theory (AT) introduces causation as a material property and establishes a metrology for objects produced by evolution and selection. The physical scale of causation is quantified by the assembly index, defined as the minimum number of recursive steps necessary to make an object. Observing countable copies of high assembly index objects indicates a mechanism producing them is persistent, such that the object's environment constructs a memory that traps causation within a contingent chain. Copy number and assembly index together underlie a standardized metrology for detecting causation (assembly index) and contingency (copy number). These allow a precise definition of an assembly threshold that demarcates life (and its derivative agential, intelligent, and technological forms and artifacts) as structures with persistent copies in regimes of deep causal possibility. In introducing a fundamental concept of material causation to quantify and measure life, AT represents a departure from prior theories of causation, such as interventional ones, which have so far proven incompatible with fundamental physics. We discuss how AT's concept of causation provides the foundation for a theory of physics that allows precise and testable concept of "life", and in which novelty, contingency and the potential for open-endedness are fundamental, and determinism is emergent from selection along assembled lineages.

Ecological interactions can dramatically alter evolutionary outcomes in complex communities. Yet, the framework of population genetics largely neglects interactions from a species-rich community. Here, we bridge this gap by using dynamical mean-field theory to integrate community ecology into classical population genetics models. We show that ecological interactions result in emergent frequency-dependent selection between parents and mutants, characterized by a single parameter measuring the strength of ecological feedbacks. This result generalizes classical population genetics models to highly diverse communities and enables predictions of mutation outcomes in these eco-evolutionary settings. We derive an analytic expression for fixation probability that extends Kimura's formula and reveals that ecological interactions strongly suppress the fixation of moderately beneficial mutations. This suppression arises because frequency-dependent selection leads to prolonged coexistence between parent and mutant lineages, which acts as a barrier to fixation. The strength of these effects increases with effective population size and the number of open niches in the ecosystem. Our study establishes a framework for integrating ecological interactions into population genetics, showing that evolutionary outcomes can be predicted using simple models even in the presence of complex community feedbacks.

Self-sustained neural activity in the absence of ongoing external input is a fundamental feature of nervous system dynamics, yet the conditions under which it can emerge in biophysically grounded network models remain incompletely understood. We studied whether a recurrent network of Hodgkin-Huxley neurons with spike-timing-dependent plasticity and intrinsic stochasticity can maintain autonomous activity after brief transient stimulation. The simulated network comprised 200 neurons (160 excitatory, 40 inhibitory) with 80% connection probability, incorporating excitatory and inhibitory STDP, probabilistic vesicle release, probabilistic synapse formation, receptor variability, and voltage-dependent inhibition. After a brief 200 ms initialization stimulus to 30 excitatory neurons, the network received no further external input. In one 1800 s simulation and two additional 500 s simulations, the network maintained sparse, irregular activity without ongoing drive. In the 1800 s run, 67% of neurons exhibited mean firing rates below 1 Hz, the population mean firing rate was 1.13 +/- 1.34 Hz, participation increased across longer observation windows, and population-mean Fano factors remained near 1-2, consistent with irregular spike timing. Raster activity also showed spontaneous qualitative reorganizations in collective firing patterns over time. These findings suggest that recurrent Hodgkin-Huxley networks with plastic and stochastic synapses can sustain long-duration autonomous activity in a sparse firing regime after brief initialization.

Brain digital twins aim to provide faithful, individualized computational representations of brains as dynamical systems, enabling mechanistic understanding and supporting prediction of clinical interventions. Yet current approaches remain fragmented across data pipelines, model classes, temporal scales, and computing platforms, which prevents the preservation of execution semantics across the end-toend workflow. This survey introduces physically constrained executability as a unifying perspective for comparing approaches at the level of execution: whether an execution state is persistent, which events are permitted to update it (simulation, measurement, actuation), and how strongly execution is temporally and causally coupled to neurobiological dynamics. Building on modeling and simulation theory, I propose a taxonomy of execution regimes ranging from isolated offline models to coordinated co-simulation, to continuously executing digital twins sustained by online data assimilation, and ultimately to neuro-neuromorphic physical systems in which biological and computational dynamics are co-executed under shared physical constraints. The executability concept clarifies why accuracy alone is insufficient, and motivates an agenda centered on semantic interoperability, hybrid-time correctness, evaluation protocols, scalable reproducible workflows, and safe closed-loop validation. This survey adopts a systems and runtime-oriented perspective, enabling comparison of heterogeneous approaches based on their execution semantics rather than on model form or application domain alone.

Theory and empirical science should be in constant dialogue, but often find it hard to understand one another. Here we describe a graduate-level university course we developed to improve matters. The course was designed to help empirically-focused biology graduate students read and understand theory papers, despite little prior mathematical training. It uses several evidence-based principles of modern teaching: backwards design, active learning, and just-in-time teaching. We believe that this or similar curricular content, emphasizing the nature of evidence and the role of theory in science, will improve critical thinking and scientific progress.

Humans and artificial agents must often learn and switch between multiple tasks in dynamic environments. Success in such settings requires cognitive flexibility: the ability to retain prior knowledge (cognitive stability) while also transferring it to novel tasks (cognitive generalization). Cognitive flexibility research has largely focused on the role of model architecture to achieve these complementary goals. However, it is less well understood how the structure of the environment itself influences cognitive flexibility, and how it interacts with model architecture. To address this gap, we design a multi-task learning environment in which tasks are defined by a combination of two cue dimensions, allowing us to characterize the environment with graph-theory methods. We also introduce gating-based (multiplicative) and concatenation-based attention models that can decompose tasks into components and can sequentially allocate attention to them. We compare the attention-based models' performance in the multi-task learning environment to multilayer perceptrons. Generalization and stability are systematically evaluated across environments that vary in richness and task connectivity. We observe that richer environments improve both generalization and stability. In addition, a critical novel observation is that (graph theory based) connectivity between the tasks in the environment strongly modulates both stability and generalization, with especially pronounced benefits for attention-based models. These findings underscore the importance of considering not only cognitive architectures but also environmental structure and their interaction in shaping multi-task learning, generalization, and stability.

Glycemic regulation is often described as maintaining glucose levels near a stable baseline. However, continuous glucose monitoring after meals displays intra-individual variability even under controlled conditions, suggesting intrinsic system dynamics beyond sensor noise, measurement error or short-term variability around a fixed set point. Therefore, we estimated pre-meal glucose baselines, tracking their changes across repeated identical meal challenges within individuals. The baseline was defined as the median glucose level in a pre-meal window, while successive displacements were computed between consecutive repetitions. Using a publicly available dataset of normoglycemic subjects, we observed systematic changes in baseline levels across repeated exposures. These displacements exceeded short-term fluctuations within the same pre-meal interval and were robust to alternative baseline definitions. Moreover, the magnitude of each baseline shifted is positively related to the size of the preceding postprandial response. This association persisted under permutation testing, indicating that it cannot be explained by random temporal ordering. Overall, these findings suggest that glycemic dynamics cannot be fully described as independent fluctuations around a fixed baseline. Instead, baseline levels evolve across repeated perturbations through history-dependent adjustments, such that each perturbation influences subsequent system states. Potential applications include refined interpretation of continuous glucose monitoring data and development of models that incorporate temporal dependence in glucose dynamics.

We study the random times between successive cases in a transmission chain of infectious diseases with asymptomatic carriers. We derive the probability distribution of this generation time (in days) from a discrete-time epidemic model with variable infectiousness both along elapsed times and across phases. The introduced non-Markovian model is a compact recursive system featuring random waiting times at each of the three infected stages: latent, asymptomatic, and symptomatic. By rearranging the terms of the basic reproduction number, which represents the expected number of secondary cases produced by an asymptomatic primary case who may eventually develop symptoms, we get to the generation-time probabilities. The expected generation time is a convex combination of the expected generation times before and after the onset of symptoms. Additionally, our analysis reveals that the n-th moment of the generation time is related to the moments up to n-th order of the weighted forward recurrence time at each phase and the moments up to n-th order of the latent period and the incubation period. These weights are the infectiousness along the elapsed times for each transmission phase. Finally, we illustrate several data-driven epidemic scenarios, assuming that infectiousness varies only across phases and discrete Weibull distributions for the waiting times. Each disease analyzed, except measles, exhibits moderate variability in its respective generation time distribution.

Nested (or reconciled) phylogenetic trees model co-evolutionary systems in which one evolutionary history is embedded within another. We introduce a geometric framework for such systems by defining $σ$-space, a moduli space of fully nested ultrametric phylogenetic trees with a fixed leaf map. Generalizing the $τ$-space of Gavryushkin and Drummond, $σ$-space is constructed as a cubical complex parametrised by nested ranked tree topologies and inter-event time coordinates of the combined host and parasite speciation events. We characterise admissible orderings via binary \textit{nesting sequences} and organise them into a natural poset. We show that $σ$-space is contractible and satisfies Gromov's cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fréchet means. We further describe its geometric structure, including boundary strata corresponding to cospeciation events, and relate it to products of ultrametric tree spaces via natural forgetful maps.

As proposed by Hebb's theory, neural assemblies are groups of excitatory neurons that fire synchronously and exhibit high synaptic density, representing external stimuli and supporting cognitive functions such as language and decision-making. Recently, a model called Assembly Calculus (AC) was proposed, enabling the formation of artificial neural assemblies through the $k$-winners-take-all selection process and Hebbian learning. Although the model is capable of forming assemblies according to Hebb's theory, the adopted selection process does not incorporate essential aspects of biological neural computation, as neural activity, which is often governed by statistical distributions consistent with power-law scaling. Given this limitation, the present work aimed to bring the model's dynamics closer to that observed in real cortical networks. To achieve this, a new selection mechanism inspired by the dynamics of gamma oscillation cycles, called E%-winners-take-all, was implemented, combined with an inhibition process based on the ratio between excitatory and inhibitory neurons observed in various regions of the cerebral cortex. The results obtained from our model (called E%-WTA model) were compared with those of the original model, and the analyses demonstrated that the introduced modifications allowed the network's own dynamics to determine the size of the formed assemblies. Furthermore, the recovery rate of these groups, through the evocation of the stimuli that generated them, became superior to that obtained in the original model.

The action potential constitutes the digital component of the signaling dynamics of neurons. But the biophysical nature of the full-time course of the action potential associated with changes in membrane potential is mathematically distinct from its representation as a discrete set of events that encode when action potentials are triggered in a collection of spike trains. In this paper, we develop from first principles a unified functional-analytic framework for neuronal spike trains, grounded in Schwartz distribution theory. We show how this representation provides an exact operational calculus for convolution, distributional differentiation, and distributional support, which enables closed-form analysis of spike train dynamics without discretization, rate approximation, or smoothing. We then analyze the framework in the context of a two-neuron reciprocal circuit with propagation latencies and refractoriness, deriving exact results for synaptic drive, spike timing sensitivity, and causal admissibility of inputs, quantities that are either ill-defined or require approximation in conventional treatments.

To maintain coherence in language, the brain must satisfy key competing temporal demands: the gradual accumulation of meaning across extended context (drift) and the rapid reconfiguration of representations at event boundaries (shift). How these processes are implemented in the human brain during naturalistic listening remains unclear. Here, we tested whether both can be captured by annotation-free drift and shift signals and whether their neural expression shows distinct regional preferences across the brain. These signals were derived from a large language model (LLM) processing the narrative input. To enable high-precision voxelwise encoding models with stable parameter estimates, we densely sampled one healthy adult across more than 7 hours of listening to crime stories while collecting 7 Tesla fMRI data. We then modeled the feature-informed hemodynamic response using a regularized encoding framework validated on independent stories. Drift predictions were prevalent in default-mode network hubs, whereas shift predictions were evident bilaterally in the primary auditory cortex and language association cortex. Together, these findings show that coherence during language comprehension is implemented through distinct but co-expressed neural regimes of slow contextual integration and rapid event-driven reconfiguration, offering a mechanistic entry point for understanding disturbances of language coherence in psychiatric disorders.

In complex ecological communities, species may self-organize into clusters or clumps where highly similar species can coexist. The emergence of such species clusters can be captured by the interplay between neutral and niche theories. Based on the generalized Lotka-Volterra model of competition, we propose a minimal model for ecological communities in which the steady states contain self-organized clusters. In this model, species compete only with their neighbors in niche space through a common interaction strength. Unlike many previous theories, this model does not rely on random heterogeneity in interactions. Even in this minimal model where only the common interaction strength is varied, we find an exponentially large set of states that exhibit a rich variety of cluster patterns with different sizes and combinations. There are sharp phase transitions into the formation of clusters. There are also multiple phase transitions between different sets of possible cluster patterns, many of which accumulate near a small number of critical points. We analyze this phase structure using both numerical and analytical methods. In addition, the special case with only nearest neighbor interactions is exactly solvable using the method of transfer matrices from statistical mechanics. We analyze the critical behavior of these systems.

Conventional colorimetric sensing methods typically rely on signal intensity at a single wavelength, often selected heuristically based on peak visual modulation. This approach overlooks the structured information embedded in full-spectrum transmission profiles, particularly in intensity-based systems where linear models may be highly effective. In this study, we experimentally demonstrate that applying a forward feature selection strategy to normalized transmission spectra, combined with linear regression and ten-fold cross-validation, yields significant improvements in predictive accuracy. Using food dye dilutions as a model system, the mean squared error was reduced from over 22,000 with a single wavelength to 3.87 using twelve selected features, corresponding to a more than 5,700-fold enhancement. These results validate that full-spectrum modeling enables precise concentration prediction without requiring changes to the sensing hardware. The approach is broadly applicable to colorimetric assays used in medical diagnostics, environmental monitoring, and industrial analysis, offering a scalable pathway to improve sensitivity and reliability in existing platforms.

Simulating glucose dynamics in individuals with type 1 diabetes (T1D) is critical for developing personalized treatments and supporting data-driven clinical decisions. Existing models often miss key physiological aspects and are difficult to individualize. Here, we introduce physiologically-constrained neural network (NN) digital twins to simulate glucose dynamics in T1D. To ensure interpretability and physiological consistency, we first build a population-level NN state-space model aligned with a set of ordinary differential equations (ODEs) describing glucose regulation. This model is formally verified to conform to known T1D dynamics. Digital twins are then created by augmenting the population model with individual-specific models, which include personal data, such as glucose management and contextual information, capturing both inter- and intra-individual variability. We validate our approach using real-world data from the T1D Exercise Initiative study. Two weeks of data per participant were split into 5-hour sequences and simulated glucose profiles were compared to observed ones. Clinically relevant outcomes were used to assess similarity via paired equivalence t-tests with predefined clinical equivalence margins. Across 394 digital twins, glucose outcomes were equivalent between simulated and observed data: time in range (70-180 mg/dL) was 75.1$\pm$21.2% (simulated) vs. 74.4$\pm$15.4% (real; P<0.001); time below range (<70 mg/dL) 2.5$\pm$5.2% vs. 3.0$\pm$3.3% (P=0.022); and time above range (>180 mg/dL) 22.4$\pm$22.0% vs. 22.6$\pm$15.9% (P<0.001). Our framework can incorporate unmodeled factors like sleep and activity while preserving key dynamics. This approach enables personalized in silico testing of treatments, supports insulin optimization, and integrates physics-based and data-driven modeling. Code: https://github.com/mosqueralopez/T1DSim_AI

Most vaccines require multiple doses, the first to induce recognition and antibody production and subsequent doses to boost the primary response and achieve optimal protection. We show that properly prioritizing the administration of first and second doses can shift the epidemic threshold, separating the disease-free from the endemic state and potentially preventing widespread outbreaks. Assuming homogeneous mixing, we prove that at a low vaccination rate, the best strategy is to give absolute priority to first doses. In contrast, for high vaccination rates, we propose a scheduling that outperforms a first-come first-served approach. We identify the threshold that separates these two scenarios and derive the optimal prioritization scheme and inter-dose interval. Agent-based simulations on real and synthetic contact networks validate our findings. We provide specific guidelines for effective resource allocation, showing that adjusting the timing between primer and booster significantly impacts epidemic outcomes and can determine whether the disease persists or disappears.

Abrupt learning is a common phenomenon in recurrent neural networks (RNNs) trained on working memory tasks. In such cases, the networks develop transient slow regions in state space that extend the effective timescales of computation. However, the mechanisms driving sudden performance improvements and their causal role remain unclear. To address this gap, we introduce the ghost mechanism, a process by which dynamical systems exhibit transient slowdown near the remnant of a saddle-node bifurcation. By reducing the high-dimensional dynamics near ghost points, we derive a one-dimensional canonical form that analytically captures learning as a process controlled by a single scale parameter. Using this model, we study a form of abrupt learning emerging from ghost points and identify a critical learning rate that scales as an inverse power law with the timescale of the learned computation. Beyond this rate, learning collapses through two interacting modes: (i) vanishing gradients and (ii) oscillatory gradients near minima. These features can lock the system into high-confidence but incorrect predictions when parameter updates trigger a no-learning zone, a region of parameter space where gradients vanish. We validate these predictions in low-rank RNNs, where ghost points precede abrupt transitions, and further demonstrate their generality in full-rank RNNs trained on canonical working memory tasks. Our theory offers two approaches to address these learning difficulties: increasing trainable ranks stabilizes learning trajectories, while reducing output confidence mitigates entrapment in no-learning zones. Overall, the ghost mechanism reveals how the computational demands of a task constrain the optimization landscape, demonstrating that well-known learning difficulties in RNNs partly arise from the dynamical systems they must learn to implement.

We propose a hierarchically modular, dynamical neural network model whose architecture minimizes a specifically designed energy function and defines its temporal characteristics. The model has an internal and an external space that are connected with a layered internetwork that consists of a pair of forward and backward subnets composed of static neurons (with an instantaneous time-course). Dynamical neurons with large time constants in the internal space determine the overall time-course. The model offers a framework in which state variables in the network relax in a warped space, due to the cooperation between dynamic and static neurons. We assume that the system operates in either a learning or an association mode, depending on the presence or absence of feedback paths and input ports. In the learning mode, synaptic weights in the internetwork are modified by strong inputs corresponding to repetitive neuronal bursting, which represents sinusoidal or quasi-sinusoidal waves in the short-term average density of nerve impulses or in the membrane potential. A two-dimensional mapping relationship can be formed by employing signals with different frequencies based on the same mechanism as Lissajous curves. In the association mode, the speed of convergence to a goal point greatly varies with the mapping relationship of the previously trained internetwork, and owing to this property, the convergence trajectory in the two-dimensional model with the non-linear mapping internetwork cannot go straight but instead must curve. We further introduce a constrained association mode with a given target trajectory and elucidate that in the internal space, an output trajectory is generated, which is mapped from the external space according to the inverse of the mapping relationship of the forward subnet.

Cancer cell populations often exhibit remarkably similar growth laws despite their heterogeneity. Explanations of universal cell population growth remain partly unresolved to this day. Here, we present a growth-law unification by investigating the connection between microscopic assumptions and the expected contact inhibition, which leads to five classical tumor growth laws: exponential, radial growth, fractal growth, generalized logistic, and Gompertzian growth. All five can be seen as manifestations of a single microscopic model. Agent-based simulations substantiate our theory, and we can explain differences in growth curves in experimental data from em in vitro cancer cell population growth. Thus, our framework offers a possible explanation for many mean-field laws used to empirically capture seemingly unrelated cancer or microbial growth dynamics. Our results highlight that the interplay between contact inhibition and other assumptions (e.g., well-mixed) can influence our quantitative understanding of how cancer cells grow and, in turn, how they may interact.

In the framework of population dynamics, the basic reproduction number R_0 is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments it is calculated as the spectral radius of the so-called next-generation operator. In continuously structured populations defined in a Banach lattice X with concentrated states at birth one cannot define the next-generation operator in X. In the present paper we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which R_0 can be computed as the spectral radius of the next-generation operator. We apply these results to some examples: the (classical) size-dependent model, a size structured cell population model, a size structured model with diffusion in structure space (under some particular assumptions) and a (physiological) age-structured model with diffusion in structure space.

Nitrous oxide (N2O) is a potent greenhouse gas emitted during biological wastewater treatment. A pseudo-mechanistic model describing three biological pathways for nitric oxide (NO) and N2O production was calibrated for mixed culture biomass from an activated sludge process using laboratory-scale experiments. The model (NDHA) comprehensively describes N2O producing pathways by both autotrophic ammonium oxidizing bacteria and heterotrophic bacteria. Extant respirometric assays and anaerobic batch experiments were designed to calibrate endogenous and exogenous processes (heterotrophic denitrification and autotrophic ammonium/nitrite oxidation) together with the associated net N2O production. Ten parameters describing heterotrophic processes and seven for autotrophic processes were accurately estimated (variance/mean < 25%). The model predicted NO and N2O dynamics at varying dissolved oxygen, ammonium and nitrite levels and was validated against an independent set of experiments with the same biomass. Aerobic ammonium oxidation experiments at two oxygen levels used for model evaluation (2 and 0.5 mg/L) indicated that both the nitrifier denitrification (42, 64%) and heterotrophic denitrification (7, 17%) pathways increased and dominated N2O production at high nitrite and low oxygen concentrations; while the nitrifier nitrification pathway showed the largest contribution at high dissolved oxygen levels (51, 19%). The uncertainty of the biological parameter estimates was propagated to N2O model outputs via Monte Carlo simulations as 95% confidence intervals. The accuracy of the estimated parameters resulted in a low uncertainty of the N2O emission factors (4.6 +- 0.6% and 1.2 +- 0.1%).

We investigate the role of migration patterns on the spread of epidemics in complex networks. We enhance the SIS-diffusion model on metapopulations to a nonlinear diffusion. Specifically, individuals move randomly over the network but at a rate depending on the population of the departure patch. In the absence of epidemics, the migration-driven equilibrium is described by quantifying the total number of individuals living in heavily/lightly populated areas. Our analytical approach reveals that strengthening the migration from populous areas contains the infection at the early stage of the epidemic. Moreover, depending on the exponent of the nonlinear diffusion rate, epidemic outbreaks do not always occur in the most populated areas as one might expect.