Automated Discovery of Spatio-Temporal Epidemic Patterns via Hybrid Evolutionary-PDE Frameworks

An Analytically Tractable Framework for Multi-Strain Epidemics: Resolving Algebraic Complexity to Map Oscillatory Dynamics

Multi-strain epidemiological systems frequently exhibit self-sustained oscillations, yet severe algebraic complexity has long obstructed a complete analytical characterization of these dynamics. Seeking to bypass these barriers, we have identified a broad, analytically tractable class of two-strain models featuring asymmetric cross-immunity that precludes secondary infections for a single strain. This targeted structural simplification allows us to derive explicit expressions for coexistence equilibria and map their stability. We show that the relative transmission advantage of secondary infections fundamentally governs the onset of robust limit-cycle oscillations, extending previous narrow-borderline results. Crucially, numerical simulations reveal a rich macroscopic landscape where small-amplitude local oscillations coexist with large-amplitude, recurrent outbreak cycles. The structural robustness of these phenomena is confirmed by incorporating effects such as waning immunity and isolation and extending the framework beyond the analytically tractable class. These findings provide a clear mechanistic explanation for multi-strain epidemic cycles and offer a highly tractable baseline for future theoretical developments.

Turing instability and bistable epidemic spread on complex networks: Effects of topology and initial seeding.

Understanding how diseases propagate through structured populations is essential for predicting and controlling epidemics. This study develops a reaction-diffusion framework on complex networks to investigate how local bistability, dispersal, and network topology jointly determine infection dynamics. Analytical conditions for Turing instability are derived and examined in Erdős-Rényi and scale-free networks using a bistable susceptible-infected model under mean-field approximation. The analysis shows that high-degree nodes become monostable, whereas low and intermediate-degree nodes exhibit bistability. Simulations confirm that infection outcomes depend strongly on degree distribution, transmission rate, and dispersal intensity. Epidemics seeded at highly connected nodes spread faster and more extensively, while structural differences between network types yield distinct thresholds and outbreak patterns. Together, these results reveal how local nonlinearities and network heterogeneity interact to shape epidemic transitions, offering a theoretical basis for understanding spatial disease persistence and designing targeted control strategies.

Applied Mathematical Modelling

Threshold dynamics and Gaussian stationary distribution of a stochastic SIQR epidemic model with spatial diffusion: a semigroup approach

The spread of infectious diseases is strongly influenced by environmental variability, random disturbances, and population mobility across geographical regions. These factors can significantly affect disease transmission and progression, making purely deterministic models insufficient for capturing realistic epidemic dynamics. Environmental noise may modify transmission rates, recovery processes, and other epidemiological parameters, while the movement of individuals contributes to the spatial propagation of infections. Reaction-diffusion epidemic models provide an effective mathematical framework for describing the spatial spread of diseases, whereas stochastic modeling captures the influence of random fluctuations. Combining stochastic effects with spatial diffusion allows epidemic models to represent more realistic biological and environmental conditions. Despite these advantages, rigorous analytical studies for stochastic reaction-diffusion epidemic models that incorporate quarantine strategies are still relatively limited in the existing literature. In this work, we formulate a stochastic SIQR reaction-diffusion epidemic model to describe the spatio-temporal dynamics of four interacting populations: susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S),$$\end{document} infected \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I),$$\end{document} quarantined \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q),$$\end{document} and recovered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R).$$\end{document} Diffusion terms are included to represent the spatial movement of individuals, while stochastic perturbations account for environmental randomness influencing disease transmission. Using stochastic transformations together with linear semigroup theory, we establish the existence and uniqueness of a global positive solution to the nonlinear stochastic system. Two threshold parameters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{e}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{s},$$\end{document} are introduced to characterize the extinction and strong persistence of the infection. For the special case of a constant transmission rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta,$$\end{document} the endemic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*}$$\end{document} is derived, and the asymptotic behavior of the system in a neighborhood of this equilibrium is investigated using linear algebra methods and Lyapunov matrix equations. The theoretical analysis shows that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{e} < 1,$$\end{document} the infected population becomes extinct at an exponential rate with probability one. In contrast, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{s} > 1,$$\end{document} the disease persists strongly in the population. Furthermore, in the vicinity of the endemic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*},$$\end{document} the joint distribution of the system states can be approximated by a multivariate normal probability density function. An explicit representation of the corresponding covariance matrix is obtained under several parameter conditions. Numerical simulations support the analytical results and demonstrate how stochastic perturbations and diffusion rates influence both the stability properties and probabilistic behavior of the epidemic system. The proposed stochastic SIQR reaction-diffusion model provides a rigorous analytical framework for studying the combined influence of spatial diffusion and environmental randomness on epidemic dynamics. The derived extinction and persistence thresholds offer clear criteria for determining long-term disease outcomes. Moreover, the probabilistic characterization near the endemic equilibrium improves understanding of fluctuations around steady states and contributes to the theoretical analysis of stochastic epidemic systems in spatially heterogeneous environments.

Rare-event sampling for stochastic dynamics in network systems using cluster updates

Understanding the stochastic evolution in complex networks is a central challenge across physics, biology, engineering, social science, and finance. The most consequential macroscopic events, like cascading failures in communication networks, widespread epidemic outbreaks, and rapid shifts in societal opinions, often emerge from a confluence of rare, localized stochastic processes and need to pass certain bottlenecks. Standard forward-time simulation algorithms like the Gillespie method are inefficient for the investigation of such phenomena due to catastrophic rejection rates. Advanced rare-event techniques like splitting methods and transition-path sampling often suffer from kinetic trapping, path degeneracy, genealogical correlations, or critical slowing down when applied to complex heterogeneous networks. We propose to overcome this challenge by establishing a novel technique called conditional-path Monte Carlo (CPMC), inspired by loop algorithms from equilibrium condensed-matter physics. By employing non-local updates on spacetime clusters without rejections, CPMC generates a Markov chain of trajectories that all strictly respect the targeted macroscopic boundary conditions like the occurrence of a massive network failure. We demonstrate the framework's potential by performing a simple risk factor analysis for rare large-scale epidemic outbreaks in SIS dynamics on kinship networks.

An exploration of finite-size effects in the dynamics of epidemic compartmental modelling

Finite-size effects arise when the dynamical behaviour of human-interacting or particle-interacting systems with inherent randomness deviates from the predictions of the corresponding mean-field equations. Despite their frequent observation in practice, the mechanisms by which finite-size effects shape and influence real-world epidemic dynamics remain largely unexplored. To this end, in this article, we study these effects in the context of both agent-based and mean-field epidemic modelling frameworks, with particular attention to dynamical regimes characterized by repeated outbreaks. Our ultimate objective is to inform and improve public strategies for epidemic prediction and prevention. As an initial step towards our broader goal, we focus on a minimal epidemic model that naturally captures recurrent epidemic waves, namely the SIHRS compartmental model, which assumes waning immunity in recovered individual. This model may be viewed as a natural extension of the classical SIHR and SIRS compartmental models. Rather than obscuring the underlying mechanisms with an overly intricate model structure, the SIHRS framework allows us to isolate, identify and track finite-size effects over time. We carry out our analysis using a martingale-based early-time-step method, in which martingale formulations of the underlying Markov processes are derived and exploited. In particular, the martingale formulations are extended beyond compartment variables to higher-order quantities, which constitutes a key step towards advancing the analysis beyond short-time approximations. Our analysis delivers a quantitative and systematic understanding of finite-size effects in epidemic dynamics, demonstrating that their impact is shaped by population size, compartment scale and temporal structure. These findings are supported by extensive numerical simulations calibrated with both national-scale and county-level COVID-19 data.