Aug 2026· Bulletin of Mathematical Biology· Vol 88· 0 citations· 44 references
Medicine
TL;DR
The proposed additive ODE–SDE model produces a close fit with coherent uncertainty quantification and a flexible seasonal reconstruction, while keeping the mechanistic transmission model interpretable.
Abstract
Deterministic dynamical models are widely used in infectious disease modelling, but they often become systematically biased when key drivers such as seasonality and random environmental variation are simplified or omitted. This paper proposes a practical way to account for structural bias while preserving the underlying mechanistic model. We model the observed time series as the sum of (i) a deterministic transmission component given by a reduced Ross malaria model and (ii) a latent stochastic seasonal component that captures unresolved seasonal forcing and other unmodelled variability. The seasonal component is defined as a mean-reverting stochastic differential equation with periodic forcing, which can be interpreted as a seasonally forced Ornstein–Uhlenbeck (OU) process and, at the same time, as a dynamically constrained model-discrepancy term. The combined model forms an additive Bayesian state-space system. A key challenge in additive decompositions is identifiability: many combinations of the deterministic and stochastic components can explain the same observations. We therefore use informative priors to stabilise this decomposition, together with a non-centred parameterisation (a reparameterisation that improves MCMC efficiency by sampling standardised noise terms instead of states directly) of the latent stochastic differential equation (SDE) states that enables efficient joint inference with Hamiltonian Monte Carlo (NUTS) in Stan. We validate the approach in three steps: (1) an OU example that contrasts parameter-only inference with latent state-space inference, (2) synthetic experiments that have a good agreement with the observed signal while highlighting the expected negative posterior dependence between components, and (3) an application to five years of monthly malaria case reports from Delta State, Nigeria. In the real-data analysis, the proposed additive ODE–SDE model produces a close fit with coherent uncertainty quantification and a flexible seasonal reconstruction, while keeping the mechanistic transmission model interpretable. Overall, the framework provides a flexible and transferable method for accounting for structural model discrepancy in misspecified dynamical models using a structured stochastic discrepancy.
A Bayesian computational framework for parsimonious inference in stochastic nonlinear dynamical systems is presented, and it is shown that inducing sparsity among the model parameters eliminates redundant interactions between compartments, equivalently revealing the optimal coupling structure between differential equations.
B. Robinson, Philippe Bisaillon, R. Sandhu et al.· PLoS ONE· 0 citations
Hierarchical Bayesian smooth transition autoregressive models provide accurate forecasts by accommodating nonlinear regime-switching dynamics while delivering robust uncertainty quantification, which makes them well suited for infectious disease surveillance and public health decision-making in resource-limited, high-uncertainty settings.
G. Singini, Samuel Manda· Frontiers in Applied Mathema...· 0 citations
We conduct a retrospective analysis of COVID-19 transmission dynamics across U.S. states using a modified population-based Susceptible-Exposed-Infectious-Asymptomatic-Recovered-Deceased (SEIARD) compartmental model. The proposed framework introduces time-varying transmission, reporting, and mortality rates to capture temporal variations in public behavior and policy interventions during the pandemic. In particular, the transmission rate is modeled as a function of population mobility (derived from Google Mobility Reports), with a residual time-decay term capturing the net effect of unobserved factors such as behavioral adaptation and control measures, while reporting is linked to nationwide testing strategies. We employ a Bayesian approach to integrate multiple data sources and quantify uncertainties in model parameters. The model explicitly distinguishes between symptomatic and asymptomatic infectious individuals and links the latent epidemic states to observable quantities, including reported cases and deaths, through a dynamic reporting function. This retrospective modeling framework provides insights into state-level epidemic trajectories and supports data-driven decision-making for optimal allocation of healthcare resources and evaluation of public health interventions during future pandemics. We further apply a clustering analysis to the posterior parameter estimates to identify groups of U.S. states exhibiting similar epidemiological characteristics, revealing substantial regional heterogeneity in transmission intensity, reproduction dynamics, and mortality burden.
A unified complete-data likelihood framework for epidemic processes evolving on partially observed dynamic networks that contributes to statistical inference for partially observed interacting stochastic systems on evolving networks and establishes a foundation for uncertainty-aware analysis of complex transmission processes.
Autocorrelated bivariate count data frequently arise in criminal, environmental and financial studies, where capturing both serial dependence and cross‐series interaction is essential for statistical modelling and inference. In many applications, such dynamics are further influenced by exogenous covariates such as policy interventions or environmental factors, leading to time‐varying dependence structures that are not adequately captured by standard models. Existing bivariate integer‐valued autoregressive (BINAR) models mainly rely on constant or observation‐driven coefficients and rarely incorporate covariate information, which restricts their ability to represent evolving dependence in multivariate count processes. To address this limitation, we propose a covariate‐driven doubly stochastic bivariate integer‐valued autoregressive process, in which the thinning mechanism evolves jointly with past observations and exogenous covariates. This formulation extends the classical BINAR framework by allowing the dependence structure to vary dynamically under both internal and external driving mechanisms. The basic statistical properties of the proposed process are derived, and two estimation methods are developed, including an EM‐based algorithm. Monte Carlo simulations and a real data application are conducted to assess finite‐sample performance and robustness under different settings.
Epidemic modelling increasingly combines routine surveillance, mechanistic transmission theory and data-intensive learning, yet the resulting approaches are often compared as though they estimate the same quantities and serve the same decisions. This critical narrative review evaluates statistical, mathematical, machine-learning and hybrid approaches to infectious disease dynamics, with emphasis on inferential purpose, data-generating and observation processes, uncertainty, validation, interpretability and public-health use. Peer-reviewed literature published from January 2000 to 31 May 2026 was identified through biomedical, multidisciplinary and computing-oriented scholarly sources, supplemented by citation searching and verification against authoritative bibliographic records. Foundational earlier papers were retained when necessary. The synthesis indicates that no model class is uniformly superior. Statistical surveillance and time-series models are often efficient for anomaly detection, nowcasting and short-horizon forecasting, but their parameters rarely support intervention counterfactuals without additional causal structure. Mechanistic compartmental, network, spatial and agent-based models make transmission assumptions explicit and can represent intervention pathways, although structural misspecification, weak identifiability and mismatch between latent infections and observed reports can dominate their uncertainty. Machine-learning models can extract nonlinear and high-dimensional patterns from heterogeneous data, but apparent accuracy is vulnerable to temporal or spatial leakage, changing surveillance systems, distribution shift, weak probabilistic calibration and limited causal meaning. Hybrid models can combine epidemiological constraints with flexible learning and data assimilation; their interpretability nevertheless depends on identifiable parameters, biologically coherent architecture and validation beyond the setting used for training. Across paradigms, the observation process, target definition and decision horizon are as consequential as model form. Credible use therefore requires question-first model selection, explicit separation of forecasts from scenarios, rolling and geographically external validation, calibrated uncertainty, versioned data and code, and transparent communication of assumptions. Progress will depend less on greater complexity alone than on prospective benchmarking, identifiable hybridisation, behaviour-aware causal designs, multimodal surveillance, equitable data systems and operational model governance.
Hamid H. Hussien, Muhammed Aljifri, Nuha Hassan Hagabdulla et al.· Asian Journal of Probability...· 0 citations
