Stochastic Modelling and Analysis of Within-Farm Highly Pathogenic Avian Influenza Dynamics in Dairy Cattle

Highly pathogenic avian influenza (HPAI) has expanded its host range with recent detections in dairy cattle, raising critical concerns regarding within-herd persistence and cross-species spillover. This study develops a stochastic $SEI_sI_aR-B$ compartmental model to analyse HPAI transmission, explicitly accounting for environmental pathogen reservoirs and noise intensities through Wiener processes. The positivity and boundedness of solutions are established, and the disease-free and endemic equilibria are analytically derived. The basic reproduction number is determined using the next-generation matrix method. Numerical simulations confirm that the model dynamics are consistent with theoretical analysis and illustrate how stochastic fluctuations significantly influence disease persistence. Furthermore, sensitivity analysis using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) identifies the transmission rate from asymptomatic infectious cattle ($β_a$) as the primary driver of transmission. The model effectively captures the dynamics of environmental variability affecting HPAI spread, suggesting that effective control strategies must prioritise the early detection and isolation of asymptomatic carriers alongside environmental management.

💡 Research Summary

This paper addresses the emerging threat of highly pathogenic avian influenza (H5N1) in dairy cattle, a host shift that has been documented in the United States during 2024. Recognizing that traditional deterministic compartmental models cannot capture the stochastic nature of disease spread in livestock systems, the authors develop a stochastic SEIsIaR‑B model that incorporates both direct transmission (from symptomatic and asymptomatic cattle) and indirect environmental transmission. The model consists of six compartments: Susceptible (S), Exposed (E), Symptomatic infectious (Is), Asymptomatic infectious (Ia), Removed (R), and an environmental virus reservoir (B). Direct transmission rates are denoted by βs and βa, while environmental transmission follows a saturating function βB S B/(K + B) to reflect diminishing marginal risk at high contamination levels. Recruitment (Λ) and natural mortality (μ) render the herd an open population, and disease‑induced mortality (d) applies only to symptomatic animals. Both infectious classes shed virus into the environment at rates ωs and ωa, and the environmental load decays at rate ε.

To capture random fluctuations in recruitment, latent period, contact patterns, shedding, and environmental persistence, multiplicative white‑noise terms (σi Xi dWi) are added to all compartments except the recovered class. The resulting system of stochastic differential equations (SDEs) is analyzed using Itô calculus. The authors prove global existence, uniqueness, positivity, and uniform boundedness of solutions, showing that the stochastic model remains biologically feasible for all time. By summing the SDEs they derive an upper bound for the expected total cattle population and for the environmental virus load, confirming that stochastic perturbations do not cause blow‑up.

A next‑generation matrix approach yields an explicit basic reproduction number:

R₀ = (βs ν σ)/(μ + d + γ) + (βa (1‑ν) σ)/(μ + d + δ) + (βB ωmax Λ)/(ε μ²),

where ν is the proportion of exposed cattle that become symptomatic, ωmax = max{ωs, ωa}, and the other symbols retain their usual epidemiological meanings. The deterministic analysis shows that R₀ > 1 guarantees a unique endemic equilibrium, while R₀ < 1 leads to disease extinction. However, stochastic simulations reveal that sufficiently large noise intensities (especially σB) can sustain infection even when the deterministic R₀ is below unity, highlighting the critical role of environmental variability.

Numerical experiments explore a range of parameter sets and noise levels. In the absence of noise, the system behaves as expected from deterministic theory. When noise is introduced, trajectories exhibit larger amplitude fluctuations, occasional epidemic bursts, and sometimes stochastic fade‑out despite R₀ > 1. The environmental noise σB is particularly influential: higher σB amplifies the variability of B, which in turn increases the force of infection from the environment, potentially driving the system into recurrent outbreaks.

A global sensitivity analysis using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) identifies βa (the transmission rate from asymptomatic cattle) as the most influential parameter on the final epidemic size and persistence probability. βs and βB also show positive correlations, whereas the environmental decay rate ε and natural mortality μ have negative PRCC values, indicating that faster virus degradation or higher turnover reduces transmission. All noise intensities have positive PRCCs, confirming that stochasticity generally promotes disease persistence.

The authors discuss practical implications: early detection and isolation of asymptomatic carriers should be a priority, as these animals are the main drivers of spread. Environmental management—regular cleaning of milking equipment, bedding, and water sources, as well as measures that reduce viral survival (e.g., temperature control, UV exposure, disinfectants)—is essential to lower βB and ωa. The stochastic framework also allows evaluation of control strategies such as vaccination, movement restrictions, and targeted culling under realistic uncertainty.

Limitations are acknowledged: parameter values are drawn from limited field data, the model assumes homogeneous mixing within a single herd, and it does not consider inter‑farm transmission or zoonotic spill‑over to humans. Future work is suggested to extend the model to a network of farms, incorporate human exposure pathways, and calibrate the stochastic model with longitudinal outbreak data.

In summary, the paper provides a rigorous stochastic modeling platform for HPAI in dairy cattle, demonstrates how random environmental fluctuations can alter disease dynamics, and offers quantitative guidance for surveillance and biosecurity policies aimed at mitigating this emerging cross‑species threat.