https://orcid.org/0000-0002-0197-2317
Figures
Abstract
Background
Highly pathogenic avian influenza (HPAI) viruses, particularly subtypes such as H5N1 and H7N9, have caused widespread outbreaks in wild birds, poultry, livestock and occasionally humans, raising concerns about cross-species transmission and pandemic potential. Effective control and surveillance strategies require a thorough understanding of HPAI transmission dynamics, which can be supported by mathematical modelling.
Objective
This scoping review aimed to identify mechanistic models used to study HPAI transmission. Specifically, we sought to categorize model types, describe their application contexts (e.g., wild birds, poultry, livestock, and humans), and highlight modelling gaps relevant to understanding and mitigating the risks of HPAI spread.
Methods
Following PRISMA guidelines and the PRISMA extension for scoping reviews (PRISMA-ScR), we conducted systematic searches of PubMed and Web of Science to identify peer-reviewed studies employing deterministic and stochastic models to analyze HPAI transmission. Eligible articles published between January 2023 and June 2025 were screened and grouped by model structure, host populations, transmission pathways, and modelling objectives.
Results
After screening, 30 studies published after 2023 were included in this scoping review. Compartmental models were the most common (26 studies), with 16 deterministic and 10 stochastic approaches. These models were primarily used to describe transmission among wild birds, poultry, livestock, and humans and to evaluate interventions such as culling, vaccination, and movement restrictions. Agent-based models (2 studies) captured individual-level interactions and spatial heterogeneity, while network models (2 studies) represented contact structures and transmission pathways between farms or species.
Conclusions
Currently, mechanistic modelling of HPAI is dominated by compartmental approaches, including both deterministic and stochastic formulations, whereas agent-based and network models remain relatively underused. Although most studies focus on transmission in wild birds and poultry, and in some cases spillover infections to humans, few explicitly examine infection dynamics in livestock or in transmission between livestock and humans, despite the importance of livestock (e.g., cattle) as potential intermediaries in human infection. Key gaps persist in the integration of empirical data, representation of multi-host interactions, and evaluation of realistic intervention strategies. Addressing these limitations is essential to improve predictive accuracy and to strengthen the role of modelling in informing HPAI surveillance and control.
Citation: Wang M, Laison EKE, Philippsen T, Ghaemi S, Liu J, Moyles I, et al. (2026) Mechanistic modelling of highly pathogenic avian influenza: A scoping review revealing critical gaps in cross-species transmission models. PLoS One 21(4): e0347929. https://doi.org/10.1371/journal.pone.0347929
Editor: Julian Ruiz-Saenz, Universidad Cooperativa de Colombia, COLOMBIA
Received: January 7, 2026; Accepted: April 9, 2026; Published: April 30, 2026
Copyright: © 2026 Wang et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC-DG(BN)), Fonds de recherche du Québec – Santé (FRQS) ResearchScholar Junior 2 (BN), and the Canadian Institutes of Health Research (CIHR-H5N1: 525197, PIs: BN and JM). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Rationale
The emergence and spread of avian influenza viruses (AIVs) threaten the health of poultry, livestock, wildlife, and humans [1–3]. Avian influenza A viruses are maintained in wild aquatic birds, which serve as the primary natural reservoir. Infected wild birds are typically asymptomatic or exhibit only mild symptoms, as they are usually infected with low pathogenic avian influenza (LPAI) viruses [4–6]. However, as wild birds migrate, they can transmit LPAI viruses to resident birds and domestic poultry. Notably, some LPAI subtypes, especially H5 and H7, have the potential to genetically mutate into highly pathogenic forms [7,8]. In contrast, highly pathogenic avian influenza (HPAI) viruses cause severe disease and high mortality in poultry, resulting in devastating economic losses and posing a significant threat to human safety [3,5].
The spread of HPAI viruses is facilitated by complex cross-species dynamics and diverse transmission pathways. The natural reservoirs of these viruses are wild aquatic birds, which become infectious very quickly after exposure [9,10]. HPAI viral spillover into new areas and susceptible populations occurs primarily from migrating infected wild birds [3,11,12]. Once introduced, the virus usually spreads rapidly within poultry farms through direct contact between birds, the fecal-oral route, or contaminated media (e.g., equipment, clothing) [13,14]. Environmental contamination plays a crucial role in indirect transmission, as HPAI viruses can survive in water, soil, and organic matter, thereby becoming a persistent source of infection [15]. The transportation and trade of poultry and livestock further expand the spread of the virus [16–19] without adequate measures in place [20].
The H5N1 viruses, descendants of the A/goose/Guangdong/1/1996 (Gs/Gd) lineage, including the widely disseminated 2.3.4.4b, have drawn sustained global attention [21]. First detected in poultry in 1996 with human cases reported in Hong Kong in 1997, H5N1 is now entrenched in parts of Asia and Africa and has spread to Europe and North America [22,23]. Since the early 2020s, a sustained H5N1 panzootic has intensified, characterized by rapid geographic expansion, large-scale wild bird mortality, and unprecedented cross-species transmission in North America [24]. In 2024, Canada reported its first domestically acquired human H5N1 infection, coinciding with widespread outbreaks in wild birds, poultry, and increasing reports in mammals, including livestock [25]. In the United States, H5N1 was first detected in dairy cattle in March 2024, and by the end of that year, it was reported in 689 herds across 15 states, with within-herd clinical illness affecting up to 20% of adult cows [26]. Several human cases, including farm workers in Texas and Michigan, were linked to occupational exposure, and by early 2025, over 70 human cases in the Americas were associated with infected dairy cattle [27,28].
Given the severe disease burden in poultry and cattle, and the ongoing concerns about zoonotic transmission to humans, risk regions worldwide have implemented a range of control measures. These include farm quarantine (movement restrictions and isolation on affected premises), strict biosecurity protocols, epidemiological investigations and contact tracing, culling of infected and exposed flocks or herds, safe disposal of animal products, and vaccination [20,29–31]. While culling infected and exposed populations is drastic, it aims to rapidly remove susceptible hosts and reduce selection pressure for viral evolution [32]. However, these measures can have substantial economic and social impacts, highlighting the need to rigorously evaluate their effectiveness across different epidemiological settings and host populations.
Mathematical modelling has become an indispensable tool in infectious disease epidemiology, providing crucial insights into the understanding and management of complex pathogens such as HPAI [33–35]. These models provide a framework to elucidate complex transmission dynamics, such as bird-to-bird and bird-to-mammal transmission pathways, and to assess the risk of viral adaptation and spillover to humans [36,37]. They can also be used to evaluate potential interventions by simulating their effectiveness before implementation, thereby optimizing resource allocation [38]. Mathematical models also help estimate key epidemiological parameters, such as the basic reproduction number () and transmission rates [39,40]. Given the still-limited but increasing number of human H5N1 cases to date, and the inherent challenges in directly observing rare spillover events or potential human-to-human transmission, mathematical models are particularly valuable. They serve as critical tools for understanding unobservable dynamics, assessing future pandemic risks, and informing preparedness strategies in data-sparse scenarios.
We identified two prior reviews on this topic. The first review encompassed both LPAI and HPAI, integrating mathematical, statistical, machine learning, and hybrid modelling approaches [41]. However, its broad scope reduced comparability of mechanistic assumptions and provided limited guidance on parameterization, model validation, or policy application. The second review synthesizes 46 mechanistic studies, with a search cutoff date of April 2023 [42]. It primarily focuses on the transmission of LPAI and HPAI in wild birds and poultry, emphasizing parameter estimation and the evaluation of culling or vaccination strategies. A few studies explicitly addressed avian–human transmission, but livestock populations were absent, the avian–livestock–human interface was not considered, and movement or trade networks were only minimally represented. Since 2023, North America has witnessed several cross-species transmission events, including spillover from infected poultry to dairy cattle and subsequent human cases [43]. These events have prompted a shift in mechanistic modelling toward multi-host, One Health–oriented frameworks that integrate wildlife, poultry, livestock, and human components, often incorporating cross-species empirical parameterization and validation. This shift requires a focused synthesis of recent mechanistic models of HPAI to characterize these emerging approaches and their remaining limitations. Accordingly, we conduct a scoping review of mechanistic HPAI models published between January 2023 and June 2025.
Objectives
The objectives of this scoping literature review are to:
- Identify and catalogue mechanistic models used to study HPAI transmission dynamics across various host species (e.g., wild birds, poultry, livestock, and humans), with particular focus on H5 and H7 subtypes.
- Summarize the data sources, modelling approaches, study purposes, and key parameters reported in these studies.
- Highlight gaps and limitations in the current literature, particularly in the One Health context, including cross-species transmission, emerging zoonotic risks, and the pandemic potential of HPAI.
Methods
Protocol and registration
This scoping review was conducted following the Preferred Reporting Items for Systematic Reviews and Meta-Analyses extension for Scoping Reviews (PRISMA-ScR) guidelines and checklist [44]. All relevant checklist items were included, except for the critical appraisal of individual sources of evidence. The primary goal of this review was to map and categorize existing mechanistic models and associated data and parameters used in studying HPAI transmission dynamics. A completed PRISMA-ScR checklist is provided in Fig 1, and a detailed description of each methodological step is outlined below.
Information sources
Two databases were searched: PubMed and Web of Science. PubMed was selected for its comprehensive coverage of literature in life sciences and biomedicine, which are central to the focus of this study. To broaden the scope and capture relevant studies beyond the biomedical field, Web of Science was also included as a complementary source.
Search
The search was conducted on 13 June 2025 using two complementary strategies to maximize retrieval. In Strategy 1, we queried PubMed and Web of Science with Boolean combinations of three keyword sets applied to titles/abstracts/keywords: set 1 = (HPAI OR “avian influenza” OR H5N1); set 2 = (model); set 3 = (mathematical OR math OR equation OR predictive OR surveillance OR forecast). Queries followed the pattern (set 1) AND (set 2) AND (set 3).
Search Strategy 2 broadened the scope by modifying Keyword set 1 to include: HPAI, avian influenza and H5. The same combinations with Keyword sets 2 and 3 were then applied.
Both search strategies were used to query PubMed and Web of Science. The specific implementation of these search strategies for each database is provided in S1 Appendix. The process of determining study eligibility was conducted collaboratively by five authors (MW, EL, TP, JM, BN).
Eligibility criteria
The following inclusion and exclusion criteria were applied to the study selection process. Studies were included if they met the following criteria: (1) original, peer-reviewed research articles published in English; (2) used mechanistic modelling approaches (compartmental models, agent-based models, networks) to describe the transmission dynamics of the HPAI virus; and (3) focused on transmission between wild birds, poultry farms, livestock farms, or humans. Studies that fell under the following exclusion criteria were rejected: (1) within-host models focusing on immune dynamics or viral evolution of avian influenza; (2) ecological niche models, multi-criteria decision analyses, spatial risk modelling with GIS, machine learning; (3) primarily or solely statistical methods; (4) epidemiological studies; (5) review papers and commentaries; (6) focus on LPAI only; (7) studies published in languages other than English; (8) studies without full text availability. No geographical or date restrictions were applied at the start of this scoping review. However, since a previous scoping review had already synthesized studies up to April 2023 [42], we focused on including studies published between January 2023 and June 2025 to capture the most recent research.
Selection of sources of evidence
The search results (n = 6,131) from each database were uploaded to the scoping review platform Covidence, where duplicates (n = 3,341) were automatically removed. In the first round of screening, titles and abstracts of the retrieved papers were reviewed. Four authors (MW, EL, TP, BN) participated in this stage. Three authors (MW, EL, TP) independently assessed each paper, with three possible voting options: ‘yes’, ‘no’, or ‘maybe’. The system required two reviewers per paper. Papers receiving two ‘yes’ votes proceeded to the second round of screening, while those with two ‘no’ votes were removed and marked as ‘irrelevant’. Papers with inconsistent votes, or where one reviewer selected ‘maybe’, were flagged as ‘conflict’ and reviewed by a senior author (BN). The first-round screening specifically assessed whether a paper addressed mechanistic models of avian influenza. Consequently, review articles, experimental studies, machine learning approaches, ecological niche modelling, and multi-criteria decision analyses with GIS were excluded.
The second round of screening was the full-text review. Four authors (MW, EL, TP, BN) participated in this phase. Three authors (MW, EL, TP) independently reviewed each paper, selecting either ‘include’ or ‘exclude’ and providing reasons for exclusion when applicable. The system required two reviewers per paper. Papers with two ‘include’ votes proceeded to the next stage (‘extraction’), while those with two ‘exclude’ votes citing the same reason were removed. If the two reviewers disagreed or provided different exclusion reasons, the paper was marked as ‘conflict’. Conflicted papers were jointly reviewed and discussed by the three authors (MW, EL, TP) to reach a consensus; if consensus could not be achieved, the fourth author (BN) participated in the discussion and made the final decision.
Data charting process and data items
For each included study, data were extracted into a standardized spreadsheet provided in S1 Table. The charting process captured bibliographic information (electronic link to the publication, year of publication), study characteristics (pathotype, subtype, spatial scale, principal region or country, and epidemic year), and study objectives. Details of the modelling approach were also recorded, including model type, equations, transmission mechanisms, host populations considered (humans, wild birds, poultry, and other animals), control measures, and calibration and validation procedures. We also documented parameter values and sources, data sources and types, and the availability of data and code links. Finally, we extracted key results along with reviewers’ notes on study limitations.
Synthesis of results
The included studies were organized according to the type of mechanistic model used to study HPAI transmission dynamics. Three main categories were identified. The first category comprised compartmental models, which describe transmission dynamics using differential equations. In the context of HPAI, these models were applied to represent infections across multiple host populations, including wild birds, domestic poultry, livestock, and humans. They were used to explore within- and between-species transmission, assess the impact of interventions such as culling, vaccination, and movement restrictions, and to examine risks of zoonotic spillover. The second category comprised agent-based models, which simulate the behaviours and interactions of individual hosts or farms, often incorporating spatial, ecological, or behavioural heterogeneity. These models capture stochasticity and complex system dynamics emerging from individual-level interactions. The third category comprised network models, which represent hosts, farms, or geographic regions as nodes connected by edges that describe potential transmission pathways, such as movements, trade, or environmental contact. Network models can be deterministic or stochastic, and may integrate features from compartmental or agent-based frameworks. In the context of HPAI, they were used to analyze transmission across poultry transportation systems or to explore stochastic processes and control strategies within networked populations.
Results
Selection of sources of evidence
The database search identified 6,131 records. After removing duplicates, 3,341 unique studies entered the screening phase. Following title and abstract screening, 322 studies were retained for full-text review. Of these, 244 met the eligibility criteria. After applying the final inclusion criteria and restricting the date range to studies published after 2023, a total of 30 studies were included in this scoping review. Fig 1 presents a PRISMA flow diagram summarizing the screening and selection process.
Characteristics of sources of evidence
Table 1 provides an overview of the modelling approaches used in the 30 HPAI studies included in this scoping review, classified by model type: compartmental, agent-based and network models. For compartmental models, we further distinguish between deterministic and stochastic approaches.
Based on this classification, Table 2 provides detailed information for each study, including the HPAI subtype(s) considered, the key geographical region(s), the epidemic year(s), the model type, and whether cross-species transmission is included (yes/no). All included studies were published after 2023. Of these studies, twelve (40%) focused on particular countries in the following regions: Europe (n = 4, 33.3%), Asia (n = 4, 33.3%), the Americas (n = 3, 25%), and Africa (n = 1, 8.3%). The remaining 18 studies (60%) were location-agnostic (Fig 2). Seventeen studies (56.7%) prioritized specific HPAI subtypes, including H5N1 (n = 7, 41.2%), H5N8 (n = 1, 5.9%), H5Nx (n = 3, 17.6%), H7N9 (n = 4, 23.5%) and multiple subtypes (n = 2, 11.8%). Fourteen studies (46.7%) incorporated multiple host populations to evaluate cross-species transmission and zoonotic spillover risk.
Map generated using the open-source Natural Earth dataset [73] with the R packages ‘rnaturalearth’ and ‘rnaturalearthdata’. Countries reflect the study locations extracted from the included articles.
Results of individual sources of evidence
The 30 studies included in this review employed three main mechanistic modelling frameworks: compartmental (n = 26, approximately 86.7%), agent-based (n = 2, approximately 6.7%) and network (n = 2, approximately 6.7%) (Table 1 and 2). Compartmental models were the most common, with 16 studies (approximately 61.5%) using deterministic approaches and 10 studies (approximately 38.5%) employing stochastic models. These models were primarily used to describe transmission dynamics among wild birds, poultry, and humans. The optimization and effect of control measures such as culling, vaccination, movement restrictions, and biosecurity interventions were evaluated in 19 studies (63.3%), while 20 studies (66.7%) provided estimates of key epidemiological parameters, including the basic reproduction number ().
Agent-based models (n = 2, 6.7%) simulated individual-level interactions within and between farms, wildlife populations, and human communities. They captured spatial heterogeneity and heterogeneous contact patterns, allowing for exploration of outbreak scenarios under varying assumptions of host behaviour, farm density, and environmental persistence.
Network models (n = 2, 6.7%) represented hosts, farms, markets or geographic regions as interconnected nodes linked by potential transmission pathways such as trade, transportation, or environmental contact. These models were used to study stochastic processes and intervention strategies in networked populations and to simulate disease spread within poultry transportation systems.
Across all modelling frameworks, common findings included the critical importance of rapid detection [45,48,53,61,63], implementation of farm-level biosecurity measures [46,48,53,59], and movement restrictions to limit the spread of HPAI [45,46,50,54,71]. A subset of studies also explored co-circulation of multiple subtypes, co-infections, or cross-species transmission, though these aspects were less frequently addressed [55,56,74].
Synthesis of results
Compartmental models (n = 26).
Among the studies included in this review, 26 studies (86.7%) employed compartmental models to describe the transmission dynamics of HPAI across multiple host populations, including wild birds, poultry, cattle and humans (Table 3). To visualize the relationships among studies, model types, modelling methods, and host or environmental components, we used an alluvial diagram (Fig 3), which highlights how different modelling frameworks capture various aspects of HPAI transmission and the proportion of studies associated with each category.
The width of each flow is proportional to the number of studies that share the connected attributes.
Of these 26 studies, 16 (61.5%) used deterministic formulations, while 10 (38.5%) adopted stochastic frameworks. The methodological distribution, including ordinary differential equations (ODEs), partial differential equations (PDEs), and delay-differential equations (DDEs), is summarized in Fig 4. Six studies (23.1%) incorporated time delays to capture incubation periods, nonlocal transmission, or immunization effects, using stochastic, neural network, or reaction–diffusion frameworks to examine outbreak dynamics, system stability, and the effectiveness of intervention strategies. Regarding host populations, most studies focused on poultry (n = 8, 30.8%), poultry–human systems (n = 6, 23.1%), wild birds and humans (n = 5, 19.2%), or wild birds, poultry, and humans (n = 2, 7.7%). Other studies examined wild birds only (n = 2, 7.7%), livestock (n = 1, 3.8%), or human-only populations (n = 1, 3.8%). Environmental compartments representing viral persistence in water, fomites, or other reservoirs were included in eight studies (30.8%). In terms of transmission scale, models spanned within-farm dynamics (n = 1, 3.8%), between-farms spread (n = 2, 7.7%), between-herds spread (n = 1, 3.8%), market-level interactions (n = 1, 3.8%), within-species and between-host transmission (n = 21, 80.8%), reflecting the diverse pathways through which HPAI can spread across species and production systems (Table 3, Fig 5).
The reviewed compartmental models were generally formulated using standard epidemiological frameworks such as susceptible–infected (SI), susceptible–infected–recovered (SIR), and susceptible–exposed–infected–recovered (SEIR) structures, with extensions tailored to the biology of HPAI. For instance, SI-type models were applied to describe continuous transmission dynamics without recovery, particularly for avian hosts with high mortality [56,63,67]. SIR frameworks, representing infection and subsequent immunity, were commonly used in poultry and mixed-species systems [52,54,65,68]. SEIR-type formulations, which include a latent (exposed) compartment to capture incubation delays, were employed to model temporal progression and environmental persistence of HPAI in both poultry and wild birds [45,46,61,69].
Many studies further introduced additional compartments to reflect interventions or host-specific epidemiological features. Vaccinated or immunized individuals (V/P) were included to capture the effects of vaccination [48,59], while quarantine (Q) and treatment (T) compartments were incorporated to model control strategies [48,60]. Mortality (D) and environmental reservoirs (C/W) were included to represent host death and persistence of the virus in water or on fomites [49,62]. Asymptomatic (A) and clinically ill or hospitalized individuals (H/C) were added to account for subclinical infections and human-specific disease outcomes [33,66]. Subpopulation stratification, for example, by species (wild birds, poultry, humans), resident versus migratory birds, or different herds and market locations, was also common [45,48,53].
Fig 6 illustrates representative examples of these model structures: an SI framework for wild birds, an SI-W/C model for poultry (W/C represents environmental viral concentration), and an SIR framework for humans. The diagram shows the flow of infection among three host populations: wild birds (Sw, Iw), poultry (Sp, Ip), and humans (Sh, Ih, Rh), as well as the environmental viral concentration (W/C). Subscripts w, p, and h denote wild birds, poultry, and humans, respectively. Transmission occurs within species at a rate and across species via explicit cross-species rates (
from wild birds to poultry, and
from poultry to humans). The environmental compartment (W/C) introduces a feedback loop, capturing viral shedding from infected hosts (
) and subsequent environmental exposure, modelled as a transmission rate from the environment to poultry (
) and humans (
). Host-specific epidemiological parameters include the death rate (
) and, for humans, the recovery rate (
).
Deterministic models (n = 16) primarily focused on threshold analysis and long-term epidemic outcomes. Several studies examined the stability of disease-free and endemic equilibria and derived expressions for the basic reproduction number (R0), identifying conditions for elimination (R0 < 1) or persistence (R0 > 1) [52,54,56,57,59,67]. Some deterministic models (n = 13, 81.3%) further incorporated intervention strategies such as culling (n = 7, 43.8%), vaccination(n = 8, 50%), and quarantine(n = 3, 18.8%), showing how their effectiveness depends on host demography and epidemiological context [48,54,57,65,69]. These intervention analyses ranged from theoretical optimal control formulations that identify mathematically optimal intervention trajectories under idealized assumptions, to simulation-based evaluations of fixed or discrete policies intended to reflect operational decision-making. Empirical data, including outbreak reports and mortality records, were also used to calibrate and validate model predictions (Tables 4 and 5). However, according to Table 5, only two studies explicitly validated their model predictions with these data [49,60].
Stochastic compartmental models (n = 10) incorporated random processes to capture temporal heterogeneity and variability in HPAI transmission dynamics. These processes included stochastic infection events and random selection of farms for removal [46], random perturbations in natural death rates of birds and humans (white noise) [66], stochastic fluctuations in transmission and disease progression [69,70], stochastic regime-switching of transmission rates via a Markov process [54], and probabilistic infection dynamics and parameter uncertainty modeled through Monte Carlo simulations or Bayesian inference [31,45,49,55,61]. By explicitly modelling these stochastic components, the models capture the inherent randomness in outbreak progression, movement, and detection, which can lead to substantial variation in epidemic size, extinction probability, and outbreak timing, particularly in small populations or during early stages of an epidemic. Among these stochastic models, five studies (50%) evaluated interventions such as culling (n = 2, 20%), vaccination (n = 2, 20%), and environmental management (n = 2, 20%), providing quantitative insights into the effectiveness and uncertainty of control strategies [61,66].
In addition to structural features, quantitative assumptions regarding transmission rates and epidemiological parameters were systematically reported across studies. Table 6 summarizes the transmission rates used in mechanistic HPAI models across different pathways (avian-to-avian, avian-to-human, human-to-human, and environmental transmission). Table 7 further synthesizes selected parameter values for different hosts, including latent and infectious periods, recovery rates, mortality rates, and intervention-related parameters such as culling rates. Together, these tables highlight both the analytical strengths of deterministic models and the probabilistic insights from stochastic models, illustrating their complementary contributions to understanding HPAI dynamics.
Agent-based Model (n = 2).
Agent-based models (ABMs) simulate the behaviours and interactions of individual hosts or entities to capture emergent epidemic dynamics in complex systems. Unlike deterministic compartmental models, ABMs allow for explicit representation of host heterogeneity, stochasticity, movement patterns, and contact structures, making them particularly suited for exploring fine-scale processes in HPAI transmission across poultry populations, wildlife, and farm networks.
Two studies applied ABMs to investigate HPAI transmission dynamics. Kirkeby et al. [58] developed an individual-based SEIR model to simulate outbreaks in large poultry flocks under different virus subtypes (H5 and H7) and poultry species. Using reported parameter estimates and literature-based assumptions, the model highlighted considerable differences in outbreak size and duration between species and subtypes, with implications for surveillance design and outbreak preparedness. A key limitation identified was the lack of detailed latency period estimates across genotypes, particularly for dominant strains such as H5N8. Takekawa et al. [47] constructed an ABM informed by telemetry data on Swan Geese in northeastern Mongolia. Their model linked host movement, environmental landscapes, and pathogen parameters to assess the likelihood of HPAI persistence during the moulting and pre-migration periods. Results emphasized that conserving undisturbed habitats for wild waterfowl could reduce transmission risk, underscoring the value of coupling ecological data with epidemiological simulations.
These two ABMs extend HPAI modelling beyond compartmental approaches by incorporating host-level variability, spatial movement, and network connectivity. They illustrate the utility of ABMs for evaluating outbreak dynamics in complex production and ecological systems, while also highlighting critical data gaps, particularly in subtype-specific epidemiological parameters. However, ABMs require detailed data on host populations, movement patterns, and environmental landscapes, as well as fine-scale parameter estimates such as latency and transmission rates. They are computationally intensive, often necessitating multiple simulation runs to capture stochastic variability and test different scenarios, which can demand high-performance computing resources, especially for large populations or extended temporal horizons.
Network Model (n = 2).
Network models capture the structure and connectivity of contacts among hosts, farms, or markets, offering a framework to study how disease transmission is influenced by network topology, mobility, and control interventions. These models are particularly useful for representing heterogeneous and spatially structured systems, where the spread of HPAI depends on complex interactions across trade or contact networks.
Two studies employed network-based approaches to explore avian influenza transmission dynamics. Yang and Wang [72] developed a stochastic reaction–diffusion avian influenza model on complex networks to investigate human–poultry transmission and near-optimal control strategies. The model incorporated multi-time delays and stochasticity to describe the co-circulation of avian and mutant human strains across networked nodes representing poultry populations and human communities. Using Ekeland’s variational principle and the Pontryagin random maximum principle, the authors derived necessary and sufficient conditions for near-optimal interventions, including culling of infected birds and treatment of human cases. Simulation results indicated that high-intensity culling and early treatment could suppress infection peaks, although the analysis was entirely theoretical and not calibrated with real outbreak data.
Pinotti et al. [71] introduced EPINEST (Epidemic Network Simulation in Poultry Transportation Systems), a modelling framework designed to simulate pathogen spread within poultry production and trade networks. The model represented interactions among farms, middlemen, vendors, and live bird markets as nodes in a transportation network, enabling exploration of how trade connectivity affects disease propagation. Simulations demonstrated that network structure and movement patterns strongly influenced outbreak size and duration. While parameter values were partly assumed and model validation was limited, EPINEST highlights the potential of network-based approaches to evaluate targeted control measures, such as trade restrictions or improved biosecurity at key nodes.
These studies demonstrate the value of network models for representing heterogeneous contact structures and identifying critical nodes or links that influence the spread of HPAI across poultry production and human–poultry interfaces. Network models rely on detailed data on contact patterns, trade movements, and farm or market connectivity. Incorporating stochasticity, delays, and intervention strategies increases computational complexity, often requiring repeated simulations to evaluate outbreak scenarios and the effectiveness of control measures. Consequently, these models demand substantial computational resources and careful data curation to ensure realistic network representations.
Discussion
This scoping review synthesized modelling approaches that have been used to investigate the transmission dynamics of HPAI across wild birds, poultry, livestock, and humans (Table 2 and 5, Figs 3 and 5). Three main methodological traditions were identified: compartmental models (deterministic and stochastic), agent-based models, and network models. Compared with earlier reviews, such as Lambert et al. [42], which included studies up to April 2023 and primarily focused on wild birds and poultry with limited consideration of avian–human transmission, the studies published from January 2023 to June 2025 show notable advances. These recent studies increasingly incorporate multi-host, “One Health”-oriented frameworks that explicitly address interspecies transmission, including the human-animal interface, and, in some cases, livestock populations. They also demonstrate enhanced empirical parameterization and validation. Given the increasing circulation of HPAI and heightened urgency for detection, prediction, mitigation, and prevention of further cross-species transmission and potential human-to-human spread, understanding these recent modelling developments is critical for informing effective outbreak preparedness and risk assessment.
Compartmental models remain the dominant approach in HPAI transmission modelling, largely due to their analytical tractability and capacity to yield threshold quantities such as the basic reproduction number (R0) [46,50,51,53,60,62,67]. Deterministic formulations were primarily used to examine long-term epidemic behaviour, equilibrium stability, and intervention effectiveness, while stochastic extensions captured variability in outbreak size, extinction probability, and timing through random infection events, demographic perturbations, and probabilistic transmission processes [46,49,61,66,72]. The flexibility of compartmental frameworks was further demonstrated through extensions incorporating latent, asymptomatic, vaccinated, and environmentally mediated states, enabling representation of complex transmission pathways linking poultry, wild birds, and humans [33,59,60,69]. Together, deterministic and stochastic formulations provide complementary insights, balancing analytical clarity with representation of uncertainty under real-world conditions.
Across these studies, model-based analyses consistently indicated that single interventions were insufficient to control HPAI transmission. Reductions in farm density alone lowered reproduction numbers but did not prevent persistence in palmiped systems [46], whereas combined strategies integrating human-targeted measures, treatment, and vaccination in poultry were most effective in reducing infections across host populations [56,65]. Environmental processes, including viral decay and spatial heterogeneity, emerged as key drivers of persistence and spread, particularly when interventions were delayed [59]. Several models further showed that outbreak severity varied substantially by virus subtype and poultry species, highlighting the importance of pathogen-specific parameterization [58].
Agent-based and network models, though less frequently applied, provided important complementary perspectives by explicitly representing heterogeneity, spatial structure, and contact networks. Agent-based models captured species-specific transmission dynamics and ecological drivers such as migration and habitat use, underscoring the value of integrating behavioural and ecological data [47,58]. Network-based approaches emphasized the role of trade connectivity, mobility, and timing of interventions in shaping outbreak trajectories [71,72]. However, both approaches were constrained by data scarcity, computational demands, and simplifying assumptions, limiting their use for large-scale or empirically calibrated policy evaluation.
While many studies evaluated intervention strategies, an important distinction exists between theoretical optimal control frameworks and simulations of operationally realistic policies. Optimal control studies provide valuable conceptual insights into the timing and intensity of interventions under idealized assumptions. Still, they often assume perfect compliance, continuous control, and unconstrained resources, which limits their direct applicability. In contrast, simulation-based studies that assess discrete, implementable measures, such as fixed vaccination strategies, culling radii, movement restrictions, or biosecurity interventions, offer more immediately actionable guidance for decision-makers. Failure to clearly distinguish between these approaches may lead to overinterpretation of theoretical optima as practical policy recommendations.
Advanced computational approaches, including neural networks and LSCM methods, demonstrated high numerical accuracy and reliability in simulating delayed or stochastic HPAI dynamics [68,70], providing promising alternatives for modelling complex epidemic systems. Similarly, the Laplace residual power series (LRPS) method has demonstrated high numerical accuracy and reliability in simulating complex epidemic [76,77]. While developed for general fractional-order systems or other health conditions, these analytical techniques provide promising alternatives for modelling the non-linear coupled initial value problems characteristic of HPAI, offering rapid convergent series approximations with reduced computational resources. Collectively, these studies indicate that HPAI transmission is shaped by a complex interplay of host species, virus subtypes, environmental conditions, spatial heterogeneity, and intervention strategies, with effective mitigation requiring multi-pronged approaches that integrate vaccination, culling, quarantine, and public health measures tailored to specific host-pathogen contexts.
Existing modelling efforts have addressed multiple scales of HPAI transmission, from within-flock and between-farm spread to interactions among wild birds, poultry, and humans. Deterministic compartmental models provide analytical clarity, stochastic and network-based models capture uncertainty and structural heterogeneity, and agent-based models represent individual-level behaviours and spatial complexity. The combined insights from these approaches highlight significant progress in understanding HPAI dynamics while revealing ongoing challenges related to data integration, model validation, and the translation of theoretical findings into actionable control strategies.
Despite these advances, several limitations were evident across the included HPAI modelling studies, which should be considered when interpreting their findings (Table 5). Data limitations were pervasive in many studies. A number of models relied heavily on assumed or literature-based parameters due to the scarcity of empirical data, leading to substantial uncertainty in the parameter values used across studies [31,45,46,48,49]. Key epidemiological parameters, such as latency periods, genotype-specific transmission rates, and vaccine-induced immunity for different poultry species or viral subtypes, were often unavailable or incomplete [45,47–49,58]. In several cases, default or symbolic values were used to fill these gaps [58,67], which may not accurately represent the complex dynamics of natural infections. Furthermore, the lack of standardized reporting for parameter values, including inconsistent or missing units for transmission rates and disease-induced death rates, makes direct comparison between models difficult and limits the reproducibility of reported findings. Data for certain poultry populations, such as geese or backyard flocks, were particularly sparse [31,49], limiting the ability of models to capture heterogeneity in susceptibility or contact patterns. Surveillance and reporting biases further complicated model calibration; for example, underreporting due to differences in sampling intensity, delayed case detection, or incomplete movement records restricted the capacity of models to reflect true outbreak dynamics [45,49]. Such parameter uncertainty raises questions about the reliability and policy relevance of model outputs, underscoring the need for robust empirical parameterization, uncertainty quantification, and ensemble modelling approaches to support evidence-based decision-making for HPAI control.
Beyond parameter uncertainty, formal model validation against empirical data was notably rare. Among the 30 included studies, only two reported any form of validation, either through qualitative comparison with field surveillance data or quantitative validation using historical outbreak records. The absence of validation in the vast majority of studies substantially limits confidence in model-based inferences, particularly when models are used to evaluate cross-species transmission dynamics or intervention effectiveness. Without validation, it remains unclear whether simulated epidemic trajectories adequately capture real-world transmission processes, host interactions, or reporting mechanisms. This widespread lack of empirical validation represents a major limitation of the current HPAI modelling literature and highlights an urgent need for closer integration between mechanistic models and available surveillance and outbreak data.
Structural and methodological simplifications were common across HPAI modelling studies, limiting model realism and practical applicability. Many models assumed homogeneous mixing within host populations, linear environmental exposure, fixed spatial boundaries, or lifelong immunity without waning [51,52]. When vaccination was included, immune protection was typically represented as binary and permanent, with little consideration of waning immunity, heterogeneity in immune responses across individuals or species, or the presence of passive immunity in young animals. Such simplifications overlook experimentally observed variation in vaccine-induced humoral immunity and maternal antibody transfer, which can strongly influence susceptibility, transmission dynamics, and the timing of outbreaks [78].
Deterministic compartmental models can be extended to include multiple host populations, intervention processes, and cross-species interfaces. However, these models typically neglect stochasticity and spatial heterogeneity [56,59], both of which are crucial for capturing variability and localized clustering in real outbreaks. While some studies implemented mechanistic representations of interspecies transmission, explicitly modelling biological interactions between host species, many relied on symbolic or exploratory couplings based on assumed parameters. This distinction is important, as models with weakly parameterized host interactions may provide limited insight into actual cross-species transmission dynamics. To better account for heterogeneity in transmission risks arising from farm organization, animal movements, and host–species interactions, these extended frameworks are often formulated as stochastic spatial or network-based models that incorporate movement networks and spatial diffusion across farms and populations [72]. While agent-based and network models offered greater flexibility in representing individual-level interactions and complex contact structures, their applications were constrained by computational demands, theoretical simplifications, or limited behavioural and empirical data for parameterization and validation [58,71]. Stochastic or delay models typically focus on theoretical aspects, such as incubation periods or delay dynamics, without empirical calibration, reducing their relevance for policy evaluation [31,46].
Furthermore, many studies did not explicitly incorporate key epidemiological processes such as co-infections, spillover from wild birds, or interspecies transmission, which could lead to over- or underestimation of outbreak risk [45,56]. In addition to mechanistic approaches, several studies used statistical or risk-based models to evaluate the likelihood of avian influenza transmission and assess relative risk at the human–animal–environment interface [75,79,80]. These frameworks typically combine spatial density, environmental exposure, and host interaction data to identify high-risk areas or transmission pathways. While such approaches provide valuable insights for targeted surveillance and early warning, they often rely on simplified assumptions or limited spatial and behavioural data, which may constrain their predictive accuracy and generalizability across regions or host systems.
These limitations reveal the challenges of accurately modelling HPAI dynamics. They highlight the critical need for improved empirical data collection across diverse host species and viral genotypes, enhanced surveillance systems that reduce reporting biases, and model designs that incorporate stochasticity, spatial heterogeneity, and realistic interventions. Addressing these gaps will improve the reliability of model predictions, support evidence-based outbreak preparedness, and inform effective policy and control strategies. Notably, despite reported outbreaks in cattle and other livestock, these settings remain largely underrepresented in mechanistic models. Among the 30 included studies, only one specifically modelled cattle, and while several considered humans and poultry farms, none explicitly examined avian–livestock or livestock–human transmission pathways. In particular, the lack of empirically informed immune processes is most pronounced in livestock settings, where age structure, vaccination practices, and passive maternal immunity may substantially shape susceptibility and transmission but are rarely represented in mechanistic models. This highlights a critical gap in modelling livestock populations and cross-species transmission among avian, livestock, and human populations, an area identified as a priority for understanding zoonotic risk. Experimental and observational datasets, such as antibody kinetics, maternal antibody transfer in milk, and seroconversion timelines, offer valuable opportunities to improve mechanistic models. Incorporating these data can help parameterize transition rates between compartments, support age-structured or immunity-stratified frameworks, and reduce reliance on assumed or literature-based [78]. Leveraging such empirical evidence, particularly in livestock populations, can enhance model fidelity, improve biological realism, and strengthen the relevance of models for informing outbreak preparedness and intervention strategies.
In addition to the limitations identified within the included studies, several modelling approaches were excluded due to the defined scope of this scoping review. Within-host models were not considered, as the focus was on population-level transmission; however, they can provide valuable insights into viral replication, immune responses, and evolutionary processes relevant to cross-species spillover. Hybrid approaches linking within-host and between-host dynamics could further enhance understanding of viral adaptation and zoonotic emergence, providing insights into viral evolution and disease progression. GIS-based multi-criteria decision analysis models were also excluded. Although these approaches do not explicitly model transmission dynamics, they offer complementary insights by highlighting spatial heterogeneity and identifying high-risk regions, supporting targeted surveillance and intervention planning. The exclusion of these frameworks reflects the review scope rather than their relevance, and future research could benefit from integrating them with mechanistic transmission models.
Conclusions
In response to the increasing focus and risk of HPAI virus, our scoping review provides a necessary updated summary of the literature published on HPAI modelling since 2023, extending the results of Lambert et al. [42]. Our study includes compartmental (deterministic and stochastic approaches), agent-based and network modelling approaches, with a focus on outbreak dynamics and intervention assessment. The review revealed that most studies relied on deterministic compartmental models with limited stochasticity or spatial heterogeneity, and often used assumed or literature-based parameters due to gaps in empirical data. Agent-based and delay models offered greater flexibility for incorporating heterogeneity and behavioural dynamics, but were often constrained by computational demands and the limited availability of data for calibration and validation. Network models, though less common, provided valuable insights into how structural connectivity, such as poultry trade routes, market linkages, and contact networks, shapes outbreak propagation and informs targeted control strategies.
Nevertheless, important gaps remain. Key epidemiological processes, including co-infections, heterogeneity across host populations, and interactions between humans and livestock, are frequently underrepresented. Intervention strategies are not consistently incorporated, and many models rely on simplified assumptions that may limit predictive accuracy. In particular, immunity is frequently oversimplified, with most models assuming lifelong or homogeneous protection following vaccination or infection. Future modelling efforts should incorporate empirically informed immune dynamics, including waning immunity, heterogeneous vaccine responses, and passive maternal antibody transfer, drawing on experimental studies of humoral immunity in livestock to improve biological realism and policy relevance [78]. Crucially, formal model validation against empirical outbreak data remains rare. This pervasive lack of validation substantially limits confidence in model predictions and represents a major constraint on the credibility and policy relevance of the current HPAI modelling literature. These limitations highlight the need for improved empirical data collection, model validation, and more realistic modelling frameworks to better inform outbreak preparedness and policy decisions. Based on these findings, future efforts should prioritize the collection of high-quality empirical data to support robust parameterization and model validation, including immunological data on antibody kinetics and maternal antibody transfer, to inform immunity-stratified or age-structured frameworks and improve model fidelity. Researchers are encouraged to develop multi-host models that explicitly include livestock, particularly cattle, to capture cross-species transmission dynamics and the human–animal interface. Such models should also account for environmental reservoirs, migratory wildlife, and seasonal fluctuations to better represent real-world transmission processes. Implementing ensemble modelling approaches across multiple frameworks can help quantify uncertainty, compare predictions, and improve confidence in outbreak forecasts, providing more reliable guidance for both short- and long-term planning.
Establishing standardized validation metrics, transparent reporting of model assumptions, and open-access parameter databases will facilitate reproducibility and enhance the credibility of model outputs. Without standardized reporting and rigorous uncertainty analysis, model outputs risk being misinterpreted by policy-makers, as the lack of comparability between studies can lead to conflicting guidance for outbreak preparedness. Policy-makers should critically assess model assumptions, validation results, and uncertainty estimates when using these models to guide decision-making. Reliance on single-model predictions should be avoided; instead, ensemble forecasts that synthesize outputs from multiple models can provide a more nuanced understanding of risk and potential outcomes. Simultaneously, investment in comprehensive surveillance and high-resolution data collection systems will generate timely information essential for informing and updating models. Funders can accelerate progress by supporting targeted data collection for parameterization, investing in the development of computationally intensive models such as agent-based and network models, and promoting transparency through requirements for open code, data sharing, and model validation studies. Together, these coordinated efforts across researchers, policy-makers, and funders will strengthen the ability of HPAI models to generate actionable insights, inform evidence-based outbreak preparedness, and guide effective interventions across wild birds, poultry farms, livestock farms, and human populations.
References
- 1. Writing Committee of the World Health Organization (WHO) Consultation on Human Influenza A/H5. Avian influenza A (H5N1) infection in humans. New England Journal of Medicine. 2005;353(13):1374–85.
- 2. What is the pandemic potential of avian influenza A (H5N1)?. The Lancet Infectious Diseases. 2024;24(5):437.
- 3. Prosser DJ, Kent CM, Sullivan JD, Patyk KA, McCool M-J, Torchetti MK, et al. Using an adaptive modeling framework to identify avian influenza spillover risk at the wild-domestic interface. Sci Rep. 2024;14(1):14199. pmid:38902400
- 4. Tian J, Bai X, Li M, Zeng X, Xu J, Li P, et al. Highly Pathogenic Avian Influenza Virus (H5N1) Clade 2.3.4.4b Introduced by Wild Birds, China, 2021. Emerg Infect Dis. 2023;29(7):1367–75. pmid:37347504
- 5. Kim E-H, Kim Y-L, Kim SM, Yu K-M, Casel MAB, Jang S-G, et al. Pathogenic assessment of avian influenza viruses in migratory birds. Emerg Microbes Infect. 2021;10(1):565–77. pmid:33666526
- 6. Peiris JSM, de Jong MD, Guan Y. Avian influenza virus (H5N1): a threat to human health. Clin Microbiol Rev. 2007;20(2):243–67. pmid:17428885
- 7. Sutton TC. The Pandemic Threat of Emerging H5 and H7 Avian Influenza Viruses. Viruses. 2018;10(9):461. pmid:30154345
- 8. Quan C, Shi W, Yang Y, Yang Y, Liu X, Xu W, et al. New threats from H7N9 influenza virus: spread and evolution of high- and low-pathogenicity variants with high genomic diversity in wave five. J Virol. 2018;92(11):e00301-18. pmid:29563296
- 9. Hénaux V, Samuel MD, Bunck CM. Model-based evaluation of highly and low pathogenic avian influenza dynamics in wild birds. PLoS One. 2010;5(6):e10997. pmid:20585637
- 10. Feare CJ. The role of wild birds in the spread of HPAI H5N1. Avian Dis. 2007;51(1 Suppl):440–7. pmid:17494603
- 11. Alkie TN, Byrne AM, Jones ME, Mollett BC, Bourque L, Lung O, et al. Recurring trans-Atlantic incursion of clade 2.3. 4.4 b H5N1 viruses by long distance migratory birds from Northern Europe to Canada in 2022/2023. Viruses. 2023;15(9):1836.
- 12. Yang Q, Wang B, Lemey P, Dong L, Mu T, Wiebe RA, et al. Synchrony of bird migration with global dispersal of avian influenza reveals exposed bird orders. Nat Commun. 2024;15(1):1126. pmid:38321046
- 13. Ahrens AK, Selinka H-C, Wylezich C, Wonnemann H, Sindt O, Hellmer HH, et al. Investigating environmental matrices for use in avian influenza virus surveillance-surface water, sediments, and avian fecal samples. Microbiol Spectr. 2023;11(2):e0266422. pmid:36700688
- 14. Koçer ZA, Obenauer J, Zaraket H, Zhang J, Rehg JE, Russell CJ, et al. Fecal influenza in mammals: selection of novel variants. J Virol. 2013;87(21):11476–86. pmid:23966381
- 15. Ly S, Vong S, Cavailler P, Mumford E, Mey C, Rith S, et al. Environmental contamination and risk factors for transmission of highly pathogenic avian influenza A (H5N1) to humans, Cambodia, 2006-2010. BMC Infectious Diseases. 2016;16(1):631.
- 16. Moyen N, Hoque MA, Mahmud R, Hasan M, Sarkar S, Biswas PK, et al. Avian influenza transmission risk along live poultry trading networks in Bangladesh. Sci Rep. 2021;11(1):19962. pmid:34620890
- 17. Kim Y, Biswas PK, Giasuddin M, Hasan M, Mahmud R, Chang Y-M, et al. Prevalence of Avian Influenza A(H5) and A(H9) Viruses in Live Bird Markets, Bangladesh. Emerg Infect Dis. 2018;24(12):2309–16. pmid:30457545
- 18. Soares Magalhães RJ, Zhou X, Jia B, Guo F, Pfeiffer DU, Martin V. Live poultry trade in Southern China provinces and HPAIV H5N1 infection in humans and poultry: the role of Chinese New Year festivities. PLoS One. 2012;7(11):e49712. pmid:23166751
- 19. Ahmed SSU, Ersbøll AK, Biswas PK, Christensen JP, Hannan ASMA, Toft N. Ecological determinants of highly pathogenic avian influenza (H5N1) outbreaks in Bangladesh. PLoS One. 2012;7(3):e33938. pmid:22470496
- 20.
FAO, World Organization for Animal Health. Global strategy for the prevention and control of high pathogenicity avian influenza (2024–2033) - Achieving sustainable, resilient poultry production systems. 2025. https://doi.org/10.4060/cd3840en
- 21. Lycett SJ, Duchatel F, Digard P. A brief history of bird flu. Philos Trans R Soc Lond B Biol Sci. 2019;374(1775):20180257. pmid:31056053
- 22. Claas EC, Osterhaus AD, van Beek R, De Jong JC, Rimmelzwaan GF, Senne DA, et al. Human influenza A H5N1 virus related to a highly pathogenic avian influenza virus. Lancet. 1998;351(9101):472–7. pmid:9482438
- 23. Yuen KY, Chan PK, Peiris M, Tsang DN, Que TL, Shortridge KF, et al. Clinical features and rapid viral diagnosis of human disease associated with avian influenza A H5N1 virus. Lancet. 1998;351(9101):467–71. pmid:9482437
- 24. Koopmans MPG, Barton Behravesh C, Cunningham AA, Adisasmito WB, Almuhairi S, Bilivogui P, et al. The panzootic spread of highly pathogenic avian influenza H5N1 sublineage 2.3.4.4b: a critical appraisal of One Health preparedness and prevention. Lancet Infect Dis. 2024;24(12):e774–81. pmid:39134084
- 25. Jassem AN, Roberts A, Tyson J, Zlosnik JEA, Russell SL, Caleta JM, et al. Critical Illness in an Adolescent with Influenza A(H5N1) Virus Infection. N Engl J Med. 2025;392(9):927–9. pmid:39740022
- 26. Peña-Mosca F, Frye EA, MacLachlan MJ, Rebelo AR, de Oliveira PSB, Nooruzzaman M, et al. The impact of highly pathogenic avian influenza H5N1 virus infection on dairy cows. Nat Commun. 2025;16(1):6520. pmid:40664659
- 27.
CDC Reports Second Human Case of H5 Bird Flu Tied to Dairy Cow Outbreak. 2024. https://www.cdc.gov/media/releases/2024/s0522-human-case-h5.html
- 28.
Pan American Health Organization, World Health Organization. Epidemiological Update Avian Influenza A(H5N1) in the Americas Region - 24 November 2025. 2025. https://www.paho.org/en/documents/epidemiological-update-avian-influenza-ah5n1-americas-region-24-november-2025
- 29. Backer JA, van Roermund HJW, Fischer EAJ, van Asseldonk MAPM, Bergevoet RHM. Controlling highly pathogenic avian influenza outbreaks: an epidemiological and economic model analysis. Prev Vet Med. 2015;121(1–2):142–50. pmid:26087887
- 30. Pramuwidyatama MG, Hogeveen H, Saatkamp HW. A systematic evaluation of measures against Highly Pathogenic Avian Influenza (HPAI) in Indonesia. Front Vet Sci. 2019;6:33. pmid:30834252
- 31. Lambert S, Fourtune L, Hobbelen PHF, Baca J, Gonzales JL, Elbers ARW, et al. Optimizing contact tracing for avian influenza in poultry flocks. J R Soc Interface. 2025;22(222):20240523. pmid:39772734
- 32. McClaughlin E, Elliott S, Jewitt S, Smallman-Raynor M, Dunham S, Parnell T, et al. UK flockdown: a survey of smallscale poultry keepers and their understanding of governmental guidance on highly pathogenic avian influenza (HPAI). Prev Vet Med. 2024;224:106117. pmid:38277819
- 33. Gourram H, Baroudi M, Labzai A, Belam M. Mathematical modeling andoptimal control strategy for the influenza (H5N1). Communications in Mathematical Biology and Neuroscience. 2023;2023:113.
- 34. Guan Y, Chen H, Li K, Riley S, Leung G, Webster R, et al. A model to control the epidemic of H5N1 influenza at the source. BMC Infect Dis. 2007;7:132. pmid:17999754
- 35. Rao ASRS. Modeling the rapid spread of avian influenza (H5N1) in India. Math Biosci Eng. 2008;5(3):523–37. pmid:18616356
- 36. Bourouiba L. Understanding the transmission of H5N1. CABI Reviews. 2013;:1–8.
- 37. Liu R, Duvvuri VRSK, Wu J. Spread pattern formation of H5N1-Avian influenza and its implications for control strategies. Math Model Nat Phenom. 2008;3(7):161–79.
- 38. Modnak C. Mathematical modelling of an avian influenza: optimal control study for intervention strategies. Appl Math Inf Sci. 2017;11(4):1049–57.
- 39. Bouma A, Claassen I, Natih K, Klinkenberg D, Donnelly CA, Koch G, et al. Estimation of transmission parameters of H5N1 avian influenza virus in chickens. PLoS Pathog. 2009;5(1):e1000281. pmid:19180190
- 40. Ward MP, Maftei D, Apostu C, Suru A. Estimation of the basic reproductive number (R0) for epidemic, highly pathogenic avian influenza subtype H5N1 spread. Epidemiol Infect. 2009;137(2):219–26. pmid:18559127
- 41. Huang S, Bragazzi NL, Nia ZM, Gillies M, Gardner E, Leung D, et al. A systematic review of mathematical and machine learning models of Avian Influenza. One Health. 2025;21:101203. pmid:41070042
- 42. Lambert S, Bauzile B, Mugnier A, Durand B, Vergne T, Paul MC. A systematic review of mechanistic models used to study avian influenza virus transmission and control. Vet Res. 2023;54(1):96. pmid:37853425
- 43.
World Health Organization. Updated joint FAO/WHO/WOAH public health assessment of recent influenza A(H5) virus events in animals and people: assessment based on data as of 1 July 2025. 2024. https://www.who.int/publications/m/item/updated-joint-fao-who-woah-assessment-of-recent-influenza-a(h5n1)-virus-events-in-animals-and-people_dec2024
- 44. Tricco AC, Lillie E, Zarin W, O’Brien KK, Colquhoun H, Levac D, et al. PRISMA Extension for Scoping Reviews (PRISMA-ScR): checklist and explanation. Ann Intern Med. 2018;169(7):467–73. pmid:30178033
- 45. Rawson T, Morgenstern C, Knock ES, Hicks J, Pham A, Morel G, et al. A mathematical model of H5N1 influenza transmission in US dairy cattle. Nat Commun. 2025;16(1):4308. pmid:40341525
- 46. Bauzile B, Durand B, Lambert S, Rautureau S, Fourtune L, Guinat C, et al. Impact of palmiped farm density on the resilience of the poultry sector to highly pathogenic avian influenza H5N8 in France. Vet Res. 2023;54(1):56. pmid:37430292
- 47. Takekawa JY, Choi C-Y, Prosser DJ, Sullivan JD, Batbayar N, Xiao X. Perpetuation of Avian Influenza from Molt to Fall Migration in Wild Swan Geese (Anser cygnoides): an agent-based modeling approach. Viruses. 2025;17(2):196. pmid:40006951
- 48. Li J, Nie B, Gao L, Zhang J, Liu L, Xu H, et al. Modelling for assessing the compulsory vaccination strategy against highly pathogenic avian influenza in a poultry farm of China. Applied Mathematical Modelling. 2025;145:116130.
- 49. Llanos-Soto SG, Yaffy D, Pavlak M, Ivanek R. Transmission dynamics of highly pathogenic avian influenza among multiple waterfowl species and backyard poultry: the impact of the stopover period. Sci Rep. 2025;15(1):5861. pmid:39966613
- 50. Lee EK, Liu Y, El-Tahawy A, Fasina FO. Analyzing strategies for containing avian influenza. AMIA Annu Symp Proc. 2023;2022:682–91. pmid:37128393
- 51. Yang L, Fan M. Reaction-advection-diffusion model of highly pathogenic avian influenza with behavior of migratory wild birds. J Math Biol. 2025;90(2):18. pmid:39821697
- 52. Chen C, Xiang Z. Analysis and optimal control strategies of epidemic models with nonlinear incidence rates. Adv Cont Discr Mod. 2025;2025(1).
- 53. Sheen JK, Rasambainarivo F, Saad-Roy CM, Grenfell BT, Metcalf CJE. Markets as drivers of selection for highly virulent poultry pathogens. Nat Commun. 2024;15(1):605. pmid:38242897
- 54. Li M, Lan G, Wei C. Threshold dynamics of stochastic H7N9 model with Markov switching and hybrid strategy. Journal of the Franklin Institute. 2024;361(2):916–32.
- 55. Kong L, Li L, Kang S, Chen F. Dynamic behavior of a stochastic avian influenza model with two strains of zoonotic virus. Mathematics. 2023;11(19):4199.
- 56. Tadmon C, Feukouo Fossi A, Tsanou B. A two-strain avian–human influenza model with environmental transmission: atability analysis and optimal control strategies. Communications in Nonlinear Science and Numerical Simulation. 2024;133:107981.
- 57. Ismail H, Hariharan S, Jeyaraj M, Shangerganesh L, Rong L. Hopf bifurcation and optimal control studies for avian-influenza virus with multi-delay model. Adv Cont Discr Mod. 2025;2025(1).
- 58. Kirkeby C, Boklund A, Larsen LE, Ward MP. Are all avian influenza outbreaks in poultry the same? The predicted impact of poultry species and virus subtype. Zoonoses Public Health. 2024;71(3):314–23. pmid:38362732
- 59. Li J, Nie L. Global dynamics analysis of non-local delayed reaction-diffusion avian influenza model with vaccination and multiple transmission routes in the spatial heterogeneous environment. Qual Theory Dyn Syst. 2024;23(5).
- 60. Malek A, Hoque A. Mathematical modeling of the infectious spread and outbreak dynamics of avian influenza with seasonality transmission for chicken farms. Comp Immunol Microbiol Infect Dis. 2024;104:102108. pmid:38070401
- 61. Planchand S, Vergne T, Guérin J-L, Rautureau S, Gerbier G, Lambert S. Surveillance strategy in duck flocks vaccinated against highly pathogenic avian influenza virus. Emerg Infect Dis. 2025;31(1):115–22. pmid:39699551
- 62. Covington R, Patton S, Walker E, Yamazaki K. Improved uniform persistence for partially diffusive models of infectious diseases: cases of avian influenza and Ebola virus disease. Math Biosci Eng. 2023;20(11):19686–709. pmid:38052620
- 63. Jiao X, Liu X. Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy. Chaos, Solitons & Fractals. 2024;182:114710.
- 64. Kang T, Zhang Q, Wang Q. Nonlinear adaptive control of avian influenza model with slaughter, educational campaigns and treatment. era. 2023;31(8):4346–61.
- 65. Olaniyi S, Lawal IA, Lebelo RS, Chuma FM, Abimbade SF. Modelling and optimal control of influenza dynamics with structured populations based on education and isolation. Int J Anal Appl. 2025;23:81.
- 66. Yang S, Zhang Q. Exponential stability and fixed-time control of a stochastic avian influenza model with spatial diffusion and nonlocal delay. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2024;34(9).
- 67. Li W, Guan Y, Cao J, Xu F. A note on global stability of a degenerate diffusion avian influenza model with seasonality and spatial Heterogeneity. Applied Mathematics Letters. 2024;148:108884.
- 68. Anwar N, Ahmad I, Fatima A, Kiani AK, Shoaib M, Raja MAZ. Design of intelligent Bayesian supervised predictive networks for nonlinear delay differential systems of avian influenza model. Eur Phys J Plus. 2023;138(10).
- 69. Zhou B, Jiang D, Dai Y, Hayat T. Threshold dynamics and probability density function of a stochastic avian influenza epidemic model with nonlinear incidence rate and psychological effect. J Nonlinear Sci. 2023;33(2).
- 70. Bao X, Han X. Dynamics and simulations of stochastic time delay avian influenza model using spectral method. Comp Appl Math. 2025;44(5).
- 71. Pinotti F, Lourenço J, Gupta S, Das Gupta S, Henning J, Blake D, et al. EPINEST, an agent-based model to simulate epidemic dynamics in large-scale poultry production and distribution networks. PLoS Comput Biol. 2024;20(2):e1011375. pmid:38381804
- 72. Yang SY, Wang JR. Near-optimal control of a stochastic avian influenza model with multi-time delays and spatial diffusion on complex networks. Nonlinear Dyn. 2024;113(9):10799–837.
- 73.
Natural Earth. Natural Earth Map Data. http://www.naturalearthdata.com/
- 74. Hassman RL, McCabe IMH, Smith KM, Allen LJS. Stochastic models of zoonotic avian influenza with multiple hosts, environmental transmission, and migration in the natural reservoir. Bull Math Biol. 2024;87(1):14. pmid:39674959
- 75. Montano Valle D de LN, Berezowski J, Delgado-Hernández B, Hernández AQ, Percedo-Abreu MI, Alfonso P, et al. Modeling transmission of avian influenza viruses at the human-animal-environment interface in Cuba. Front Vet Sci. 2024;11:1415559. pmid:39055861
- 76. Liu A, Yasin F, Afzal Z, Nazeer W. Analytical solution of a non-linear fractional order SIS epidemic model utilizing a new technique. Alexandria Engineering Journal. 2023;73:123–9.
- 77. Afzal Z, Alshehri M. Fractional-order SIR model for ADHD as a neurobiological and genetic disorder. Sci Rep. 2025;15(1):22992. pmid:40594728
- 78. Abousenna MS, Shafik NG, Abotaleb MM. Evaluation of humoral immune response and milk antibody transfer in calves and lactating cows vaccinated with inactivated H5 avian influenza vaccine. Sci Rep. 2025;15(1):4637. pmid:39920177
- 79. Yoo D-S, Lee K, Beatriz ML, Chun BC, Belkhiria J, Lee K-N. Spatiotemporal risk assessment for avian influenza outbreak based on the dynamics of habitat suitability for wild birds. Transbound Emerg Dis. 2022;69(4):e953–67. pmid:34738338
- 80. Alvarez J, Boklund A, Dippel S, EFSA Panel on Animal Health, Animal Welfare (AHAW), ECDC. Preparedness, prevention and control related to zoonotic avian influenza. EFSA Journal. 2025;23(1):e9191.