Modeling and control of highly pathogenic avian influenza in poultry using network disease dynamics

Abstract

In light of the ongoing 2022-2025 HPAI outbreak in the U.S., which affects millions of commercial and backyard flocks, disrupts egg and meat production, and causes significant economic losses, it is important to develop biological models and optimization algorithms that conform to the U.S.-specific patterns of HPAI transmission and reflect control and prevention measures adopted in the USA. In this study, we introduce a partially stochastic network compartmental model to ascertain the progression and potential containment of HPAI virus in commercial flocks and wildlife. Parameters of the model get estimated using available data on wild bird migration, HPAI poultry outbreaks, and poultry farm inventory in different states of the U.S. The new model simulates HPAI virus transmission driven by wildlife dynamic, seasonality, and farm-to-farm relations. Unlike many prior global models, this framework is closely tailored to the U.S. HPAI statistics, farm structure, and current mitigation practices. The proposed network model, along with HPAI surveillance data, are used to analyze optimal control strategies aimed at lowering HPAI spread from wild birds to poultry. Our numerical experiments illustrate that the above control strategy is very powerful. In the absence of prevalent vaccination, these relatively inexpensive separation measures, such as covered runs and secure housing, help to prevent environmental contamination and the risk of HPAI transmission to domestic birds, thus protecting the flock and reducing depopulation.

1. Introduction

Highly Pathogenic Avian Influenza (HPAI), or ”bird flu”, is a contagious viral disease of wild birds, poultry, and other species. It's a major threat to agriculture, public and animal health, and international trade (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025d). Wild birds, especially waterfowl and shorebirds, form natural reservoirs for avian influenza (AI) viruses, including HPAI. While not completely immune to HPAI, many wildlife birds hardly exhibit any symptoms and their migratory trends are instrumental in spreading the virus globally among domestic poultry, dairy cattle, and other mammals (Krammer et al., 2024; Tarbuck et al., 2024).

To domestic poultry, highly pathogenic influenza strains are extremely dangerous, with up to 100% mortality rate in 48 hours, and it can wipe out entire flocks in a matter of days (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025d; Lycett et al., 2019). Since the last decade, HPAI has been changing from a primarily poultry-centered pathogen to a multi-host panzootic lineage spanning wild birds, commercial flock, and livestock (Brüssow, 2024; Peacock et al., 2025; Spackman et al., 2023). Genomic surveillance shows rapid evolution within clade 2.3.4.4b of the Highly Pathogenic Avian Influenza H5 subtypes evidenced by repeated reassortments as viruses move along migratory flyways and into dense poultry-production systems. These processes created genotypes with enhanced fitness in both wild and domestic hosts, complicating control strategies that were originally designed for earlier, more geographically restricted waves of HPAI (Brüssow, 2024; Spackman et al., 2023; Van Borm et al., 2025).

The HPAI outbreak of 2014-2015 in the USA infected 211 commercial poultry farms and 21 backyard premises within 15 states (Biondo and Congressional Research Service (CRS), 2025). The present U.S. HPAI outbreak in poultry started in February 2022 following its emergence in commercial turkey meat birds in Indiana (CDC Avian Influenza (Bird Flu), 2025; Biondo and Congressional Research Service (CRS), 2025). A month before that, there were multiple detections of HPAI H5N1 strain in the South Carolina waterfowl (Biondo and Congressional Research Service (CRS), 2025; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c). As of December 2025, cases of HPAI have been confirmed in all 50 USA states and in Puerto Rico, affecting more than 174 million wild birds, commercial poultry, and backyard flocks since early months of 2022 (CDC Avian Influenza (Bird Flu), 2025).

Mathematical modeling has become a cornerstone of epidemiology, providing the necessary framework to simulate transmission dynamics, estimate disease parameters, and assess intervention strategies for a broad range of viral outbreaks. Compartmental and spatial models have been instrumental in mapping population flows between various stages of infection and across multiple physical locations (Huang et al., 2025; Rorres et al., 2011). Over the years, variants of susceptible-infected-recovered (SIR) compartmental and agent-based (AB) spatial models have been adapted to different biological contexts, including the spread of a highly pathogenic avian influenza (Kuo et al., 2009).

In a recent study (Pinotti et al., 2024), a version of SEIR model was used to investigate the transmission of H9N2 avian influenza viruses within live bird markets in Bangladesh. The authors leveraged Bayesian techniques for the estimation of important model parameters such as transmission rate, host-specific latent periods, and the likelihood of prior H9N2 exposure. By accounting for both direct contacts and environmental transmission pathways, the paper offers valuable insights into how the virus persists and amplifies in a market setting.

Addressing specific challenges of stable parameter estimation, a robust numerical procedure for the reconstruction of a time-dependent bird-to-human transmission rate of H5N1 has been developed in (Smirnova & Tuncer, 2012). This work tackles the inherent instability of inverse problems by introducing a local regularization algorithm for model validation given discrete cumulative data on H5N1 human cases in Indonesia from July 2008 to June 2011. This approach effectively filters noise without the need for smooth interpolation, allowing for a reliable reconstruction of viral transmission patterns (Smirnova & Tuncer, 2012). Further studies (Tuncer et al., 2016; Tuncer & Martcheva, 2013; Vaidya & Wahl, 2015) addressed important questions of seasonality in bird-to-bird and bird-to-human influenza dynamics, looked into potential drivers of H5N1 periodic behavior, and incorporated partial immunity towards HPAI for wild and domestic birds recovered from low pathogenic avian influenza (LPAI). In (Liu et al., 2017), a study of threshold values for the prevalence of avian influenza viruses was carried out, motivated by the alarming H7N9 outbreaks in China from 2013 to 2016.

Overall, given a complicated interplay of various HPAI transmission routes, different models trace the spread of the virus between individual birds, between individual birds and humans, between domestic and wild birds (Chong et al., 2014; Prosser et al., 2024), between individual poultry farms (Dorigatti et al., 2010; Malek & Hoque, 2020), and oftentimes several of these pathways in combination (Vergne et al., 2021; Lambert et al., 2023; Malek & Hoque, 2021).

To capture the inherent randomness of real-world outbreaks, stochastic elements have been frequently introduced into classical influenza models. These stochastic variations add terms, which account for the uncertainty of disease transmission, incubation period, case detection, and data processing (Dorea et al., 2010; Stevens et al., 2013). Furthermore, the complexity of avian influenza transmission is driven by a multitude of anthropogenic and ecological factors, including spatial organization (Hill et al., 2017; Kuo et al., 2009; Pinotti et al., 2024), temperature fluctuations (Huang et al., 2025; Vaidya & Wahl, 2015), and human behavior (Dorea et al., 2010; Huang et al., 2025). Statistical models bridge the gap between theory and reality by using probabilistic frameworks and real data to simulate random transmission dynamics (Hsu & Shih, 2010; Mannelli et al., 2007; Yang et al., 2015). A notable application of this was seen in the analysis of the 2006 H5N1 outbreak in India, where a statistical transmission model integrated data on poultry trade and environmental factors to assess the efficiency of control measures (Upadhyay et al.). Sensitivity analyses are often paired with influenza models to evaluate their robustness. By varying parameters such as latent periods, researchers can identify which factors are the primary drivers of disease spread in different ecological settings (Hsu & Shih, 2010; Penny et al., 2010; Liu and Fang, 2015).

HPAI control and prevention measures vary by country. While countries like China, Mexico, Egypt and Vietnam deploy vaccination in conjunction with culling, other countries, such as the UK and Brazil, adhere to a strategy of depopulation upon detection, emphasizing trade and surveillance clarity (Kandeil et al., 2023; Shi et al., 2023; Minh et al., 2011). In February 2025, the U.S. Department of Agriculture (USDA) granted conditional approval to Zoetis vaccine, which contains a killed version of an H5N2 subtype designed to work against currently circulating variants of the H5N1 virus (USDA conditionally approves H5N1 poultry, 2025). Yet, widespread poultry vaccination hasn't been fully authorized in the USA due to international trade barriers (U.S. Department of Agriculture & Census of Agriculture, 2017; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025a). As of 2023, most member states of the World Organization of Animal Health (WOAH) would not accept imports of products from poultry vaccinated against HPAI. The primary concern is that it may not be easy to tell infected birds against those who have antibodies due to vaccination (U.S. Department of Agriculture & Census of Agriculture, 2017; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025a). Furthermore, domestic birds partially protected by vaccine can still spread the virus while showing no clinical signs of illness. To date, to prevent the spread of HPAI in poultry, the U.S. relies on strict biosecurity measures along with vigilant monitoring and euthanizing of affected flocks (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025d; Biondo and Congressional Research Service (CRS), 2025).

Theoretical and numerical studies of HPAI control strategies focus on preemptive and reactive culling, surveillance and vaccination, both individually and in combination (Lambert et al., 2023). Preemptive culling, which involves depopulation of all farms within a certain distance from affected premisses, has been reported to be very effective in reducing the number of HPAI cases, but at a very high economic cost (Jeon et al., 2023). Reactive culling is often included as a default component in HPAI transmission models to ensure their consistency with the current government regulations (Shi et al., 2023; Shih et al., 2022). Vaccination strategies were studied with the use of real data (Walker et al., 2010a) and by exploring hypothetical vaccination scenarios (Lambert et al., 2023; Malek & Hoque, 2021) assuming that vaccination is applied either within a certain radius of an outbreak (Backer et al., 2015; Hill et al., 2018; Pelletier et al., 2012) or to a certain percentage of the flock (Lee & Lao, 2018; Retkute et al., 2018; Walker et al., 2010b). All this research has been crucial for assessing the scope of the virus in order to ensure better diagnostics, biosecurity, and vaccination practices, to guarantee food safety, and to protect lives in the face of a high-risk poultry outbreak posing mutation and pandemic threats.

However, rigorous control and prevention did not stop the emergence of HPAI H5N1 virus that the U.S. has been dealing with since February 2022. There is a pressing need for further research on HPAI transmission dynamics that is closely aligned with the current USA production systems, surveillance intensity, and depopulation policies (Brüssow, 2024; Van Borm et al., 2025). In light of the ongoing HPAI outbreak in the U.S., which affects millions of commercial and backyard flocks, disrupts egg and meat production, and causes significant economic losses, it is important to develop biological models and optimization algorithms that conform to the U.S.- specific patterns of HPAI transmission and reflect control and prevention measures adopted in the USA.

In Section 2, we propose a partially stochastic network compartmental model to ascertain the progression and potential containment of HPAI virus in the U.S. commercial flocks and wildlife. Parameters of the model get estimated in Sections 3, 4 using available data on wild bird migration (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c), HPAI poultry outbreaks (CDC Avian Influenza (Bird Flu), 2025), and poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022) in the U.S. The new model simulates HPAI virus transmission driven by wildlife dynamic, seasonality, and farm-to-farm relations. Unlike many prior global models, this framework is closely tailored to the HPAI incidence, farm structure, and current mitigation practices in the United States of America (CDC Avian Influenza (Bird Flu), 2025; U.S. Department of Agriculture & Census of Agriculture, 2022; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c). In our study, we investigate how culling, spatial proximity between farms, and wild bird exposure influence outbreak duration and bird losses.

Comparing reported data on HPAI outbreaks in poultry farms (CDC Avian Influenza (Bird Flu), 2025) with the data on HPAI detections in wild birds (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c) across multiple states in the U.S., one can observe that the correlation is striking. See, for example, Fig. 1, Fig. 2, showing poultry cases vs. wild bird detections in California and Minnesota, USA. These two states have seen a considerable number of HPAI infections in poultry and wild birds during the HPAI virus spread of 2022-2025. Consequently, in the absence of fully adopted vaccination practices, separating flocks from wild birds (especially waterfowl) to prevent both direct and indirect HPAI transmission is critical. In Section 5, we use our proposed network model, along with HPAI surveillance data (CDC Avian Influenza (Bird Flu), 2025; U.S. Department of Agriculture & Census of Agriculture, 2022; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c), to analyze optimal control strategies aimed at lowering HPAI spread from wild birds to poultry. We show that this form of control results in a considerable decline in HPAI poultry cases, thus reducing the need for culling and bringing down economic losses associated with culling. Discussion and future plans are offered in Section 6. All codes and data sets used in this paper are available in the repository https://github.com/hkarami-GSU/Avian.git.

Fig. 1.

California: poultry outbreaks vs. wild bird detections.

Fig. 2.

Minnesota: Poultry outbreaks vs. wild bird detections.

2. Model formulation

In this section, we introduce a partially stochastic network compartmental model to study the spread of Highly Pathogenic Avian Influenza (HPAI) at the level of poultry farms and individual wild birds. The model incorporates HPAI transmission between backyard and commercial farms, culling interventions, restocking, and the impact of wild bird migration.

While some wild birds like ducks and gulls are nearly immune to avian influenza, the current strain of HPAI is causing mortality even to wild birds that were historically less vulnerable. In poultry, HPAI is highly contagious with death rate nearly 90–100% in 48 hours. Because there is no effective treatment for HPAI in birds, euthanizing the entire affected flock is the only way to stop the disease from spreading further. For this reason, while studying HPAI dynamic in poultry in the U.S., we consider susceptible, infected and culled population in unit number of farms rather than individual birds.

In system (2.1) below, we introduce two sets of model compartments, one for poultry farms and another for wild birds, along with three host groups of poultry farms of different sizes. Wild-bird-to-poultry-farm disease spread inside each group of farms is coupled with within-group and between-group influenza dynamics. Let $S_p^{(i)}(t)$ represent the number of poultry farms of size 1 to 49 (i = 1), of size 50 to 399 (i = 2), and of size 400 and up (i = 3), susceptible to avian influenza in a particular state in the USA at the time t, t ∈ [0, T], corresponding to 2022-2025 HPAI outbreak (U.S. Department of Agriculture & Census of Agriculture, 2022). Assume also that $I_p^{(i)}(t)$ stands for the number of poultry farms in group i infected with HPAI through either direct or indirect contact with infected wild birds or through farm-to-farm movement. Finally, let $C_p^{(i)}(t)$ be the number of culled/depopulated farms in group i consistent with the United States Department of Agriculture (USDA) response policy requiring timely destruction of affected flocks to contain the virus (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2016). Since wild birds, especially migratory waterfowl, are known to be the primary carriers of HPAI, in our biological model the second set of compartments accounts for HPAI transmission in wild birds and its significant impact on poultry farms within the USA. In what follows, we presume that S_w(t), I_w(t), and D_w(t) are susceptible, infected, and deceased (due to HPAI) wild bird populations, respectively (see Fig. 3).

Fig. 3.

Flowchart of HPAI dynamics among poultry farms and wild birds.

HPAI in poultry typically occurs when low pathogenic forms introduced from wild waterbirds evolve into highly pathogenic strains that prove lethal to poultry farms. In disease progression, HPAI strain can circle back from poultry to wild birds (particularly through carcasses, excreta, etc.). Yet, especially in the U.S., wild birds play the leading role in spatial dissemination of HPAI virus, with poultry-to-wild birds' transmission being less significant compared to the transmission from wild birds to poultry (Azeem et al., 2025; Couty et al., 2026; Rao, 2008). Indeed, wild birds mainly get infected by poultry via spillover from high-density poultry operations or via wet markets. In the U.S., farm biosecurity measures and the USDA HPAI culling policies noticeably reduce the danger of this transmission. And live bird markets are not common or popular in the U.S. in the same way they are in Asia. Many wet markets in the U.S. focus on seafood over live birds and animals. Thus, in our model, poultry-to-wild birds’ transmission is omitted. In line with (The U.S. Geological Survey (USGS), 2026), it is assumed in model (2.1) below that indirect transmission from poultry occurs mainly through viral contamination of farm environment (feed, equipment, boots, clothes) and via between-farm movement. In our proposed HPAI network, this type of transmission is accounted for as poultry-to-poultry. Given the above assumptions, we arrive at the network system of differential equations in the form:

$$\begin{array}{ll}\frac{{\mathrm{dS}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}}{\mathrm{dt}}& =-{\mathrm{\beta }}^{\left(\mathrm{i}\right)}\left(\mathrm{t}\right){\mathrm{S}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}\left(\mathrm{t}\right){\mathrm{I}}_{\mathrm{w}}\left(\mathrm{t}\right)-{\mathrm{S}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}\left(\mathrm{t}\right)\sum _{\mathrm{j}=1}^3{\mathrm{\alpha }}^{(\mathrm{i},\mathrm{j})}\frac{{\mathrm{I}}_{\mathrm{p}}^{\left(\mathrm{j}\right)}\left(\mathrm{t}\right)}{{\mathrm{N}}_{\mathrm{p}}^{\left(\mathrm{j}\right)}}+\mathrm{\delta }{\mathrm{C}}_{\mathrm{p}}^{\left(\mathrm{i}\right)}\left(\mathrm{t}\right)\\ \frac{d{I}_p^{\left(i\right)}}{dt}& ={\beta }^{\left(i\right)}\left(t\right)S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)+S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}-\phi I_p^{\left(i\right)}\left(t\right)\\ \frac{d{C}_p^{\left(i\right)}}{dt}& =\phi I_p^{\left(i\right)}\left(t\right)-\delta C_p^{\left(i\right)}\left(t\right),i=1,2,3,\\ \frac{d{S}_w}{dt}& =-\sigma \frac{S_w\left(t\right)I_w\left(t\right)}{N_w}+{\gamma }_r{I}_w\left(t\right)+f\left(t\right)\\ \frac{d{I}_w}{dt}& =\sigma \frac{S_w\left(t\right)I_w\left(t\right)}{N_w}-{\gamma }_r{I}_w\left(t\right)-{\gamma }_d{I}_w\left(t\right)\\ \frac{d{D}_w}{dt}& ={\gamma }_d{I}_w\left(t\right).\end{array}$$ (2.1)

In ODE system (2.1), the product, β⁽ⁱ⁾(t)I_w(t), quantifies the flow of poultry farms in group i from susceptible, $S_p^{(i)}$, to infected, $I_p^{(i)}$, stage as the result of wild-bird-to-poultry-farm virus spread. A time-dependent parameter, β⁽ⁱ⁾(t), stands for the average number of contacts per bird per time, multiplied by the probability of disease transmission in a contact between a susceptible poultry farm and an infected wild bird. The network parameters, α^(i,j), measure the intensity of farm-to-farm HPAI spread within-group and between-group of farms of a comparable size. The three remaining parameters characterizing HPAI transmission in poultry are $N_p^{(i)}$, the total number of poultry farms in the ith group, φ, the rate of culling/removal due to the detection of HPAI at a property, and δ, the restocking rate in poultry farms (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b).

In the subsystem of (2.1) illustrating avian influenza dynamics in wild birds, σ is the average number of contacts per time, multiplied by the probability of HPAI transmission in a contact between susceptible and infected wild birds. The corresponding product, $\sigma \frac{S_w(t)I_w(t)}{N_w}$, is scaled by N_w, the estimated mean number of wild birds in the state (KUHL, 2025). To account for the wild bird migration in the area, the time-dependent function f(t) stands for the rate of change in the wild bird population, thus bringing seasonality in the transmission dynamics, which can be clearly seen from incidence data presented in Fig. 1, Fig. 2. The remaining two parameters, γ_r and γ_d, are average recovery and HPAI-related death rates within the wild bird population.

The values and the meaning of disease parameters in (2.1) are summarized in Table 1 below. Some parameters, like $N_p^{(i)}$, N_w, and f(t), can be pre-estimated or modelled from data on poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022) and wild bird count (fall and spring migration) (KUHL, 2025) available for all states in the USA. Parameter values for φ (deterministic) and δ (stochastic) follow from USDA regulations on culling and restocking, respectively (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2016; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b). The remaining system parameters, β⁽ⁱ⁾(t), α^(i,j), σ, γ_r, and γ_d must be estimated from the reported incidence data on poultry outbreaks (CDC Avian Influenza (Bird Flu), 2025) and HPAI detections in wild birds (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c).

Table 1.

Parameter definitions and baseline values.

Symbol	Definition	Value	Reference
β(i)(t)	Transmission rate from wild birds to poultry farms in group i	birds−1months−1	Estimated
α(i,j)	Transmission rate between poultry farms in groups i and j	months−1	Estimated
φ	Culling/removal rate in poultry farms	(30/1) months−1	U.S. Department of Agriculture and Animal and Plant Health Inspection Service (2016)
δ	Restocking rate in poultry farms	$\sim U\left(\frac{30}{60},\frac{30}{21}\right){\text{months}}^{-1}$	U.S. Department of Agriculture and Animal and Plant Health Inspection Service (2025b)
$N_p^{(i)}$	Number of poultry farms in group i	farms	U.S. Department of Agriculture and Census of Agriculture (2022)
σ	Within-wild-bird transmission rate	months−1	Estimated
Nw	Average population of wild birds in the region	birds	KUHL (2025)
γr	Recovery rate of infected wild birds	months−1	Estimated
γd	HPAI-induced death rate of wild birds	months−1	Estimated
f(t)	Exogenous inflow of susceptible wild birds	birds ×months−1	KUHL (2025)

In Sections 3, 4, we confront HPAI transmission model (2.1) with data on wild bird migration (KUHL, 2025; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c), HPAI poultry outbreaks (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2016; U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b; CDC Avian Influenza (Bird Flu), 2025), and poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022) in the U.S. in order to estimate unknown disease parameters β⁽ⁱ⁾(t), α^(i,j), σ, γ_r, and γ_d by solving the corresponding constrained minimization problems by regularized computational algorithms. As one can see, the last three equations for the wild bird compartments, S_w(t), I_w(t), and D_w(t), can be uncoupled from the rest of system (2.1). Thus, in Section 3, we fit the observable part of the S_wI_wD_w submodel to the reported data on wild bird detections (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c) to infer parameter values (with quantified uncertainty) for σ, γ_r, and γ_d. Then, we employ these parameters to solve the forward S_wI_wD_w problem for the state variables, S_w(t), I_w(t), and D_w(t).

In Section 4, the bundle of reconstructed curves, I_w(t), along with available data on HPAI poultry outbreaks (CDC Avian Influenza (Bird Flu), 2025) and poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022), are used to estimate the unknown parameters β⁽ⁱ⁾(t) and α^(i,j) from the subsystem for $S_p^{(i)}$, $I_p^{(i)}$, and $C_p^{(i)}$. In the network HPAI model (2.1), the matrix $\left[{\alpha }^{(i,j)}\right]$, 1 ≤ i, j ≤ 3, is assumed to be symmetric, where each entry α^(i,j), 1 ≤ i < j ≤ 3, is an average of (i, j) and (j, i) transmissions. As the result, the six unknown farm-to-farm transmission parameters, α^(i,j), 1 ≤ i ≤ j ≤ 3, must be estimated from incidence data (CDC Avian Influenza (Bird Flu), 2025) in addition to 3n(n > 1) parameters for the discrete approximation of β⁽ⁱ⁾(t), i = 1, 2, 3. The discretization parameter, n, is chosen individually for different states in a data-specific manner.

3. Parameter estimation: wild bird transmission dynamics

In this section, we use biological model (2.1) to estimate three unknown disease parameters (within-wild-bird transmission rate, σ, the average recovery rate of infected wild birds, γ_r, and the average HPAI-induced death rate of wild birds, γ_d) for the states of California and Minnesota, USA. These two states have seen a considerable number of HPAI cases in poultry and wild birds during the outbreak of 2022-2025. While the risk to public health remained relatively low, the unfolding dynamic had serious negative implications on wildlife and agriculture. To extract parameter values for σ, γ_r, and γ_d, we fit the last three equations in (2.1),

$$\begin{array}{ll}\frac{d{S}_w}{dt}& =-\sigma \frac{S_w\left(t\right)I_w\left(t\right)}{N_w}+{\gamma }_r{I}_w\left(t\right)+f\left(t\right)\\ \frac{d{I}_w}{dt}& =\sigma \frac{S_w\left(t\right)I_w\left(t\right)}{N_w}-{\gamma }_r{I}_w\left(t\right)-{\gamma }_d{I}_w\left(t\right)\\ \frac{d{D}_w}{dt}& ={\gamma }_d{I}_w\left(t\right),\end{array}$$ (3.1)

to reported data on highly pathogenic avian influenza detections in wild birds (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c) and to the available estimates of the wild bird populations in California and Minnesota (KUHL, 2025). The site (KUHL, 2025) provides an approximate count of wild birds during fall and spring migrations in each state of the USA. Based on these data, we model the wild bird population as a periodic sine function:

$$y(t)=\frac{A_1+A_2}{2}+\frac{A_2-A_1}{2}\sin \left(\frac{\pi }{6}t+\mu \right),t\in [0,T],$$ (3.2)

where A₁ and A₂ are the population lower and upper bounds, respectively, and the angular velocity, $\frac{\pi }{6}$, reflects the 12-month periodic nature of the wild bird migration. The phase shift, μ, allows to properly adjust the timing of population peaks in fall and spring. According to (KUHL, 2025), the total bird count in California changes from A₁ = 117, 959, 600 (fall migration) to A₂ = 215, 292, 600 (spring migration). In the state of Minnesota, the range is from A₂ = 760, 565, 900 (fall migration) to A₁ = 470, 250, 900 (spring migration) (KUHL, 2025). As one can easily see from (3.1), the external inflow of susceptible wild birds, f(t), is equal to $\frac{dy}{dt}$. Thus, f(t) and the population average, $N_w=\frac{A_1+A_2}{2}$, are uniquely defined by (3.2).

Given f(t) and N_w in (3.1), we estimate the unknown parameters, σ, γ_r, and γ_d, using Levenberg–Marquardt algorithm (lsqcurvefit built-in function in MATLAB Optimization Toolbox). At each iteration step, we employ ode23s to solve ODE system (3.1) for the state variables S_w(t), I_w(t), and D_w(t).

The USDA statistics (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c) on wild bird HPAI detections allows to comprise both cumulative and incidence data on avian influenza cases. Commonly, for parameter estimation, incidence data is preferred since the errors in reported incidence data are independent and identically distributed (i.i.d.) potentially leading to better inversion results. Besides, by its very nature, incidence data is more sensitive to even minor variations in the underlying transmission process and in the associated parameters, thus making it easier to learn from incidence data.

Cumulative data, on the other hand, tends to over-smooth temporal variations and to produce parameter estimates biased toward earlier cases. Yet, if combined with case incidence and properly weighted, cumulative data can help to stabilize parameter estimation and to avoid possible over-fitting.

To that end, for our numerical simulations, we employ USDA data (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c) to generate both incidence, ${\mathbf{D}}_w={[D_{w,1},D_{w,2},\dots ,D_{w,m}]}^{\top }$, and cumulative data, ${\mathbf{G}}_w={[G_{w,1},G_{w,2},\dots ,G_{w,m}]}^{\top }$, with m being the total number of months in the study period. We let incidence data play the main part in parameter estimation, all the while using cumulative data to carefully penalize the algorithm.

According to the first equation in (3.1), the number of newly infected wild birds over the month t is given by $\frac{\sigma S_w(t)I_w(t)}{N_w}$. Thus, we define the model-predicted observation operator for new HPAI detections, ${\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d]$, to be fitted to incidence monthly data, D_w, as follows

$${\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d]={[{\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d](t_1),{\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d](t_2),\dots ,{\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d](t_m)]}^{\top },$$ (3.3)

$${\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d](t_i):=\frac{\sigma S_w[\sigma ,{\gamma }_r,{\gamma }_d](t_i)I_w[\sigma ,{\gamma }_r,{\gamma }_d](t_i)}{N_w},i=1,2,\dots ,m,$$ (3.4)

which yields the nonlinear least squares problem (NLSP)

$${\min }_{\sigma ,{\gamma }_r,{\gamma }_d}{‖{\mathcal{F}}_w(\sigma ,{\gamma }_r,{\gamma }_d)‖}^2,{\mathcal{F}}_w(\sigma ,{\gamma }_r,{\gamma }_d):={[{\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d]-{\mathbf{D}}_w,{\zeta }_w(L{\mathcal{D}}_w[\sigma ,{\gamma }_r,{\gamma }_d]-{\mathbf{G}}_w)]}^{\top }.$$ (3.5)

In the operator ${\mathcal{F}}_w$ defined by (3.5), $L={\left[l^{(i,j)}\right]}_{m\times m}$ is a lower triangular matrix such that l^(i,j) = 1 for i ≥ j and l^(i,j) = 0 for i < j, a positive constant, ζ_w, is an appropriate weighting factor, and ‖ ⋅‖ stands for the Euclidian norm in ${\mathbb{R}}^{2m}$.

To quantify uncertainty in the approximate parameters, we repeat the inversion (using parallel programming loop parfor in MATLAB) with M = 100 additional calibration data sets for incidence cases assuming Gaussian error structure (and clipping negative values). This works better than Poisson noise for discrete data sets with a significant number of zeros and small positive values. The resulting M best-fit parameter sets are used to build a histogram for each unknown parameter, showing the frequency distribution for reconstructed values along with 95% confidence intervals (CI), see Fig. 4, Fig. 6.

Fig. 4.

California: parameters histograms for HPAI dynamics in wild birds; columns refer to the estimated parameters; the yellow and red vertical lines represent initial values and mean estimates, respectively, and the dashed black vertical lines denote the 95% confidence intervals.

Fig. 6.

Minnesota: parameters histograms for HPAI dynamics in wild birds; columns refer to the estimated parameters; the yellow and red vertical lines represent initial values and mean estimates, respectively, and the dashed black vertical lines denote the 95% confidence intervals.

Fig. 4 illustrates the extracted parameter values for the wild bird HPAI transmission dynamics in the state of California, 2022-2025. By confronting submodel (3.1) with reported data (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c; KUHL, 2025), we estimate within-wild-bird transmission rate, σ = 2.388 (95% CI: 1.276, 5.746), the average recovery rate of infected wild birds, γ_r = 2.067 (95% CI: 1.308, 3.510), and the average HPAI-induced death rate of wild birds, γ_d = 0.361 (95% CI: 0.000, 2.250). While skewed right (most notably, for γ_d), all three histograms show that the reconstruction is stable and the confidence intervals are consistent with the level of uncertainty in the reported data. Fig. 5 represents wild bird HPAI detections for the state of California vs. simulated model fit for monthly cumulative and incidence cases along with 100 bootstrap reconstructions of I_w(t).

Fig. 5.

California: state data (dots) and model fit (solid line) for cumulative (right) and incidence (center) wild bird HPAI detections; 100 bootstrap reconstructions of I_w(t) (left).

Fig. 6, Fig. 7, contain numerical results for the state of Minnesota. Fig. 6 shows approximate values for within-wild-bird transmission rate, σ = 3.923 (95% CI: 2.113, 8.480), the average recovery rate of infected wild birds, γ_r = 3.012 (95% CI: 1.917, 5.785), and the average HPAI-induced death rate of wild birds, γ_d = 0.974 (95% CI: 0.186, 3.178). Again, we see stable parameter estimates with reasonable uncertainty. The data fit and the bundle of reconstructed functions, I_w(t), for the state of Minnesota are illustrated in Fig. 7, which highlights a good data coverage for incidence and cumulative sets.

Fig. 7.

Minnesota: state data (dots) and model fit (solid line) for cumulative (right) and incidence (center) wild bird HPAI detections; 100 bootstrap reconstructions of I_w(t) (left).

4. Parameter estimation: poultry farm network

The next step in our inversion procedure is to use the previously reconstructed bundle, I_w(t), along with available data on HPAI poultry outbreaks (CDC Avian Influenza (Bird Flu), 2025) and poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022), to estimate the unknown parameters β⁽ⁱ⁾(t) and α^(i,j) from the subsystem for $S_p^{(i)}$, $I_p^{(i)}$, and $C_p^{(i)}$:

$$\begin{array}{ll}\frac{d{S}_p^{\left(i\right)}}{dt}& =-{\beta }^{\left(i\right)}\left(t\right)S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)-S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}+\delta C_p^{\left(i\right)}\left(t\right)\\ \frac{d{I}_p^{\left(i\right)}}{dt}& ={\beta }^{\left(i\right)}\left(t\right)S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)+S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}-\phi I_p^{\left(i\right)}\left(t\right)\\ \frac{d{C}_p^{\left(i\right)}}{dt}& =\phi I_p^{\left(i\right)}\left(t\right)-\delta C_p^{\left(i\right)}\left(t\right),i=1,2,3.\end{array}$$ (4.1)

To account for the farm-to-farm transmission, we employ a symmetric matrix $\left[{\alpha }^{(i,j)}\right]$, 1 ≤ i, j ≤ 3, with six unknown entries, where each entry α^(i,j), 1 ≤ i < j ≤ 3, is an average of (i, j) and (j, i) transmissions. As the result, the six unknown parameters, α^(i,j), 1 ≤ i ≤ j ≤ 3, must be estimated from the reported HPAI data (CDC Avian Influenza (Bird Flu), 2025; U.S. Department of Agriculture & Census of Agriculture, 2022) in addition to 3n(n > 1) parameters used for the discrete approximation of β⁽ⁱ⁾(t), i = 1, 2, 3.

The discretization algorithm for β⁽ⁱ⁾(t), i = 1, 2, 3, must be carefully selected to ensure that the dynamic of wild-bird-to-poultry transmission is properly quantified. To adequately estimate β⁽ⁱ⁾(t), we project these functions onto the subspace spanned by shifted Legendre polynomials, P_j(t), of degree j = 0, 1, …, n − 1, that are orthogonal on the interval [0, T] with respect to L₂ inner product, defined recursively as follows

$$x=(2t-T)/T,P_0(x)=1,P_1(x)=x,(j+1)P_{j+1}(x)=(2j+1)x{P}_j(x)-j{P}_{j-1}(x),$$ (4.2)

t ∈ [0, T], j = 1, 2, …, n − 2. This gives rise to the finite-dimensional analog of the wild bird-to-poultry transmission rate in the inversion algorithm:

$${\beta }^{(i)}(t)\approx \sum _{j=0}^{n-1}{\tilde{\beta }}_{j+1}^{(i)}{P}_j(t),i=1,2,3.$$ (4.3)

The number of base polynomials needs to be identified from the goodness-of-fit, which ensures that the model is consistent with reported data. At the same time, over-fitting noisy data tends to destabilize parameter estimation. Due to a significant ill-posedness of the inversion, increasing the size of the solution space in order to learn more from corrupt values and outliers leads to an excessive noise propagation and a less accurate reconstruction.

While Table 2 clearly shows that with larger n, the Sum of Squared Errors (SSE), measuring the total squared differences between observed case incidence and model-predicted values, are going down, the estimated transmission rates, β⁽ⁱ⁾(t), tend to take on negative values and become extremely erratic as n increases. This underlines that goodness-of-fit alone cannot be a reliable criterion for the approximation of n. To that end, in Tables 2 and in addition to computing SSE (to show that the fit does go down with n, and the discretization process is convergent), we employ the Corrected Akaike Information Criterion (AIC_c)

$$AI{C}_c=\frac{3m}{2}\ln \left(\frac{SSE}{3m}\right)+2K+\frac{2K(K+1)}{3m-K-1},$$

which can be viewed as a regularized goodness-of-fit, rewarding models for their accurate representation of data, while penalizing them for too many parameters. In the above, m is the number of incidence data points in each farm group (that is, 3m is the total number of points in the incidence set), and K is the size of the solution space (i.e., K = 3n + 6). Unlike SSE, the AIC_c fluctuates initially, but starts growing when the tendency to over-fit is detected. The quantitative measure provided by the Akaike Criterion remarkably reconfirms our qualitative observation that going further than n = 10 is not advisable for both California and Minnesota data sets. Thus, n = 10, is identified as the near-optimal value in Legendre polynomial approximation of β⁽ⁱ⁾(t) (see Table 2, Table 3), which ensures the best balance between accuracy and stability as measured by AIC_c. This choice of n leads to a good data fit for the states of California and Minnesota, without over-fitting, as shown in Fig. 10, Fig. 11, Fig. 14, Fig. 15. For California, the reconstructed functions, β⁽ⁱ⁾(t), exhibit the kind of behavior that is consistent with reported HPAI detections in each of the three farm groups, see Fig. 8, Fig. 10.

Table 2.

California: discretization size selection metrics for β⁽ⁱ⁾(t).

n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
SSE	1214	1052	1033	1016	920	697	608	502	356	333	298	278	271	269	253	202	182	152	136
AICc	158	158	166	174	178	174	177	179	174	185	194	208	225	246	266	280	304	328	360

Table 3.

Minnesota: discretization size selection metrics for β⁽ⁱ⁾(t).

n	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
SSE	2569	2525	1994	1894	1970	1921	1801	1052	739	656	566	480	418	359	316	281	215	196	168
AICc	214	220	214	219	231	239	245	223	213	219	223	228	235	243	253	266	272	291	309

Fig. 10.

California: state data (dots) and model fit (solid line) for incidence HPAI outbreaks in poultry farms (100 bootstrap reconstructions).

Fig. 11.

California: state data (dots) and model fit (solid line) for cumulative HPAI outbreaks in poultry farms (100 bootstrap reconstructions).

Fig. 14.

Minnesota: state data (dots) and model fit (solid line) for incidence HPAI outbreaks in poultry farms (100 bootstrap reconstructions).

Fig. 15.

Minnesota: state data (dots) and model fit (solid line) for cumulative HPAI outbreaks in poultry farms (100 bootstrap reconstructions).

Fig. 8.

California: reconstructed transmission rate, β⁽ⁱ⁾(t), from wild birds to poultry farms in group i (100 bootstrap reconstructions).

As evident from Fig. 12, Fig. 14, for the state of Minnesota, the reconstructed wild bird-to-poultry transmission rates, β⁽ⁱ⁾(t), are also in sync with the dynamic of the incidence data. The figures illustrate that the uncertainty levels in the reconstructed functions, β⁽ⁱ⁾(t), are consistent with the uncertainty in the reported data (CDC Avian Influenza (Bird Flu), 2025; U.S. Department of Agriculture & Census of Agriculture, 2022), and the data fit, for both incidence and cumulative sets, is reasonably accurate.

Fig. 12.

Minnesota: reconstructed transmission rate, β⁽ⁱ⁾(t), from wild birds to poultry farms in group i (100 bootstrap reconstructions).

Apart from data (U.S. Department of Agriculture & Census of Agriculture, 2022; CDC Avian Influenza (Bird Flu), 2025), provided by the US Centers for Disease Control and Prevention (CDC) and the United States Department of Agriculture (USDA), as well as the data on I_w(t), obtained in the course of parameter estimation for the wild bird submodel (3.1), to reconstruct β⁽ⁱ⁾(t) and α^(i,j), 1 ≤ i ≤ j ≤ 3, from (4.1) we use pre-estimated parameter values $N_p^{(i)}$, the number of poultry farms in each of the three groups, φ, the rate of culling/removal due to HPAI infection at a property, and δ, the restocking rate in poultry farms (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b).

As shown in Tables 1 and in our model, we assume that φ = (30/1) months⁻¹ since USDA's goal is to get each farm depopulated within 24 hours of first detecting HPAI.

Pre-estimation of δ is more challenging. Per USDA regulations (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b), poultry farms must remain empty for a minimum of 21 days following mandatory cleaning, disinfection, and environmental testing, but the exact timeline varies depending on factors like flock size and the disposal methods used. Moreover, due to heavy losses and increased demand, hatching chicks and pullets are not always available, which can further delay restocking (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025b). So, for large egg farms, returning to full production may be particularly difficult and take longer. Considering all that, we introduce stochasticity into the model by setting $\delta \sim U\left(\frac{30}{60},\frac{30}{21}\right){\text{months}}^{-1}$ to indicate that it may take, on average, somewhere between 3 weeks and 2 months for a farm to restock.

Using the USDA data on state-by-state poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022), we divide all farms in California and Minnesota in three groups based on the flock size: (1) 1-49, (2) 50-399, (3) 400 and up. Table 4 is used to calculate the number of poultry farms, $N_p^{(i)}$, i = 1, 2, 3, in each group for both states.

Table 4.

Number of poultry farms by inventory size group in California and Minnesota (U.S. Department of Agriculture & Census of Agriculture, 2022).

State	1–49	50–99	100–399	400–3199	3200–9999	10,000–19,999	20,000–49,999	50,000–99,999	100,000 or more
California	5575	434	187	56	6	4	16	6	26
Minnesota	5412	472	161	36	8	24	4	5	24

As it follows from the first equation in (4.1), the number of newly infected poultry farms in group i over the month of t_k can be approximated by

$${\mathcal{D}}_{p,k}^{(i)}[{\tilde{\beta }}^{(i)},{\alpha }^{(i,j)}]≔S_p^{(i)}(t_k)I_w(t_k)\sum _{j=0}^{n-1}{\tilde{\beta }}_{j+1}^{(i)}{P}_j(t_k)+S_p^{(i)}(t_k)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{(j)}(t_k)}{N_p^{(j)}},$$

k = 1, 2, …, m, i = 1, 2, 3. Thus, in order to estimate the unknown parameters, ${\tilde{\beta }}^{(i)}$ and α^(i,j), one has to fit the observation operator, ${\mathcal{D}}_p={[{\mathcal{D}}_{p,1}^{(1)},{\mathcal{D}}_{p,2}^{(1)},\dots ,{\mathcal{D}}_{p,m}^{(1)},{\mathcal{D}}_{p,1}^{(2)},{\mathcal{D}}_{p,2}^{(2)},\dots ,{\mathcal{D}}_{p,m}^{(2)},{\mathcal{D}}_{p,1}^{(3)},{\mathcal{D}}_{p,2}^{(3)},\dots ,{\mathcal{D}}_{p,m}^{(3)}]}^{\top },$ to the available data on monthly new HPAI detections in poultry farms in each of the three groups (CDC Avian Influenza (Bird Flu), 2025): ${\mathbf{D}}_p={[D_{p,1}^{(1)},D_{p,2}^{(1)},\dots ,D_{p,m}^{(1)},D_{p,1}^{(2)},D_{p,2}^{(2)},\dots ,D_{p,m}^{(2)},D_{p,1}^{(3)},D_{p,2}^{(3)},\dots ,D_{p,m}^{(3)}]}^{\top }.$ This yields the primary, fidelity, term in the nonlinear least squares problem (NLSP) for parameter estimation in poultry submodel (4.1). As in the case of wild birds, we penalize the functional by also fitting to the cumulative data, G_p, which brings about

$${\min }_{{\tilde{\beta }}^{(i)},{\alpha }^{(i,j)}}{‖{\mathcal{F}}_p({\tilde{\beta }}^{(i)},{\alpha }^{(i,j)})‖}^2,{\mathcal{F}}_p({\tilde{\beta }}^{(i)},{\alpha }^{(i,j)}):={[{\mathcal{D}}_p[{\tilde{\beta }}^{(i)},{\alpha }^{(i,j)}]-{\mathbf{D}}_p,{\zeta }_p({\mathcal{G}}_p[{\tilde{\beta }}^{(i)},{\alpha }^{(i,j)}]-{\mathbf{G}}_p)]}^{\top },$$

where ${\mathcal{G}}_p={[{\mathcal{G}}_{p,1}^{(1)},{\mathcal{G}}_{p,2}^{(1)},\dots ,{\mathcal{G}}_{p,m}^{(1)},{\mathcal{G}}_{p,1}^{(2)},{\mathcal{G}}_{p,2}^{(2)},\dots ,{\mathcal{G}}_{p,m}^{(2)},{\mathcal{G}}_{p,1}^{(3)},{\mathcal{G}}_{p,2}^{(3)},\dots ,{\mathcal{G}}_{p,m}^{(3)}]}^{\top }$. The ith subvector of ${\mathcal{G}}_p$ is the observation operator for the cumulative HPAI detections in the poultry group i defined as ${[{\mathcal{G}}_{p,1}^{(i)},{\mathcal{G}}_{p,2}^{(i)},\dots ,{\mathcal{G}}_{p,m}^{(i)}]}^{\top }=L{[{\mathcal{D}}_{p,1}^{(i)},{\mathcal{D}}_{p,2}^{(i)},\dots ,{\mathcal{D}}_{p,m}^{(i)}]}^{\top }$, and $L={\left[l^{(i,j)}\right]}_{m\times m}$ is a lower triangular matrix such that l^(i,j) = 1 for i ≥ j and l^(i,j) = 0 for i < j. In the operator ${\mathcal{F}}_p$, a positive constant, ζ_p, is an appropriate weighting factor, m is the total number of months in the study period, and ‖ ⋅‖ stands for the Euclidian norm in ${\mathbb{R}}^{6m}$. The wild-bird transmission parameter, ${\tilde{\beta }}^{(i)}$, i = 1, 2, 3, is comprised of 3n expansion coefficients for the finite dimensional approximation of the transmission rate, β⁽ⁱ⁾(t), by the linear combination of basis functions (4.3).

Under this framework, parameter estimation for the network model (4.1) comes down to estimating 3n + 6 unknown parameters from 3m data points (counting incidence only). With n = 10 and m = 36 for the state of California, and n = 10 and m = 41 for the state of Minnesota, both nonlinear least squares problems are over-determined (as intended). However, due to a considerable number of zeros in each incidence data set coupled with reporting irregularities, the problems are still very unstable. Thus, statistical analysis of the extracted parameters is important.

To find the mean values of the estimated β⁽ⁱ⁾(t) and α^(i,j) along with the corresponding 95% confidence intervals (CI), we employ the Levenberg–Marquardt algorithm (lsqcurvefit built-in function in MATLAB Optimization Toolbox) combined with the subfunction ode23s for solving (4.1) to approximate state variables $S_p^{(i)}$, $I_p^{(i)}$, and $C_p^{(i)}$.

The inversion is carried out by generating M = 100 additional data sets while adding Gaussian error structure to the reported data (and setting possible negative values to zero), and sampling from the bundle of infection curves, I_w(t), obtained during parameter estimation for the wild bird model (3.1). As mentioned before, for discrete data with many zeros and small positive values, Gaussian noise is preferred to Poisson. The resulting M best-fit parameter sets are used to quantify the uncertainty in β⁽ⁱ⁾(t), i = 1, 2, 3 (see Fig. 8, Fig. 12), and to build histograms for the six unknown parameters, α^(i,j), 1 ≤ i ≤ j ≤ 3, as shown in Fig. 9, Fig. 13.

Fig. 9.

California: reconstructed transmission rate, α^(i,j), between poultry farms in groups i and j, assuming α^(i,j) = α^(j,i).

Fig. 13.

Minnesota: reconstructed transmission rate, α^(i,j), between poultry farms in groups i and j, assuming α^(i,j) = α^(j,i).

Comparing estimates of wild bird-to-poultry transmission rates, β⁽ⁱ⁾(t) (Fig. 8, Fig. 12), to the estimates of poultry-to-poultry HPAI transmission, α^(i,j), 1 ≤ i ≤ j ≤ 3 (Fig. 9, Fig. 13), for the states of California and Minnesota, one can observe that, overall, the wild-bird-to-poultry transmission dominates over the poultry-to-poultry one. This is understandable, considering that wild birds, especially waterfowl, are known long-distance carriers of HPAI, spreading the virus to poultry through migration, contaminated water, and shared spaces. Thus, restricting direct and indirect contacts between poultry and wild birds and keeping poultry away from potentially contaminated environments (respiratory secretions and feces of wild birds), is the necessary first step in HPAI control and prevention. In the next section, we present a mathematical study of optimal control scenarios aimed at the reduction of wild bird-to-poultry HPAI transmission.

5. The optimal control problem

In the absence of vaccination, biosecurity is crucial, and isolating wild birds from poultry is an important part of it. The goal of the optimal control problem is to see how much a real-life scenario can be improved through the implementation of control strategies aimed at reducing wild bird-to-poultry HPAI transmission. To that end, in the ”real-life” model (2.1), we replace β⁽ⁱ⁾(t) with β⁽ⁱ⁾(t)(1 − u⁽ⁱ⁾(t)), where β⁽ⁱ⁾(t) are the HPAI transmission rates from wild birds to poultry and u⁽ⁱ⁾(t), i = 1, 2, 3, are the control functions intended to scale down the spread of HPAI virus from wild birds to poultry farms.

To optimize the choice of u⁽ⁱ⁾(t), i = 1, 2, 3, we aim to lower the force of infection, $\mathcal{I}(t)$, while keeping the cost of preventive measures at bay. From the model of HPAI transmission in domestic birds, it follows that the force of infection in poultry farms is given by

$$\mathcal{I}(t)=\sum _{i=1}^3\left\{{\beta }^{(i)}(t)(1-u^{(i)}(t))S_p^{(i)}(t)I_w(t)+S_p^{(i)}(t)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{(j)}(t)}{N_p^{(j)}}\right\}.$$

Considering the nature of u⁽ⁱ⁾(t), we define the feasible set for these functions as

$$\mathcal{U}=\left\{u^{(i)}\in {\mathcal{L}}^1[0,T],0\le u^{(i)}(t)<1,i=1,2,3\right\},$$ (5.1)

and introduce the controlled flow of disease transmission:

$$\begin{array}{ll}\frac{d{S}_p^{\left(i\right)}}{dt}& =-{\beta }^{\left(i\right)}\left(t\right)(1-u^{\left(i\right)}\left(t\right))S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)-S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}+\delta C_p^{\left(i\right)}\left(t\right)\\ \frac{d{I}_p^{\left(i\right)}}{dt}& ={\beta }^{\left(i\right)}\left(t\right)(1-u^{\left(i\right)}\left(t\right))S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)+S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}-\phi I_p^{\left(i\right)}\left(t\right)\\ \frac{d{C}_p^{\left(i\right)}}{dt}& =\phi I_p^{\left(i\right)}\left(t\right)-\delta C_p^{\left(i\right)}\left(t\right),i=1,2,3.\end{array}$$ (5.2)

With that in mind, we propose the objective functional

$$\begin{array}{ll}\mathcal{J}(\mathrm{x},\mathrm{u})& =\sum _{i=1}^3{\int }_0^T\left\{{\beta }^{\left(i\right)}\left(t\right)(1-u^{\left(i\right)}\left(t\right))S_p^{\left(i\right)}\left(t\right)I_w\left(t\right)+S_p^{\left(i\right)}\left(t\right)\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}+{\lambda }^{\left(i\right)}c\left(u^{\left(i\right)}\left(t\right)\right)\right\}\mathrm{d}t\\ & =\sum _{i=1}^3\left(S_p^{\left(i\right)}\left(0\right)-S_p^{\left(i\right)}\left(T\right)+{\int }_0^T\left\{{\lambda }^{\left(i\right)}c\left(u^{\left(i\right)}\left(t\right)\right)+\delta C_p^{\left(i\right)}\left(t\right)\right\}\mathrm{d}t\right),\end{array}$$ (5.3)

where $\mathbf{x}(t):={[S_p^{(1)}(t),I_p^{(1)}(t),C_p^{(1)}(t),\dots ,S_p^{(3)}(t),I_p^{(3)}(t),C_p^{(3)}(t)]}^{\top }$ combines all state variables of controlled model (5.2), the vector $\mathbf{u}(t)≔{[u^{(1)}(t),u^{(2)}(t),u^{(3)}(t)]}^{\top }$ represents control strategies applied to each group of farms, the vector $\mathbf{c}(\mathbf{u})≔{[c(u^{(1)}),c(u^{(2)}),c(u^{(3)})]}^{\top }$ stands for the cost of control u(t), and $\mathit{\lambda }≔{[{\lambda }^{(1)},{\lambda }^{(2)},{\lambda }^{(3)}]}^{\top }$ are the weights associated with the components of the cost function, c(u). Thus, the optimal control problem is to minimize the objective functional (5.3) subject to the ODE system (5.2).

According to the Pontryagin's Minimum Principle [43, 44], if a function $\mathbf{u}\in \mathcal{U}$ is an optimal control strategy with respect to the ODE system $\dot{\mathbf{x}}=f(\mathbf{x},\mathbf{u})$, x(0) = x₀, and the objective functional $\mathcal{J}(\mathbf{x},\mathbf{u})=h(\mathbf{x}(T))+{\int }_0^{T}L(\mathbf{x}(t),\mathbf{u}(t))dt$, then there is a trajectory p(t) such that

$$\dot{\mathbf{p}}(t)={\left(-{{\partial }_{\mathbf{x}}H(\mathbf{x},\mathbf{u},\mathbf{p})}^{\top }\right|}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)},\mathbf{p}(T)={\left({\partial }_{\mathbf{x}}h{(\mathbf{x})}^{\top }\right|}_{\mathbf{x}(T)},$$ (5.4)

$$\mathbf{u}\left(t\right)={argmin}_{\mathbf{v}\in \mathcal{U}}H(\mathbf{x}\left(t\right),\mathbf{v}\left(t\right),\mathbf{p}\left(t\right)),H(\mathbf{x},\mathbf{v},\mathbf{p})≔L(\mathbf{x},\mathbf{v})+{\mathbf{p}}^{\top }f(\mathbf{x},\mathbf{v}).$$ (5.5)

As the result, solving the optimal control problem comes down to minimizing the Hamiltonian, $H(\mathbf{x},\mathbf{u},\mathbf{p})=\delta {\sum }_{i=1}^3{C}_p^{(i)}+{\mathit{\lambda }}^{\top }\mathbf{c}(\mathbf{u})+{\mathbf{p}}^{\top }f(\mathbf{x},\mathbf{u})$, with respect to u (treating x, u and p as independent variables) under the assumption that $\dot{\mathbf{x}}=f(\mathbf{x},\mathbf{u})$, x(0) = x₀, is the controlled biological model (5.2) and x, u and p satisfy the costate system (5.4) with $h(\mathbf{x}(T))={\sum }_{i=1}^3\left(S_p^{(i)}(0)-S_p^{(i)}(T)\right)$.

The complexity of this minimization problem largely depends on the choice of the control cost function, c(u), in the objective functional (5.3). In the numerical algorithm below, we employ a twice continuously differentiable cost function, $\mathbf{c}(\mathbf{u})≔{[c(u^{(1)}),c(u^{(2)}),c(u^{(3)})]}^{\top }$, with the following key properties (Smirnova et al., 2024; Smirnova & Ye, 2024):

• •c(u), are defined in $\mathcal{D}\supseteq \left(\left[0,1\right)\right)$ ensuring that each domain contains the feasible set;
• •c(0) = 0, implying zero cost when no control is applied;
• •${\lim }_{u\to 1^{-}}c(u)=\infty$, guaranteeing that the cost becomes prohibitive as u approaches the upper bound of control range;
• •$\frac{dc}{du}>0$ when u > 0, and $\frac{dc}{du}<0$ when u < 0, suggesting that c(u) is increasing for positive control values and preventing u from becoming negative;
• •$\frac{d^{2}c}{d{u}^2}>0$ for all $u\in \mathcal{D}$, indicating that c(u) is strictly convex.

Note that c(u) satisfying the above conditions are necessarily nonnegative in their domain, $\mathcal{D}$. In numerical simulations presented in this paper, we consider the following three cost functions (Smirnova & Ye, 2024):

$$c_1(u)=-u-\ln (1-u),c_2(u)=-\ln (1-u^2),\text{and}{c}_3(u)=-u\ln (1-u).$$ (5.6)

It is important to mention that in many optimal control problems, the function c(u) = u² is used to measure the cost associated with control u(t) (Diagne et al., 2024; Tuncer et al., 2022). For this function all partial derivatives of the Hamiltonian with respect to u⁽ⁱ⁾, i = 1, 2, 3, are linear in u⁽ⁱ⁾, thus leading to a straightforward minimization of the Hamiltonian with respect to u. Yet, the requirement ${\lim }_{u\to 1^{-}}c(u)=\infty$ is not met for c(u) = u². In our experiments, this requirement, along with other assumptions listed above, proves to be crucial because it guarantees that the global minimum of H(x, u, p) with respect to u subject to state and costate systems is feasible, i.e., the inequality constraints, u⁽ⁱ⁾ ≥ 0 and u⁽ⁱ⁾ < 1, don't have to be enforced in the optimization algorithm. From practical standpoint, condition ${\lim }_{u\to 1^{-}}c(u)=\infty$ ensures that ”ultimate” control gets infinitely expensive (and, consequently, impossible) as it is usually the case in real life. As such, this is the right property to enforce. For c₁(u), c₂(u), and c₃(u), all aforementioned assumptions are fulfilled, and the global minimum of H(x, u, p) with respect to u stays between 0 and 1 as shown below.

As one concludes from (5.2), (5.3) and (5.5), the Hamiltonian of the objective functional (5.3) combined with the controlled model (5.2) is given by the equation:

$$\begin{array}{ll}H(\mathbf{x},\mathbf{u},\mathbf{p})& =\sum _{i=1}^3\left(p_1^{\left(i\right)}\left(t\right)\left[\delta C_p^{\left(i\right)}\left(t\right)-S_p^{\left(i\right)}\left(t\right)\left\{{\beta }^{\left(i\right)}\left(t\right)(1-u^{\left(i\right)}\left(t\right))I_w\left(t\right)+\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}\right\}\right]\right)\\ & +p_2^{\left(i\right)}\left(t\right)\left[S_p^{\left(i\right)}\left(t\right)\left\{{\beta }^{\left(i\right)}\left(t\right)(1-u^{\left(i\right)}\left(t\right))I_w\left(t\right)+\sum _{j=1}^3{\alpha }^{(i,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}\right\}-\phi I_p^{\left(i\right)}\left(t\right)\right]\\ & \left(+p_3^{\left(i\right)}\left(t\right)\left[\phi I_p^{\left(i\right)}\left(t\right)-\delta C_p^{\left(i\right)}\left(t\right)\right]+{\lambda }^{\left(i\right)}c\left(u^{\left(i\right)}\right)+\delta C_p^{\left(i\right)}\left(t\right)\right).\end{array}$$ (5.7)

Taking into account Pontryagin's Minimum Principle (Lenhart & Workman, 2007; Pontryagin, 2018), we conclude that the costate system of equation (5.4) takes the form: $p_1^{(k)}(T)=-1$, $p_2^{(k)}(T)=p_3^{(k)}(T)=0$,

$$\begin{array}{ll}\frac{d{p}_1^{\left(k\right)}}{dt}& =\left\{{\beta }^{\left(k\right)}\left(t\right)(1-u^{\left(k\right)}\left(t\right))I_w\left(t\right)+\sum _{j=1}^3{\alpha }^{(k,j)}\frac{I_p^{\left(j\right)}\left(t\right)}{N_p^{\left(j\right)}}\right\}\left[p_1^{\left(k\right)}\left(t\right)-p_2^{\left(k\right)}\left(t\right)\right]\\ \frac{d{p}_2^{\left(k\right)}}{dt}& =\sum _{i=1}^3{\alpha }^{(i,k)}\frac{S_p^{\left(i\right)}\left(t\right)\left[p_1^{\left(i\right)}\left(t\right)-p_2^{\left(i\right)}\left(t\right)\right]}{N_p^{\left(k\right)}}+\phi \left[p_2^{\left(k\right)}\left(t\right)-p_3^{\left(k\right)}\left(t\right)\right]\\ \frac{d{p}_3^{\left(k\right)}}{dt}& =\delta \left[p_3^{\left(k\right)}\left(t\right)-p_1^{\left(k\right)}\left(t\right)-1\right],k=1,2,3.\end{array}$$ (5.8)

To find $\mathbf{u}\left(t\right)={argmin}_{\mathbf{v}\in \mathcal{U}}H(\mathbf{x}\left(t\right),\mathbf{v}\left(t\right),\mathbf{p}\left(t\right))$, subject to equations (5.2), (5.8), we note that ${{\partial }_{\mathbf{u}}H(\mathbf{x},\mathbf{u},\mathbf{p})}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)}=\left[{\left(\frac{\partial H(\mathbf{x},\mathbf{u},\mathbf{p})}{\partial u^{(1)}}\right|}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)},{\left(\frac{\partial H(\mathbf{x},\mathbf{u},\mathbf{p})}{\partial u^{(2)}}\right|}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)},{\left(\frac{\partial H(\mathbf{x},\mathbf{u},\mathbf{p})}{\partial u^{(3)}}\right|}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)}\right]$,

$${\left(\frac{\partial H(\mathbf{x},\mathbf{u},\mathbf{p})}{\partial u^{(k)}}\right|}_{\mathbf{x}(t),\mathbf{u}(t),\mathbf{p}(t)}={\beta }^{(k)}(t)S_p^{(k)}(t)I_w(t)\left(p_1^{(k)}(t)-p_2^{(k)}(t)\right)+{\lambda }^{(k)}{\left(c^{\prime }(u^{(k)})\right|}_{u^{(k)}(t)}.$$ (5.9)

To discretize u(t), we project each u⁽ⁱ⁾(t) onto the subspace spanned by shifted Legendre polynomials, P_j(t), of degree j = 0, 1, …, n − 1, defined on [0, T]. Our experiments suggest that for 42 ≤ n ≤ 47, the reconstructed control functions and the corresponding data fit are virtually identical. If n < 42, the fit is less accurate due to a higher discretization error, while for n > 47, the numerical algorithm becomes unstable. Taking this into account, we use n = 45 for every u⁽ⁱ⁾(t), which yields Levenberg-Marquardt minimization algorithm for the optimal control problem (5.2), (5.3), (5.8) in the following form:

Algorithm 1

Numerical method for solving the optimal control problem

Require: Cost function c(u), weight λ, finite dimensional approximation u[θ], initial guess θ.

Ensure: Optimal control u[θ] with estimated θ.

repeat

Solve (5.2) for x forward in time.

Solve (5.8) for p backward in time.

$\theta \leftarrow \theta -\varrho {(J^{\top }(\theta )J(\theta )+{ϵ}_{k}E)}^{-1}{J}^{\top }(\theta )F(\theta )$.

until converged.

In Algorithm 1 above, θ are the expansion coefficients for the control function, u(t), F(θ) is a discrete analog of ∂_uH(x,u,p)^⊤, J(θ) is the Jacobian of F(θ), E is the identity matrix in the solution space, ϱ is the step size, and ɛ_k is the regularization sequence. We point out that ∂_uH(x, u, p) exists, since c_i(u), i = 1, 2, 3, are twice continuously differentiable by our assumption. In all simulation results presented in this section, Matlab built-in function ”ode23s” was employed to solve both ODE systems, (5.2) and (5.8), while ”lsqnonlin” implemented the trust-region optimization procedure.

Table 5 illustrates cumulative incidence reduction for the state of California using cost of control functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u) with different weights, λ⁽ⁱ⁾, i = 1, 2, 3. With a higher cost of control, the drop in cumulative cases, as compared to the actual reported data, is small. One can see from Table 5 that λ⁽¹⁾ = 1.3, λ⁽²⁾ = 0.47 and λ⁽³⁾ = 4.5 yield a high cost of control and a relatively small case reduction, while a lower cost of control, with λ⁽¹⁾ = 0.03, λ⁽²⁾ = 0.04 and λ⁽³⁾ = 0.15, allows to reduce the total number of cases by nearly 90%. Table 6 provides similar information for the state of Minnesota. Low cost of control, such as, for example, λ⁽¹⁾ = 0.05, λ⁽²⁾ = 0.05 and λ⁽³⁾ = 0.40, gives the reduction at the level of about 90%. At the same time, larger values of λ⁽ⁱ⁾, for instance, λ⁽¹⁾ = 1.40, λ⁽²⁾ = 1.00 and λ⁽³⁾ = 15.00, cause a much lower case drop (20%, on average).

Table 5.

California: cumulative incidence reduction for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²), and c₃(u) = −u ln(1 − u) under different weights λ⁽ⁱ⁾, i = 1, 2, 3.

Weight functions	Control Cost 1	Control Cost 2	Control Cost 3
λ(1) = 0.03, λ(2) = 0.04, λ(3) = 0.15	90.64%	90.85%	89.61%
λ(1) = 0.08, λ(2) = 0.09, λ(3) = 0.30	87.43%	85.43%	83.96%
λ(1) = 0.21, λ(2) = 0.11, λ(3) = 0.60	79.20%	78.82%	75.27%
λ(1) = 0.34, λ(2) = 0.16, λ(3) = 1.00	72.12%	70.35%	64.19%
λ(1) = 0.44, λ(2) = 0.22, λ(3) = 1.40	66.53%	63.30%	56.84%
λ(1) = 0.55, λ(2) = 0.29, λ(3) = 2.00	52.86%	46.42%	40.18%
λ(1) = 0.73, λ(2) = 0.34, λ(3) = 2.90	47.75%	41.14%	31.94%
λ(1) = 0.93, λ(2) = 0.40, λ(3) = 3.60	40.35%	32.72%	23.27%
λ(1) = 1.30, λ(2) = 0.47, λ(3) = 4.50	31.42%	22.84%	12.94%

Table 6.

Minnesota: cumulative incidence reduction for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²), and c₃(u) = −u ln(1 − u) under different weights λ⁽ⁱ⁾, i = 1, 2, 3.

Weight functions	Control Cost 1	Control Cost 2	Control Cost 3
λ(1) = 0.05, λ(2) = 0.05, λ(3) = 0.40	93.54%	90.30%	92.30%
λ(1) = 0.10, λ(2) = 0.10, λ(3) = 0.90	89.42%	83.24%	82.03%
λ(1) = 0.13, λ(2) = 0.11, λ(3) = 2.00	78.01%	74.44%	68.97%
λ(1) = 0.20, λ(2) = 0.20, λ(3) = 3.30	67.04%	63.09%	57.30%
λ(1) = 0.34, λ(2) = 0.35, λ(3) = 4.50	59.36%	53.99%	47.84%
λ(1) = 0.50, λ(2) = 0.45, λ(3) = 6.20	51.71%	45.17%	38.85%
λ(1) = 0.70, λ(2) = 0.60, λ(3) = 8.00	45.02%	37.39%	31.29%
λ(1) = 1.00, λ(2) = 0.80, λ(3) = 10.50	37.69%	29.27%	23.60%
λ(1) = 1.40, λ(2) = 1.00, λ(3) = 15.00	29.10%	19.98%	15.44%

In the state of California, the values of λ⁽ⁱ⁾ for the 90% reduction are, roughly, one third of the corresponding values of λ⁽ⁱ⁾ for the 80% reduction. In Minnesota, reducing each λ⁽ⁱ⁾ by half, increases the case reduction from about 80% to 90%. In group 3, California and Minnesota have 26 and 24 poultry farms, respectively, with the inventory of more than 100,000 birds. There are also 56 and 36 farms, respectively, with the flock size from 400 to 3199. In light of that, the same percentage reduction in farm group 3 comes at a considerably higher cost (that is, at a considerably higher value of λ⁽³⁾) as compared to the reduction in groups 1 and 2.

Fig. 17, Fig. 19 show the optimal control functions, u⁽¹⁾(t), u⁽²⁾(t) and u⁽³⁾(t), for the control cost functions, c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u) that allow to reduce cumulative incidence over the total period of time by about 50-60% in the states of California and Minnesota. For California, this level of control is reached with weights λ⁽¹⁾ = 0.44, λ⁽²⁾ = 0.22 and λ⁽³⁾ = 1.4, while for Minnesota, a 50% drop is achieved with λ⁽¹⁾ = 0.34, λ⁽²⁾ = 0.35 and λ⁽³⁾ = 4.5, which can be explained by a higher number of cases in the state of Minnesota in all three farm groups, see Fig. 16, Fig. 18.

Fig. 17.

California: optimal control functions u⁽¹⁾(t), u⁽²⁾(t) and u⁽³⁾(t) for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u), and weights λ⁽¹⁾ = 0.44, λ⁽²⁾ = 0.22 and λ⁽³⁾ = 1.40.

Fig. 19.

Minnesota: optimal control functions u⁽¹⁾(t), u⁽²⁾(t) and u⁽³⁾(t) for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u), and weights λ⁽¹⁾ = 0.34, λ⁽²⁾ = 0.35 and λ⁽³⁾ = 4.5.

Fig. 16.

California: cumulative incidence for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u), and weights λ⁽¹⁾ = 0.44, λ⁽²⁾ = 0.22 and λ⁽³⁾ = 1.40 versus real data.

Fig. 18.

Minnesota: cumulative incidence for control cost functions c₁(u) = −u − ln(1 − u), c₂(u) = − ln(1 − u²) and c₃(u) = −u ln(1 − u), and weights λ⁽¹⁾ = 0.34, λ⁽²⁾ = 0.35 and λ⁽³⁾ = 4.5 versus real data.

Overall, our numerical study confirms that separating wild birds from poultry is an effective control strategy. Blocking contacts with wild birds helps to avoid (or at least reduce) the need for reactive culling of the entire affected flock. It brings down the number of poultry outbreaks thus making the virus more manageable and localized by suppressing HPAI transmission from wild birds’ feathers, droppings, and saliva. This helps to maintain overall flock health and prevent potential human exposure.

6. Discussion and future plans

As of February 2025, the U.S. Department of Agriculture (USDA) authorized the use of conditional vaccines, but their implementation is still limited due to international trade regulations (USDA conditionally approves H5N1 poultry, 2025). In the absence of prevalent vaccination, it is crucial to contain the initial introduction of HPAI virus from the infected wildlife.

In our study, we introduce a partially stochastic network compartmental model (2.1) to ascertain the progression and potential containment of HPAI virus in commercial flocks and wildlife. Parameters of the model get estimated using available data on wild bird migration (U.S. Department of Agriculture & Animal and Plant Health Inspection Service, 2025c), HPAI poultry outbreaks (CDC Avian Influenza (Bird Flu), 2025), and poultry farm inventory (U.S. Department of Agriculture & Census of Agriculture, 2022) in the U.S. The new model simulates HPAI virus transmission driven by wildlife dynamic, seasonality, and farm-to-farm relations. Unlike many prior global models, this framework is closely tailored to the U.S. HPAI statistics, farm structure, and current mitigation practices. The proposed network model, along with HPAI surveillance data, is used to analyze optimal control strategies aimed at lowering HPAI spread from wild birds to poultry. Our numerical experiments illustrate that this control strategy is very powerful. Relatively inexpensive separation measures, such as covered runs and secure housing, help to prevent environmental contamination and the risk of HPAI transmission to domestic birds, thus protecting the flock and bypassing depopulation. Additionally, the experiments illustrate that in order to reduce cumulative incidence in all farm groups, 75-80% of resources must be dedicated to enhanced protection of group 3 (large farms, including layer farms).

To successfully optimize mitigation and prevention, an important topic of future research is the development of reliable forecasting algorithms capable to provide short-term, spatially resolved prognoses to enable timely decisions about movement restrictions, resource allocation, and targeted surveillance around infected premises. Models that can link real-time wild bird influenza pulses with subsequent poultry outbreaks are especially valuable for early warnings, because they can turn wildlife surveillance data into actionable risk estimates for different regions in the United States (Malek & Hoque, 2021; Peacock et al., 2025; Spackman et al., 2023; Van Borm et al., 2025).

To generate operational forecasts, mechanistic modeling needs to be combined with statistical and time-series approaches to processing HPAI incidence data. These methods can assimilate noisy surveillance counts and provide probabilistic predictions of future outbreak trajectories. When calibrated to the U.S. data on wild-bird detections and poultry outbreaks, these forecasting tools can complement mechanistic frameworks and help to identify regions where the risk of explosive spread is the highest and provide actionable insights for outbreak preparedness.

CRediT authorship contribution statement

Hamed Karami: Validation, Software, Methodology, Investigation. Sifur Safuka Chowdhury: Visualization, Investigation, Data curation. Alexandra Smirnova: Writing – review & editing, Writing – original draft, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization.

Conflict of interest

The authors have no conflict of interest.

Handling Editor: Dr Daihai He