Mathematical Modeling of Avian Influenza Transmission with Neural Network-Based Simulation

Abstract

Avian influenza (AI) remains a serious threat to poultry and public health worldwide due to its zoonotic nature and pandemic potential. This paper develops and analyzes a coupled system of nonlinear ordinary differential equations and an SEIR-SEIR model that describes the transmission dynamics of avian influenza in both human and bird populations. The model incorporates multiple transmission routes (bird-to-bird, bird-to-human, human-to-human), exposed/latent compartments in both hosts, disease-induced mortality, and demographic processes. From a mathematical perspective, we present a rigorous analysis of this eight-dimensional dynamical system. We prove positivity and boundedness of solutions in ${\mathbb{R}}_{+}^8$, characterize the equilibrium points, and derive the basic reproduction numbers $R_0^b$ and $R_0^h$ using the next-generation matrix method. Local asymptotic stability of the disease-free equilibrium is established via the Routh–Hurwitz criterion. A composite Lyapunov function is constructed to prove global asymptotic stability when both reproduction numbers are less than unity—a result that exploits the cascade structure of the system and provides a template for analyzing similar multi-host models. Sensitivity analysis using normalized forward sensitivity indices identifies critical parameters. In addition, we use neural network models to validate both models and provide error analysis. These results emphasize the crucial role of controlling cross-species transmission and improving recovery efforts, which have significant implications for the design of effective intervention and surveillance programs in the context of the One Health framework.

1. Introduction

Avian influenza, also known as bird flu, is a highly contagious viral disease among avian species, which has resulted in significant economic losses in the poultry sector because of high mortality rates and culling of flocks [1]. However, the economic significance of avian influenza is not the only public health concern; it is also zoonotic, capable of transmitting from birds to humans [2]. The highly pathogenic H5N1 influenza A viruses continue to be endemic in bird populations, and human cases continue to increase [3]. The biggest threat is the possibility of the virus mutating and becoming capable of efficient human-to-human transmission, which would increase the risk of a global pandemic [4]. The biggest pandemic in recorded history occurred between 1918 and 1919, when there were 21 million reported deaths worldwide. It was one of the most deadly events in recorded history. After that, there were three other pandemics in the 20th century: the 1957 H2N2 pandemic, the 1968 H3N2 pandemic, and the 2009 influenza A (H1N1) virus (pH1N1) pandemic [5]. From 1 January to 1 July 2025, Cambodia’s International Health Regulations (IHR) National Focal Point (NFP) notified the World Health Organization (WHO) of 11 laboratory-confirmed cases of avian influenza A(H5N1) virus human infection. Seven of the 11 cases occurred in June, a rare monthly surge [6]. According to the WHO, from 2003 to 2025, a total of 948 laboratory-confirmed human cases of avian influenza A(H5N1) have been reported globally, of which 464 were fatal (case fatality rate of approximately 49%). The economic losses from avian influenza outbreaks between 2003 and 2020 were estimated at over $20 billion USD. In 2024 alone, more than 50 million poultry birds were culled in Europe and North America due to H5N1 outbreaks, marking one of the largest avian influenza epidemics in history [1,7].

Recent data highlight the ongoing threat and emphasize the urgent need for effective tools to understand and predict the spread of avian influenza. In this regard, mathematical modeling provides a powerful framework to study disease dynamics and assess intervention strategies [8]. Compartmental models, particularly the $SIR$ and $SEIR$ frameworks, have been widely applied in epidemiology, as they capture essential processes such as infection, incubation, transmission, recovery, and disease-induced mortality [9]. These models allow researchers to simulate outbreak scenarios, determine critical thresholds for disease persistence, and evaluate the potential impact of control measures [10].

Mathematical modeling serves as an indispensable tool for understanding and controlling infectious disease outbreaks [8,11,12,13]. By translating biological processes into mathematical equations, these models enable researchers to predict disease trajectories, identify critical intervention points, and evaluate the potential impact of control strategies before implementation [14]. Furthermore, mathematical models provide quantitative frameworks for estimating key epidemiological parameters, such as the basic reproduction number $R_0$, which determine whether an outbreak will persist or die out [15]. This predictive capability is essential for public health authorities to allocate resources effectively, design optimal intervention policies, and prepare for potential pandemics.

The development of mathematical models for avian influenza has progressed significantly over time [16]. Early efforts were based on classical epidemic models such as SIR and SEIR, introduced by Kermack and McKendrick in 1927 [17]. With the emergence of H5N1 in 1997 and widespread outbreaks in the early 2000s, disease-specific models were developed to estimate transmission dynamics and test strategies like culling, vaccination, and movement restrictions [18]. As human cases appeared, models expanded to include cross-species transmission and pandemic risk. Later, network and spatial models incorporated poultry trade, live bird markets, and wild bird migration to capture regional spread [7]. Today, avian influenza models follow a One Health perspective, linking birds, humans, and the environment, and play a vital role in predicting outbreaks, guiding surveillance, and shaping effective control strategies [19].

Recently, fractional-order delay differential epidemic models have gained significant attention due to their ability to capture memory effects and time delays inherent in disease transmission, with stability and bifurcation analysis revealing complex dynamical behaviors such as Hopf bifurcations and chaos [20]. Additionally, neural network-based numerical approaches, particularly Physics-Informed Neural Networks (PINNs), have emerged as powerful tools for solving and validating complex epidemic models [21,22]. These advancements complement traditional ODE models and provide new avenues for understanding disease dynamics.

M. Derouich [23] proposed an integrated two-population model that links human and bird populations through zoonotic transmission. In this framework, the human population is represented by an SIR-type structure, incorporating recruitment, natural mortality, recovery, and disease-induced death. The bird population, on the other hand, is modeled using two compartments: susceptible and infectious. The overall system is formulated as follows.

Human population

$$\left\{\begin{array}{c}{\displaystyle \frac{dS}{dt}}=\Lambda -\left(\mu +{\displaystyle \frac{\beta I_0}{N_0}}\right)S+\delta R,\hfill \\ {\displaystyle \frac{dI}{dt}}={\displaystyle \frac{\beta I_0}{N_0}}S-(\mu +\gamma +\alpha )I,\hfill \\ {\displaystyle \frac{dR}{dt}}=\gamma I-(\mu +\delta )R.\hfill \end{array}\right.$$

Bird population

$$\left\{\begin{array}{c}{\displaystyle \frac{d{S}_0}{dt}}={\mu }_0{N}_0-\left({\mu }_0+{\displaystyle \frac{{\beta }_0{I}_0}{N_0}}\right)S_0,\hfill \\ {\displaystyle \frac{d{I}_0}{dt}}={\displaystyle \frac{{\beta }_0{I}_0}{N_0}}{S}_0-{\mu }_0{I}_0.\hfill \end{array}\right.$$

2. Problem Formulation

To describe the transmission dynamics of avian influenza between bird and human populations, we divide each population into four epidemiological compartments.

For the bird population, the susceptible class $S_b\left(t\right)$ consists of birds that are free from infection but are at risk of infection through contact with infectious birds. The exposed class $E_b\left(t\right)$ represents infected birds in the incubation period who are not yet infectious. The infectious class $I_b\left(t\right)$ includes birds capable of transmitting the disease. The recovered class $R_b\left(t\right)$ consists of birds that have recovered and acquired immunity.

For the human population, the susceptible class $S_h\left(t\right)$ represents individuals who are not infected but may acquire the disease through contact with infectious birds or infectious humans. The exposed class $E_h\left(t\right)$ consists of infected individuals in the incubation stage who are not yet infectious. The infectious class $I_h\left(t\right)$ includes individuals capable of transmitting the infection. The recovered class $R_h\left(t\right)$ represents individuals who have recovered and developed immunity.

The model is unidirectionally coupled from birds to humans, meaning that infectious birds transmit the virus to susceptible humans through zoonotic transmission, while humans do not transmit the infection back to birds. This assumption is consistent with the epidemiology of avian influenza, where birds act as reservoir hosts.

The proposed model is given by the following system of ordinary differential equations.

2.1. Bird Population

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_b}{dt}}& ={\Pi }_b-{\beta }_b{I}_b{S}_b-{\mu }_b{S}_b,\hfill \\ \hfill {\displaystyle \frac{d{E}_b}{dt}}& ={\beta }_b{I}_b{S}_b-({\sigma }_b+{\mu }_b)E_b,\hfill \\ \hfill {\displaystyle \frac{d{I}_b}{dt}}& ={\sigma }_b{E}_b-({\gamma }_b+{\mu }_b+{\alpha }_b)I_b,\hfill \\ \hfill {\displaystyle \frac{d{R}_b}{dt}}& ={\gamma }_b{I}_b-{\mu }_b{R}_b.\hfill \end{array}$$

(1)

2.2. Human Population

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{S}_h}{dt}}& ={\Pi }_h-\left({\beta }_{bh}{I}_b+{\beta }_h{I}_h+{\mu }_h\right)S_h,\hfill \\ \hfill {\displaystyle \frac{d{E}_h}{dt}}& =\left({\beta }_{bh}{I}_b+{\beta }_h{I}_h\right)S_h-({\sigma }_h+{\mu }_h)E_h,\hfill \\ \hfill {\displaystyle \frac{d{I}_h}{dt}}& ={\sigma }_h{E}_h-({\gamma }_h+{\mu }_h+{\alpha }_h)I_h,\hfill \\ \hfill {\displaystyle \frac{d{R}_h}{dt}}& ={\gamma }_h{I}_h-{\mu }_h{R}_h.\hfill \end{array}$$

(2)

The initial conditions are given by

$$S_b\left(0\right),S_h\left(0\right)>0,E_b\left(0\right),I_b\left(0\right),R_b\left(0\right),E_h\left(0\right),I_h\left(0\right),R_h\left(0\right)\ge 0.$$

(3)

2.3. Compartmental Flow Diagram and Flow Chart of the Model

This section present the compartmental flow diagram showing in Figure 1 while the flow chart diagram in Figure 2 which has showing the entire methodology for the aforementioned model along with description of the used parameters in

Table 1. Model parameters, descriptions, units, and biological justification.

Symbol	Explanation of Parameters	Units
${\Pi }_b$	Recruitment rate of birds into $S_b$. Inflow of new susceptible birds via birth, hatching, or restocking. Constant rate reflecting steady supply from farming.	birds/day
${\mu }_b$	Natural death rate of birds. Per-capita natural mortality excluding disease. Reciprocal $1/{\mu }_b$ is the average natural lifespan of a bird.	1/day
${\beta }_b$	Bird-to-bird transmission rate. Effective contact rate for transmission between infectious and susceptible birds. Higher in crowded farms or with virulent strains.	1/(bird·day)
${\sigma }_b$	Progression rate $E_b\to I_b$ in birds. Rate at which exposed birds become infectious.	1/day
${\gamma }_b$	Recovery rate of infectious birds. Rate at which infectious birds recover. Target for treatment and culling interventions.	1/day
${\alpha }_b$	Disease-induced death rate for $I_b$. Additional mortality caused by infection. High for highly pathogenic avian influenza (HPAI).	1/day
${\Pi }_h$	Recruitment rate of humans into $S_h$. Inflow of new susceptible humans via birth or immigration. Balances mortality at demographic equilibrium.	humans/day
${\mu }_h$	Natural mortality rate in humans. Per-capita natural death rate excluding disease-induced mortality.	1/day
${\beta }_{bh}$	Bird-to-human transmission coefficient. Zoonotic transmission rate from infectious birds to susceptible humans. Depends on human–poultry contact frequency. Couples both populations.	1/(bird·day)
${\beta }_h$	Human-to-human transmission rate. Effective contact rate between infectious and susceptible humans. Key indicator of pandemic potential.	1/(human·day)
${\sigma }_h$	Progression rate $E_h\to I_h$ in humans. Rate at which exposed humans become infectious. Important for quarantine strategies.	1/day
${\gamma }_h$	Recovery rate of infectious humans. Rate at which infectious humans recover. Can be increased by antiviral treatment.	1/day
${\alpha }_h$	Disease-induced mortality rate in humans. Additional mortality due to avian influenza infection. Represents potentially preventable deaths.	1/day

.

2.4. Improvements over Previous Models

Compared to the SIR–SI framework proposed by Derouich and Boutayeb [23], the present model introduces the following improvements:

• Exposed compartments ($E_b$, $E_h$): The inclusion of latent classes accounts for the incubation period observed in avian influenza, improving the realism of disease progression.
• Recovered class for birds ($R_b$): Unlike earlier models, recovered birds are explicitly considered, allowing analysis of immunity and recovery.
• Human-to-human transmission (${\beta }_h$): This component captures limited but documented transmission among humans and enables assessment of potential outbreak scenarios.
• Refined zoonotic transmission (${\beta }_{bh}$): The bird-to-human transmission pathway is explicitly modeled within the SEIR structure.
• Disease-induced mortality (${\alpha }_b$, ${\alpha }_h$): Mortality due to infection is incorporated for both populations.

2.5. Biological Justification of Model Components

The assumptions of the model are supported by epidemiological observations:

• Bird-to-bird transmission (${\beta }_b$): Occurs through direct contact and contaminated environments.
• Bird-to-human transmission (${\beta }_{bh}$): Arises from exposure to infected birds or contaminated surroundings.
• Human-to-human transmission (${\beta }_h$): Limited transmission has been observed in close-contact settings.
• Incubation periods ($1/{\sigma }_b$, $1/{\sigma }_h$): Reflect the delay between infection and infectiousness.

The model therefore provides a comprehensive framework for analyzing avian influenza transmission under a One Health perspective.

2.6. Methodological Workflow

Figure 2 presents a schematic overview of the methodology employed in this study. The workflow consists of six sequential steps: (i) model formulation of the coupled SEIR-SEIR system (Equations (1) and (2)); (ii) analytical analysis including positivity, boundedness, equilibrium points, and derivation of the basic reproduction numbers $R_0^b$ and $R_0^h$ using the next-generation matrix method; (iii) stability analysis establishing local asymptotic stability via the Routh–Hurwitz criterion and global asymptotic stability via composite Lyapunov functions; (iv) numerical simulations using MATLAB ode45 to validate analytical findings; (v) sensitivity analysis using normalized forward sensitivity indices to identify key intervention targets; and (vi) a neural network consistency check trained on ode45 solutions to confirm numerical accuracy. Finally, the results are interpreted to provide epidemiological insights and policy recommendations within the One Health framework.

3. Basic Properties of the Model

The topics of positivity, boundedness, reproductive number, stability analysis, sensitivity analysis, and numerical simulation are covered in this section. Before presenting the said properties of the model, we give some definitions which explore the idea of neural network (NN) and the need for its application to the said model along with a fractional operator.

Definition 1 

(Neural Network Performance). In the concept of the neural network, the performance refers to the model’s goal achievements in the context of accuracy and by best approaches. For the performances, the accuracy, mean square error, absolute error, recalling, and precision are required for the concerned data. Furthermore, for improvement in performance, the training, networking design, and placement of the hyperparameter should be modified.

Definition 2 

(Average Square Error). The measurement of efficiency is termed as the mean square error. It is used for the computation of the mean square deviation between the exact and estimated or approximate result which implies the correctness of network predictions.

Definition 3 

($Mu\left(\mu \right)$). For the rate of learning or the step size interval, we usually chose μ. A hyperparameter is used for the controlling biases and weight and age when network variation occurs in the learning rate of the training. The learning rate μ refers to the magnitude of step sizes in loss quantities in the optimization technique under gradient.

3.1. Positivity

Theorem 1. 

Let the initial conditions of the system be $(S_b\left(0\right),E_b\left(0\right),I_b\left(0\right),R_b\left(0\right),S_h\left(0\right),E_h\left(0\right),$$I_h\left(0\right),R_h\left(0\right))\in {\mathbb{R}}_{+}^8$. Then the solutions of the system remain in the positive orthant ${\mathbb{R}}_{+}^8$ for all $t>0$.

Proof. 

To establish the non-negativity of the solutions, we apply the results for essentially non-negative systems as established by De Leenheer and Aeyels [24]. A dynamical system $\dot{x}=f\left(x\right)$ is essentially non-negative if, for every i, $f_i\left(x\right)\ge 0$ whenever $x\ge 0$ and $x_i=0$. This condition ensures that the positive orthant ${\mathbb{R}}_{+}^8$ is a forward invariant.

We evaluate the direction of the vector field on each bounding hyperplane:

$$\left\{\begin{array}{c}\dot{S_b}{|}_{S_b=0}={\Pi }_b>0,\hfill \\ \dot{E_b}{|}_{E_b=0}={\beta }_b{S}_b{I}_b\ge 0,\hfill \\ \dot{I_b}{|}_{I_b=0}={\sigma }_b{E}_b\ge 0,\hfill \\ \dot{R_b}{|}_{R_b=0}={\gamma }_b{I}_b\ge 0,\hfill \\ \dot{S_h}{|}_{S_h=0}={\Pi }_h>0,\hfill \\ \dot{E_h}{|}_{E_h=0}=({\beta }_h{I}_h+{\beta }_{bh}{I}_b)S_h\ge 0,\hfill \\ \dot{I_h}{|}_{I_h=0}={\sigma }_h{E}_h\ge 0,\hfill \\ \dot{R_h}{|}_{R_h=0}={\gamma }_h{I}_h\ge 0.\hfill \end{array}\right.$$

Given that the right-hand side functions are locally Lipschitz continuous in ${\mathbb{R}}^8$ and the vector field at the boundaries of the positive orthant points inward ($\dot{x_i}\ge 0$ when $x_i=0$), the region ${\mathbb{R}}_{+}^8$ is an invariant set. Therefore, all trajectories starting in ${\mathbb{R}}_{+}^8$ remain non-negative for all $t>0$. □

3.2. Boundedness and Positive Invariance

Theorem 2. 

The region

$$D=\left\{(S_b,E_b,I_b,R_b,S_h,E_h,I_h,R_h)\in {\mathbb{R}}_{+}^8:N_b\le {\displaystyle \frac{{\Pi }_b}{{\mu }_b}},N_h\le {\displaystyle \frac{{\Pi }_h}{{\mu }_h}}\right\}$$

(4)

is positively invariant and bounded for the systems (1) and (2).

Proof. 

Let $N_b=S_b+E_b+I_b+R_b$ and $N_h=S_h+E_h+I_h+R_h$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{d{N}_b}{dt}}={\Pi }_b-{\mu }_b{N}_b-{\alpha }_b{I}_b\le {\Pi }_b-{\mu }_b{N}_b,{\displaystyle \frac{d{N}_h}{dt}}={\Pi }_h-{\mu }_h{N}_h-{\alpha }_h{I}_h\le {\Pi }_h-{\mu }_h{N}_h.\end{array}$$

By comparison, we obtain

$$N_b\left(t\right)\le {\displaystyle \frac{{\Pi }_b}{{\mu }_b}},N_h\left(t\right)\le {\displaystyle \frac{{\Pi }_h}{{\mu }_h}}\mathrm{as}t\to \infty .$$

(5)

Since all state variables are non-negative, it follows that $0\le S_b,E_b,I_b,R_b\le N_b$ and $0\le S_h,E_h,I_h,R_h\le N_h$; hence, all components are bounded.

Moreover, on the boundary $N_b={\displaystyle \frac{{\Pi }_b}{{\mu }_b}}$ and $N_h={\displaystyle \frac{{\Pi }_h}{{\mu }_h}}$, we have ${\displaystyle \frac{d{N}_b}{dt}}\le 0$ and ${\displaystyle \frac{d{N}_h}{dt}}\le 0$, implying that D is positively invariant. Hence, D is bounded and invariant. The result follows from standard dynamical systems arguments [25]. □

3.3. Basic Reproduction Number

Now we find the basic threshold number for our proposed problem (1) and (2). For this purpose, we use the next-generation method. We consider the infectious classes in the proposed model. Let $Y=(E_b,I_b,E_h,I_h)$.

$$\mathbf{f}=\left[\begin{array}{c}{\beta }_b{I}_b{S}_b\\ 0\\ {\beta }_{bh}{I}_b{S}_h+{\beta }_h{I}_h{S}_h\\ 0\end{array}\right],\mathbf{v}=\left[\begin{array}{c}({\sigma }_b+{\mu }_b)E_b\\ -{\sigma }_b{E}_b+({\gamma }_b+{\mu }_b+{\alpha }_b)I_b\\ ({\sigma }_h+{\mu }_h)E_h\\ -{\sigma }_h{E}_h+({\gamma }_h+{\mu }_h+{\alpha }_h)I_h\end{array}\right].$$

The Jacobian matrix of $\mathbf{F}$ at the DFE is

$$\begin{array}{c}\hfill \mathbf{F}=\left[\begin{array}{cccc}0& {\beta }_b{S}_b^0& 0& 0\\ 0& 0& 0& 0\\ 0& {\beta }_{bh}{S}_h^0& 0& {\beta }_h{S}_h^0\\ 0& 0& 0& 0\end{array}\right].\end{array}$$

The Jacobian matrix of $\mathbf{v}$ at the DFE is

$$\begin{array}{c}\hfill \mathbf{V}=\left[\begin{array}{cccc}({\sigma }_b+{\mu }_b)& 0& 0& 0\\ -{\sigma }_b& ({\gamma }_b+{\mu }_b+{\alpha }_b)& 0& 0\\ 0& 0& ({\sigma }_h+{\mu }_h)& 0\\ 0& 0& -{\sigma }_h& ({\gamma }_h+{\mu }_h+{\alpha }_h)\end{array}\right].\end{array}$$

The inverse of $\mathbf{V}$ is given by

$${\mathbf{V}}^{-1}=C\left[\begin{array}{cccc}{a}_2{a}_3{a}_4& 0& 0& 0\\ {\sigma }_b{a}_3{a}_4& a_1{a}_3{a}_4& 0& 0\\ 0& 0& a_1{a}_2{a}_4& 0\\ 0& 0& {\sigma }_h{a}_1{a}_2& a_1{a}_2{a}_3\end{array}\right],$$

(6)

where

$$\begin{array}{c}\hfill a_1=({\sigma }_b+{\mu }_b),a_2=({\gamma }_b+{\mu }_b+{\alpha }_b),a_3=({\sigma }_h+{\mu }_h),a_4=({\gamma }_h+{\mu }_h+{\alpha }_h),\end{array}$$

(7)

and

$$C={\displaystyle \frac{1}{\left|\mathbf{V}\right|}}={\displaystyle \frac{1}{a_1{a}_2{a}_3{a}_4}}={\displaystyle \frac{1}{({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}}.$$

(8)

The next-generation matrix is

$$\mathbf{F}{\mathbf{V}}^{-1}=\left[\begin{array}{cccc}{\displaystyle \frac{{\beta }_b{S}_b^0{\sigma }_b}{a_1{a}_2}}& {\displaystyle \frac{{\beta }_b{S}_b^0}{a_2}}& 0& 0\\ 0& 0& 0& 0\\ {\displaystyle \frac{{\beta }_{bh}{S}_h^0{\sigma }_b}{a_1{a}_2}}& {\displaystyle \frac{{\beta }_{bh}{S}_h^0}{a_3}}& {\displaystyle \frac{{\beta }_h{S}_h^0{\sigma }_h}{a_3{a}_4}}& {\displaystyle \frac{{\beta }_h{S}_h^0{\sigma }_h}{a_4}}\\ 0& 0& 0& 0\end{array}\right].$$

(9)

The basic reproduction number $R_0$ is the spectral radius of $\mathbf{F}{\mathbf{V}}^{-1}$:

$$R_0=\rho \left(\mathbf{F}{\mathbf{V}}^{-1}\right)=max\left\{{\displaystyle \frac{{\beta }_b{S}_b{\sigma }_b}{({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)}},{\displaystyle \frac{{\beta }_h{S}_h{\sigma }_h}{({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}}\right\}.$$

(10)

At the disease-free equilibrium, $S_b^0={\Pi }_b/{\mu }_b$ and $S_h^0={\Pi }_h/{\mu }_h$. Thus, the basic reproduction numbers for the bird and human populations are given by

$$R_0^b={\displaystyle \frac{{\beta }_b{\Pi }_b{\sigma }_b}{{\mu }_b({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)}},R_0^h={\displaystyle \frac{{\beta }_h{\Pi }_h{\sigma }_h}{{\mu }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}}.$$

(11)

The quantities $R_0^b$ and $R_0^h$ represent the average number of secondary infections generated by a single infected individual in the bird and human populations, respectively. These threshold parameters determine whether the disease will die out or persist in the population.

3.3.1. 3D Graphs of $R_0^h$

The parameter domains in Figure 3 are centered around the baseline values reported in

Table 2. Sensitivity indices of $R_h^0$ with parameter values and sources.

Parameter	Value	Source	SI
${\beta }_h$	$0.5$	[26]	$1.00$
${\Pi }_h$	100	[27]	$1.00$
${\sigma }_h$	$0.01$	Estimated	$0.038$
${\mu }_h$	$0.0004$	[28]	$-1.04$
${\gamma }_h$	$0.08$	[29]	$-0.72$
${\alpha }_h$	$0.03$	[26]	$-0.27$

. Each plot varies two parameters over a biologically plausible range while holding all other parameters constant at their baseline values.

Discussion of $R_0^h$ vs. Parameters

The plots reveal the sensitivity of $R_0^h$ to various parameters. As shown in Figure 3a–d, $R_0^h$ exhibits a direct and approximately linear relationship with parameters in the numerator, namely the contact rate (${\beta }_h$) and the recruitment rate (${\Pi }_h$). Increasing either of these parameters leads to a proportional increase in $R_0^h$. In contrast, $R_0^h$ is inversely proportional to the parameters in the denominator, such as the mortality rate (${\mu }_h$), recovery rate (${\gamma }_h$), and disease-induced death rate (${\alpha }_h$). Figure 3a,b, and especially Figure 3d, illustrate that as these parameters decrease, the value of $R_0^h$ increases sharply. The most dramatic effect is seen in Figure 3d, where a simultaneous decrease in ${\gamma }_h$ and ${\alpha }_h$ causes a sudden and significant spike in $R_0^h$, indicating a high risk of sustained disease transmission when recovery and mortality rates are low.

3.3.2. 3D Graphs of $R_0^b$

All parameter ranges in Figure 4 are chosen using baseline parameter values given in

Table 3. Sensitivity indices of $R_b^0$ with parameter values and sources.

Parameter	Value	Source	SI
${\beta }_b$	$0.4$	[29]	$1.00$
${\Pi }_b$	30	[29]	$1.00$
${\sigma }_b$	$0.01$	Estimated	$0.12$
${\mu }_b$	$0.00137$	[28]	$-1.10$
${\gamma }_b$	$0.3$	Assumed	$-0.68$
${\alpha }_b$	$0.137$	Assumed	$-0.31$

,where two parameters are varied over a biologically realistic range while the others remain fixed to their baseline values.

Discussion of $R_0^b$ vs. Parameters

The plots collectively illustrate how the basic reproduction number, $R_0^b$, for the bird population is influenced by its parameters. The value of $R_0^b$ is directly proportional to the contact rate (${\beta }_b$), the recruitment rate (${\Pi }_b$), and the maturation rate (${\sigma }_b$), as these are either in the numerator or contribute positively to the overall value. In contrast, $R_0^b$ is inversely proportional to parameters in the denominator, namely the natural mortality rate (${\mu }_b$), recovery rate (${\gamma }_b$), and disease-induced death rate (${\alpha }_b$). The plots confirm these relationships, showing sharp increases in $R_0^b$ as the inverse parameters approach zero. The most significant and linear increases are seen with the contact and recruitment rates, which directly drive the disease’s transmission potential.

3.4. Equilibrium Points

3.4.1. Disease-Free State

When a community is in a disease-free state (DFS), there is no disease infection. The symbol for it is $E^0$. The infectious classes are taken to be equal to zero in order to reach this state. We observe that the DFS is given by $E^0=(S_b^0,0,0,0,S_h^0,0,0,0)$, where

$$E^0=\left({\displaystyle \frac{{\Pi }_b}{{\mu }_b}},0,0,0,{\displaystyle \frac{{\Pi }_h}{{\mu }_h}},0,0,0\right).$$

(12)

3.4.2. Disease-Endemic State

Let $E^{\ast }=(S_b^{\ast },E_b^{\ast },I_b^{\ast },R_b^{\ast },S_h^{\ast },E_h^{\ast },I_h^{\ast },R_h^{\ast })$ represent the disease-endemic state, which indicates that the disease is present in the population. The endemic equilibrium point for the system is given by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill S_b^{\ast }& ={\displaystyle \frac{{\Pi }_b}{{\mu }_b{R}_0^b}},\hfill \\ \hfill E_b^{\ast }& ={\displaystyle \frac{({\gamma }_b+{\mu }_b+{\alpha }_b)}{{\sigma }_b}}{I}_b^{\ast },\hfill \\ \hfill I_b^{\ast }& ={\displaystyle \frac{{\mu }_b(R_0^b-1)}{{\beta }_b}},\hfill \\ \hfill R_b^{\ast }& ={\displaystyle \frac{{\gamma }_b}{{\mu }_b}}{I}_b^{\ast },\hfill \\ \hfill S_h^{\ast }& ={\displaystyle \frac{({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}{{\sigma }_h({\beta }_{bh}{I}_b^{\ast }+{\beta }_h{I}_h^{\ast })}}{I}_h^{\ast },\hfill \\ \hfill E_h^{\ast }& ={\displaystyle \frac{({\gamma }_h+{\mu }_h+{\alpha }_h)}{{\sigma }_h}}{I}_h^{\ast },\hfill \\ \hfill R_h^{\ast }& ={\displaystyle \frac{{\gamma }_h}{{\mu }_h}}{I}_h^{\ast }.\hfill \end{array}\right.\end{array}$$

(13)

Thus, $I_b^{\ast }>0$ if and only if $R_0^b>1$. When $R_0^b\le 1$, the only non-negative solution is the disease-free equilibrium.

The human infectious compartment $I_h$ satisfies the quadratic equation

$$a{I}_h^2+b{I}_h+c=0,$$

(14)

where

$$\begin{array}{c}\hfill a={\beta }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h),c=-{\Pi }_h{\sigma }_h{\beta }_{bh}{I}_b^{\ast },\end{array}$$

(15)

$$\begin{array}{c}\hfill b=({\beta }_{bh}{I}_b^{\ast }+{\mu }_h)({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)-{\Pi }_h{\sigma }_h{\beta }_h.\end{array}$$

(16)

Solving for $I_h$ will yield two possible equilibrium points (or one if the discriminant is zero).

Biological feasibility summary: For the bird subsystem, $I_b^{\ast }>0$ if and only if $R_0^b>1$; when $R_0^b\le 1$, the only non-negative solution is the disease-free equilibrium. The human infectious compartment $I_h^{\ast }$ satisfies a quadratic equation, which yields two possible equilibrium points (or one if the discriminant is zero). The endemic equilibrium exists and is biologically feasible (all components positive) if and only if $R_0^b>1$ and $R_0^h>1$. The expressions $S_b^{\ast }={\Pi }_b/\left({\mu }_b{R}_0^b\right)$ and $I_b^{\ast }=({\mu }_b/{\beta }_b)(R_0^b-1)$ clearly show that as $R_0^b$ increases above 1, the susceptible bird population decreases while the infected bird population increases.

3.5. Stability Analysis

Local Stability of the Disease-Free Equilibrium

Applying the Routh–Hurwitz criterion, we first consider the local stability of equilibria by investigating eigenvalues of the Jacobian matrices of the given model at equilibria.

Theorem 3. 

The disease-free equilibrium (DFE) of systems (1) and (2) is locally asymptotically stable when $R_0^b<1$ and $R_0^h<1$, but it is unstable when $R_0^b>1$ and $R_0^h>1$.

Proof. 

Jacobian matrix of systems (1) and (2) is as follows:

$$\begin{array}{c}\hfill J^0=\left[\begin{array}{cccccccc}-{\mu }_b& 0& -{\beta }_b{S}_b^0& 0& 0& 0& 0& 0\\ 0& -({\sigma }_b+{\mu }_b)& {\beta }_b{S}_b^0& 0& 0& 0& 0& 0\\ 0& {\sigma }_b& -({\gamma }_b+{\mu }_b+{\alpha }_b)& 0& 0& 0& 0& 0\\ 0& 0& {\gamma }_b& -{\mu }_b& 0& 0& 0& 0\\ 0& 0& -{\beta }_{bh}{S}_h^0& 0& -{\mu }_h& 0& -{\beta }_h{S}_h^0& 0\\ 0& 0& {\beta }_{bh}{S}_h^0& 0& 0& -({\sigma }_h+{\mu }_h)& {\beta }_h{S}_h^0& 0\\ 0& 0& 0& 0& 0& {\sigma }_h& -({\gamma }_h+{\mu }_h+{\alpha }_h)& 0\\ 0& 0& 0& 0& 0& 0& {\gamma }_h& -{\mu }_h\end{array}\right]\end{array}$$

The matrix $J^0$ is a block lower triangular matrix.

$$\begin{array}{c}\hfill J^0=\left[\begin{array}{cc}A& O\\ B& C\end{array}\right]\end{array}$$

(17)

where

$$A=\left[\begin{array}{cccc}-{\mu }_b& 0& -{\beta }_b{S}_b^0& 0\\ 0& -({\sigma }_b+{\mu }_b)& {\beta }_b{S}_b^0& 0\\ 0& {\sigma }_b& -({\gamma }_b+{\mu }_b+{\alpha }_b)& 0\\ 0& 0& {\gamma }_b& -{\mu }_b\end{array}\right]C=\left[\begin{array}{cccc}-{\mu }_h& 0& -{\beta }_h{S}_h^0& 0\\ 0& -({\sigma }_h+{\mu }_h)& {\beta }_h{S}_h^0& 0\\ 0& {\sigma }_h& -({\gamma }_h+{\mu }_h+{\alpha }_h)& 0\\ 0& 0& {\gamma }_h& -{\mu }_h\end{array}\right]O=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]B=\left[\begin{array}{ccc}0& -{\beta }_{bh}{S}_h^0& 0\\ 0& {\beta }_{bh}{S}_h^0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(18)

The eigenvalues of this $J^0$ matrix are given by

$$\begin{array}{c}\hfill \left\{\mathrm{Eigenvalues}\mathrm{of}A\right\}\cup \left\{\mathrm{Eigenvalues}\mathrm{of}C\right\}\end{array}$$

For matrices A and C, the eigenvalues are ${\lambda }_{1,4}=-{\mu }_b$ and ${\lambda }_{1,4}=-{\mu }_h$;

The characteristic equation for the remaining matrices A and C with eigenvalue $\lambda$ is

$$\begin{array}{c}\hfill |A-\lambda I|=\left[\begin{array}{cc}-({\sigma }_b+{\mu }_b+\lambda )& {\beta }_b{S}_b^0\\ {\sigma }_b& -({\gamma }_b+{\mu }_b+{\alpha }_b+\lambda )\end{array}\right]\end{array}$$

(19)

$$\begin{array}{c}\hfill |C-\lambda I|=\left[\begin{array}{cc}-({\sigma }_h+{\mu }_h+\lambda )& {\beta }_h{S}_h^0\\ {\sigma }_h& -({\gamma }_h+{\mu }_h+{\alpha }_h+\lambda )\end{array}\right]\end{array}$$

(20)

Now, to determine the nature of the eigenvalues of the above square matrix, we use the Routh–Hurwitz criteria. For the bird subsystem (matrix A), the characteristic polynomial is

$$\begin{array}{c}\hfill {\lambda }^2+a_1\lambda +a_2=0\end{array}$$

(21)

Here

$$\begin{array}{c}\hfill a_1=({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)\end{array}$$

(22)

and

$$a_2=({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)-{\beta }_b{S}_b^0{\sigma }_b$$

or

$$\begin{array}{c}\hfill a_2=({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)(1-R_0^b)\end{array}$$

(23)

For the human subsystem (matrix C), the characteristic polynomial is

$$\begin{array}{c}\hfill {\lambda }^2+c_1\lambda +c_2=0\end{array}$$

(24)

Here

$$\begin{array}{c}\hfill c_1=({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)\end{array}$$

(25)

and

$$c_2=({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)-{\beta }_h{S}_h^0{\sigma }_h$$

or

$$\begin{array}{c}\hfill c_2=({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)(1-R_0^h)\end{array}$$

(26)

Thus, in both cases, $a_1>0$, $a_2>0$ if $R_0^b<1$ and $c_1>0$, $c_2>0$ when $R_0^h<1$.

This demonstrates that when $R_0^b<1$ and $R_0^h<1$, our SEIR system for the human and avian populations is locally asymptotically stable. □

3.6. Global Asymptotic Stability of the Disease-Free Equilibrium

It is important to emphasize that although the system admits multiple equilibria in general, these equilibria do not coexist under the same parameter regime. In particular, we show that the endemic equilibrium is not feasible when the basic reproduction numbers satisfy $R_0^b<1$ and $R_0^h<1$.

Uniqueness of the DFE when $R_0^b,R_0^h<1$: We verify that no endemic equilibrium exists under this condition. From the bird subsystem, the endemic equilibrium satisfies

$$I_b^{\ast }={\displaystyle \frac{{\mu }_b}{{\beta }_b}}(R_0^b-1).$$

Thus, if $R_0^b<1$, then $I_b^{\ast }\le 0$. Since $I_b^{\ast }\ge 0$ in the feasible region D, it follows that $I_b^{\ast }=0$.

Substituting $I_b^{\ast }=0$ into the human equilibrium Equation (14), we obtain $c=0$, and the quadratic reduces to

$$I_h(a{I}_h+b)=0,$$

where $a={\beta }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)>0$ and b reduce to

$$b={\mu }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)(1-R_0^h)>0$$

whenever $R_0^h<1$. Hence, $I_h=0$ is the only biologically feasible solution.

Therefore, no endemic equilibrium exists in D when $R_0^b<1$ and $R_0^h<1$, and the disease-free equilibrium $E^0$ is the unique equilibrium in this region. This allows us to establish its global asymptotic stability in D.

Theorem 4. 

The disease-free equilibrium $E^0=\left({\displaystyle \frac{{\Pi }_b}{{\mu }_b}},0,0,0,{\displaystyle \frac{{\Pi }_h}{{\mu }_h}},0,0,0\right)$ is globally asymptotically stable in the feasible region D whenever $R_0^b<1$ and $R_0^h<1$.

Proof. 

We construct the Lyapunov function

$$\begin{array}{c}\hfill V={\sigma }_b{E}_b+({\sigma }_b+{\mu }_b)I_b+c\left[{\sigma }_h{E}_h+({\sigma }_h+{\mu }_h)I_h\right],\end{array}$$

(27)

where $c>0$ is a constant to be determined.

Differentiating V along the solutions of the system gives

$$\begin{array}{cc}\hfill \dot{V}& =\left[{\sigma }_b{\beta }_b{S}_b-({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)\right]I_b\hfill \\ \hfill & +c\left[{\sigma }_h{\beta }_h{S}_h-({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)\right]I_h\hfill \\ \hfill & +c{\sigma }_h{\beta }_{bh}{S}_h{I}_b.\hfill \end{array}$$

Using the bounds $S_b\le S_b^0={\displaystyle \frac{{\Pi }_b}{{\mu }_b}}$ and $S_h\le S_h^0={\displaystyle \frac{{\Pi }_h}{{\mu }_h}}$, we obtain

$$\begin{array}{cc}\hfill \dot{V}& \le ({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)(R_0^b-1)I_b\hfill \\ \hfill & +c({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)(R_0^h-1)I_h\hfill \\ \hfill & +c{\sigma }_h{\beta }_{bh}{S}_h^0{I}_b.\hfill \end{array}$$

Let

$$A_1=({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b),A_2=({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h).$$

Then,

$$\begin{array}{c}\hfill \dot{V}\le \left[A_1(R_0^b-1)+c{\sigma }_h{\beta }_{bh}{S}_h^0\right]I_b+c{A}_2(R_0^h-1)I_h.\end{array}$$

Choose

$$c={\displaystyle \frac{A_1(1-R_0^b)}{2{\sigma }_h{\beta }_{bh}{S}_h^0}}.$$

Substituting this value of c, we obtain

$$\begin{array}{c}\hfill \dot{V}\le {\displaystyle \frac{1}{2}}{A}_1(R_0^b-1)I_b+{\displaystyle \frac{A_1{A}_2}{2{\sigma }_h{\beta }_{bh}{S}_h^0}}(1-R_0^b)(R_0^h-1)I_h.\end{array}$$

(28)

Since $R_0^b<1$ and $R_0^h<1$, it follows that $\dot{V}\le 0$ for all solutions in D, and $\dot{V}=0$ if and only if $I_b=0$ and $I_h=0$.

From the system equations, this implies $E_b=0$ and $E_h=0$. Hence, the largest invariant set contained in $\{\dot{V}=0\}$ is

$$\{E_b=I_b=E_h=I_h=0\}.$$

On this set, the system reduces to

$$S_b\to {\displaystyle \frac{{\Pi }_b}{{\mu }_b}},S_h\to {\displaystyle \frac{{\Pi }_h}{{\mu }_h}},$$

which corresponds to the disease-free equilibrium $E^0$.

Therefore, by LaSalle’s invariance principle, the disease-free equilibrium $E^0$ is globally asymptotically stable in D whenever $R_0^b<1$ and $R_0^h<1$. □

3.7. Sensitivity Analysis

To determine the relative importance of each parameter in driving disease transmission, we compute the normalized forward sensitivity index of $R_0^b$ and $R_0^h$ showing in Figure 5 with respect to each parameter p using the formula:

$$\begin{array}{c}\hfill {{\rm Y}}_p^{R_0}={\displaystyle \frac{\partial R_0}{\partial p}}\times {\displaystyle \frac{p}{R_0}}.\end{array}$$

(29)

A positive index indicates that increasing the parameter increases $R_0$, while a negative index indicates the opposite effect. The magnitude of $|{{\rm Y}}_p^{R_0}|$ quantifies the proportional change in $R_0$ for a proportional change in p.

$$\begin{array}{c}\hfill R_h^0={\displaystyle \frac{{\beta }_h{\Pi }_h{\sigma }_h}{{\mu }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}}{R}_b^0={\displaystyle \frac{{\beta }_b{\Pi }_b{\sigma }_b}{{\mu }_b({\sigma }_b+{\mu }_b)({\gamma }_b+{\mu }_b+{\alpha }_b)}}\end{array}$$

(30)

1.

Sensitivity of ${\beta }_h$:

$${\displaystyle \frac{\partial R_0^h}{\partial {\beta }_h}}\times {\displaystyle \frac{{\beta }_h}{R_0^h}}={\displaystyle \frac{{\Pi }_h{\sigma }_h}{{\mu }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}}\times {\displaystyle \frac{{\beta }_h{\mu }_h({\sigma }_h+{\mu }_h)({\gamma }_h+{\mu }_h+{\alpha }_h)}{{\beta }_h{\Pi }_h{\sigma }_h}}=1$$

The same approach can be used to calculate the sensitivity of the basic reproduction number with respect to the other parameters along with sensitivity indices as mentioned in

Table 4. Sensitivity indices of basic reproduction numbers.

Sensitivity of ${\mathit{R}}_{\mathit{h}}^0$	Sensitivity of ${\mathit{R}}_{\mathit{b}}^0$
1. Sensitivity of ${\beta }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\beta }_h}}\times {\displaystyle \frac{{\beta }_h}{R_h^0}}=1$$	1. Sensitivity of ${\beta }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\beta }_b}}\times {\displaystyle \frac{{\beta }_b}{R_b^0}}=1$$
2. Sensitivity of ${\Pi }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\Pi }_h}}\times {\displaystyle \frac{{\Pi }_h}{R_h^0}}=1$$	2. Sensitivity of ${\Pi }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\Pi }_b}}\times {\displaystyle \frac{{\Pi }_b}{R_b^0}}=1$$
3. Sensitivity of ${\sigma }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\sigma }_h}}\times {\displaystyle \frac{{\sigma }_h}{R_h^0}}\approx 0.038$$ 4. Sensitivity of ${\mu }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\mu }_h}}\times {\displaystyle \frac{{\mu }_h}{R_h^0}}\approx -1.04$$ 5. Sensitivity of ${\gamma }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\gamma }_h}}\times {\displaystyle \frac{{\gamma }_h}{R_h^0}}\approx -0.72$$ 6. Sensitivity of ${\alpha }_h$: $${\displaystyle \frac{\partial R_h^0}{\partial {\alpha }_h}}\times {\displaystyle \frac{{\alpha }_h}{R_h^0}}\approx -0.27$$	3. Sensitivity of ${\sigma }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\sigma }_b}}\times {\displaystyle \frac{{\sigma }_b}{R_b^0}}\approx 0.12$$ 4. Sensitivity of ${\mu }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\mu }_b}}\times {\displaystyle \frac{{\mu }_b}{R_b^0}}\approx -1.1$$ 5. Sensitivity of ${\gamma }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\gamma }_b}}\times {\displaystyle \frac{{\gamma }_b}{R_b^0}}\approx -0.68$$ 6. Sensitivity of ${\alpha }_b$: $${\displaystyle \frac{\partial R_b^0}{\partial {\alpha }_b}}\times {\displaystyle \frac{{\alpha }_b}{R_b^0}}\approx -0.31$$

.

3.8. Parameter Justification and Policy Implications

The parameter values listed in Table 2 and Table 3 were determined through a combination of a literature review and biological estimation. Parameters such as natural mortality (${\mu }_h$) and recruitment (${\Pi }_h$) are based on standard demographic data, while progression and recovery rates ($\sigma ,\gamma$) are estimated from observed incubation and infectious periods of highly pathogenic avian influenza (HPAI) strains.

Epidemiological Interpretation and Policy Implications

The sensitivity analysis identifies the most critical parameters for disease control. The human and bird recovery rates (${\gamma }_h$ and ${\gamma }_b$) exhibit strong negative sensitivity (SI = $-0.72$ and $-0.68$), indicating that a 10% increase in treatment efficiency or antiviral administration can reduce $R_0$ by approximately 7%. Conversely, transmission parameters (${\beta }_h,{\beta }_b$) and recruitment rates (${\Pi }_h,{\Pi }_b$) show a direct positive sensitivity (SI = $+1.00$). This implies that a 10% reduction in contact rates—achieved through biosecurity, social distancing, or vaccination—will lead to a 10% reduction in the reproduction number.

The relatively low sensitivity for progression rates (${\sigma }_h,{\sigma }_b$) suggests that interventions focusing on the latent period are less effective than those targeting active transmission and recovery. Therefore, an integrated strategy combining transmission reduction (hygiene, biosecurity) with enhanced clinical recovery under the One Health framework provides the most robust approach for controlling the avian influenza epidemic.

3.9. Numerical Simulation

The analytical results obtained from working on the model’s numerical simulation using the MATLAB ode45 solver are discussed in this section.

3.9.1. Disease-Free Behavior When Both $R_0^b,R_0^h<1$

We obtain $R_0^h=0.35$ by selecting the specific values of parameters

$${\beta }_h=0.0009;{\Pi }_h=60;{\sigma }_h=0.05;{\mu }_h=0.15;{\gamma }_h=0.003;{\alpha }_h=0.137.$$

Similarly, by selecting

$${\beta }_b=0.001;{\Pi }_b=60;{\sigma }_b=0.02;{\mu }_b=0.15;{\gamma }_b=0.003;{\alpha }_b=0.137,$$

we obtain $R_0^b=0.166$.

Using the initial values in both cases, $S\left(0\right)=200$, $E\left(0\right)=200$, $I\left(0\right)=180$, and $R\left(0\right)=80$, the graphs below are produced. It indicates that the infection is approaching zero when $R_0^h<1$ and $R_0^b<1$.

Figure Analysis: Disease-Free Equilibrium

The provided Figure 6 illustrates the population dynamics of a coupled human and bird system where the disease, avian influenza, is unable to persist. A direct result of the basic reproduction number, $R_0$, being smaller than one ($R_0^h<1,R_0^h<1$), is the graphs’ obvious shift towards a disease-free equilibrium (DFE). In this state, the number of susceptible individuals ($S_h$ and $S_b$) steadily increases as the disease fails to spread effectively. Simultaneously, the exposed ($E_h$ and $E_b$) and infected ($I_h$ and $I_b$) populations decrease over time, eventually approaching zero. This decrease implies that every infected person infects fewer than one other person on average, and the epidemic dies out naturally.

3.9.2. Endemic Behavior When Both $R_0^b,R_0^h>1$

We obtain

$$R_0^b=13.94and{R}_0^h=13.22$$

by selecting the specific values of parameters

$${\beta }_b=0.017;{\Pi }_b=60;{\sigma }_b=0.62;{\mu }_b=0.093;{\gamma }_b=0.5;{\alpha }_b=0.091;$$

and

$${\beta }_h=0.019;{\Pi }_h=60;{\sigma }_h=0.64;{\mu }_h=0.095;{\gamma }_h=0.6;{\alpha }_h=0.095;$$

Figure Analysis: Endemic Equilibrium

Figure 7 demonstrates a clear movement towards an endemic equilibrium, which is a direct consequence of the basic reproduction number, $R_0$, being greater than one ($R_0^h>1,R_0^b>1$). In this state, the number of susceptible individuals ($S_h$ and $S_b$) decreases and then stabilizes at a new constant level. Simultaneously, the exposed ($E_h$ and $E_b$), infected ($I_h$ and $I_b$), and recovered ($R_h$ and $R_b$) populations increase and then stabilize at a constant, non-zero value. This persistence signifies that each infected individual transmits the disease to more than one other individual on average, causing the epidemic to reach a stable state rather than dying out.

3.9.3. Effect of Different Parameters on Human Infected Individuals

Figure Figure 8 illustrates the influence of various epidemiological parameters on the human infected class $I_h\left(t\right)$. Panel (a) shows that an increase in the bird-to-human transmission rate ${\beta }_{bh}$ leads to a rapid growth in human infections, with higher values of ${\beta }_{bh}$ producing a steeper rise in the infection curve. Panel (b) demonstrates the effect of the human recovery rate ${\gamma }_h$, where higher recovery rates result in a noticeable decline in the infected class, indicating that faster recovery substantially reduces disease prevalence. Panel (c) presents the impact of the progression rate ${\sigma }_h$ from exposed to infected peoples, showing that larger values of ${\sigma }_h$ accelerate the transition into the infectious state, thereby increasing the number of human infections. Panel (d) highlights the role of the human-to-human transmission rate ${\beta }_h$, where increasing ${\beta }_h$ amplifies the growth and peak level of infections, emphasizing its critical contribution to sustaining the outbreak. Collectively, the results reveal that transmission parameters (${\beta }_{bh},{\beta }_h$) and the progression rate (${\sigma }_h$) drive infection growth, while the recovery rate (${\gamma }_h$) mitigates it, underscoring their importance in designing effective control strategies.

4. Neural Networking

The methodology of a neural network (NN) resembles an artificial network that operates through hidden layers, utilizing maximum and minimum operations along with multiplication and addition, based on matrix vector rules from linear algebra, probability, and data science. Various researchers have extensively explored neural network-based approaches through a variety of methodologies, examining and analyzing a broad spectrum of both linear and nonlinear mathematical models, such as [30,31,32,33]. The pre-saved model parameters are embedded into the NN code within the MATLAB package. This package tests all data for validation, training, performance evaluation, histogram analysis, fitting values, and error metrics such as root mean square error (RMSE), mean square error (MSE), and absolute error (AE). The code can search through data using built-in functions to extract all these properties, as illustrated in the figures below. The flowchart representing the NN process is shown in Figure 9.

For biological justification, the neural network is linked to numerical simulations of local stability for both $R_0^b<1$ and $R_0^h<1$, using saved data (Data 1 and Data 2) in MATLAB 2020 software as standard. Subsequently, these data are utilized in the neural network code of the MATLAB package, where the target and output built-in functions are pre-stored in the form of hidden layers for the five compartments, running up to 1000 epochs.

The performance and correlations of the data are then tested, and the saved data are employed to train, test, and validate all data sets. This process includes comparisons, regression analysis, absolute error, and mean square error relative to the already stored data. We further extend the discussion of the neural network section. The NN analysis—similar to artificial intelligence—integrates previously saved work related to the said model or its modified versions, applied here to verify the comparison or correlation with the saved numerical simulation expressed as a system of differential equations.

This analysis indicates that previous work on avian influenza infection is comparable with the present model. Moreover, the model yields small absolute and mean square errors, as the neural network curves overlap with the earlier simulation curves. The neural network is executed twice, applied to the saved simulation of the model for both conditions: $R_0^b<1$ and $R_0^h<1$. This section is devoted to the neural networking of both the models representing the bird flu dynamics for human and bird populations which depends on the saved data of both the models. Neural networks have two or three hidden layers which process the saved data and return the output in the form of graphical dynamics having all, train, valid, and test data along with mean errors, performance, regression, and fitting values. The graphs are shown in Figure 10 and Figure 11. Figure 10e is the comparison of the human model and neural network dynamics, while Figure 10f is the absolute error.

The graphs are shown in Figure 12 and Figure 13. Figure 12e is the comparison of the bird model and neural network dynamics, while Figure 12f is the absolute error. In both cases, the NN curves lie on the curves of human and bird curves showing very small mean square errors, which demonstrates the correctness of the obtained scheme.

5. Model Limitations and Future Work

While the proposed model captures essential transmission dynamics of avian influenza, several limitations should be acknowledged. First, the model assumes homogeneous mixing of populations and does not account for spatial heterogeneity, such as differences between rural and urban areas or the role of live bird markets and wild bird migration routes. Second, demographic processes are modeled as constant rates, whereas birth and death rates may vary seasonally or in response to outbreaks. Third, the model does not incorporate behavioral dynamics, including vaccination hesitancy, risk perception, social distancing compliance, or public health interventions that change over time in response to disease prevalence. Fourth, the neural network validation is performed on simulated data generated from the same ODE system; future work should train neural networks on real epidemiological data once available.

Future research directions:

• Extending the model to fractional-order delay differential equations to capture memory effects and time delays in disease transmission.
• Incorporating optimal control theory to identify time-varying intervention strategies (e.g., culling, vaccination, treatment).
• Adding behavioral feedback loops where human behavior changes in response to perceived risk.
• Developing spatial or network models to capture heterogeneous transmission patterns.
• Applying Physics-Informed Neural Networks (PINNs) to directly estimate parameters from real outbreak data.

6. Conclusions

In the current study, we proposed and analyzed an SEIR-type model to describe the avian influenza transmission dynamics between human and bird populations. The model takes into account bird-to-bird, bird-to-human, and human-to-human transmission routes, as well as natural and disease-induced mortality. We demonstrated the important theoretical properties of the system, including positivity, boundedness, and stability. The basic reproduction numbers were derived using the next-generation matrix method, where $R_0$ is defined as the spectral radius of $F{V}^{-1}$ evaluated at the disease-free equilibrium. The disease-free equilibrium is locally and globally asymptotically stable when the basic reproduction numbers for both populations satisfy $R_0^b<1$ and $R_0^h<1$. Numerical simulations confirm this result, showing susceptible populations converging to their equilibrium values while infected populations decay to zero. When $R_0^b>1$ and $R_0^h>1$, the system exhibits an endemic equilibrium, indicating that the disease persists in both populations. Sensitivity analysis reveals that transmission rates and recruitment rates are the most influential parameters in increasing the basic reproduction numbers (sensitivity index $+1.00$), while recovery rates act as strong suppressors (sensitivity index approximately $-0.70$). Numerical simulations further illustrate that increasing the recovery rate reduces human infections, whereas higher bird-to-human or human-to-human transmission rates amplify outbreaks. The neural network validation shows very small mean squared errors on the order of ${10}^{-13}$, confirming the accuracy of the numerical solutions. These results highlight the significance of minimizing bird-to-human transmission, maximizing treatment and recovery in humans, and implementing effective bird population surveillance. In totality, this research adds to the One Health framework by linking human and avian health dynamics, and it presents a theoretical foundation for the development of effective control and intervention measures against avian influenza. The key quantitative findings’ $R_0$ thresholds, sensitivity indices, and convergence behavior provide actionable insights for public health planning.