To analyze the dynamics of opinion dissemination under media interventions, we propose a compartmental model inspired by epidemiological frameworks. The model classifies the population into five compartments, where S (Susceptible Individuals), E (Exposed Individuals), I p (Positive Communicators), I n (Negative Communicators), and I m (Mixed-Emotion Communicators) represent the number of netizens in each state. The terms “positive”, “negative”, and “mixed-emotion” communicators are used to represent individuals who actively express views in support of, against, or engaging with both sides of a topic. These labels do not imply internal affective states but rather the public orientation of the opinions being communicated.

The governing equations of our model, presented below, describe the dynamical transitions between opinion states, while the corresponding flow diagram (Figure 1) visually represents these transitions S ′ = u t − λ S I p + I n + I m − μ 1 S , E ′ = λ S I p + I n + I m − α 1 + α 2 + α 3 E − μ 2 E , I p ′ = α 1 E + ω I n I p + δ 1 I m − μ 3 I p , I n ′ = α 2 E − ω I n I p + δ 2 I m − μ 3 I n , I m ′ = α 3 E − δ 1 + δ 2 I m − μ 3 I m . (1)

We assume that the disengagement rates for positive communicators ( I p ) , negative communicators ( I n ) , and mixed-emotion individuals ( I m ) are identical, denoted by μ 3 . This assumption is based on the idea that disengagement is primarily influenced by psychological fatigue, cognitive overload, and external platform mechanics rather than ideological stance. Empirical studies suggest that user disengagement from online discourse is often driven by information saturation rather than ideological conviction [18, 25, 27]. Furthermore, in many controlled environments—such as state-regulated media platforms or algorithmically moderated online spaces—users are subject to similar disengagement pressures, reinforcing our choice of a uniform μ 3 . While this study assumes a homogeneous disengagement rate, future work could explore the effects of asymmetrical disengagement rates, particularly in environments where ideological entrenchment plays a stronger role in sustained engagement.

In contrast, susceptible ( S ) and exposed ( E ) individuals have distinct disengagement rates ( μ 1 , μ 2 ) , as their exit behaviors are fundamentally different. Political science research suggests that those who never engage in discourse disengage due to low motivation or cognitive effort [10]. Meanwhile, cognitive overload studies indicate that exposed individuals ( E ) , faced with conflicting opinions, are more prone to decision fatigue rather than ideological frustration [31]. By incorporating these distinctions, our model more accurately represents how individuals disengage at different stages of opinion formation.

In system (1), each derivative denotes the rate of change with respect to time ( t ) , which is modeled as a continuous and dimensionless variable. While the model does not assume a specific unit of time, it is designed to reflect the fast-paced nature of discourse evolution in digital environments. Depending on the application context, one unit of time can be interpreted as a few hours, a day, or a full cycle of social media engagement or news coverage. This flexible treatment allows the model to capture short-term fluctuations and longer-term opinion shifts within a unified analytical framework.

Here, λ represents the probability of susceptible individuals transitioning into the exposed state after encountering active communicators, effectively becoming passive observers or ‘lurkers’. α 1 , α 2 and α 3 denote the probabilities of lurkers transitioning into different communicator states, specifically becoming positive, negative and mixed-emotion communicators, respectively. The transition rates α 1 , α 2 and α 3 are assumed to be constant, reflecting an internal predisposition of exposed individuals toward specific opinion states. These rates are not influenced by the current distribution of opinionated communicators in the system. ω refers to the probability of negative communicators shifting under positive influence, reflecting the interaction between opposing perspectives. Since I p and I n play structurally similar roles, the definition of ω as influencing either group remains a modeling choice, ensuring flexibility without affecting the fundamental structure or solution of the system.

The disengagement probabilities are given by μ 1 , μ 2 and μ 3 , corresponding to different stages of engagement in the system. μ 1 represents the probability of uninformed individuals disengaging from discourse entirely, while μ 2 accounts for the probability of exposed individuals exiting the discussion before forming an opinion. Finally, μ 3 denotes the probability of active communicators ( I p , I n , I m ) disengaging, ceasing to contribute to opinion formation. Here, δ 1 and δ 2 represent the rate at which mixed-emotion individuals transition into positive and negative communicators, respectively. The inclusion of I m introduces a dynamic mediator that indirectly affects the balance between I p and I n , allowing for more complex trajectories and equilibrium behavior than in traditional SEIR-type opinion models.

In our model, the influx of new individuals into the susceptible population is represented by the function u ( t ) , defined as u t = 0 , if  t < t int A , if  t int ≤ t ≤ t int + T int 0 , if  t > t int + T int . Here, t int denotes the initiation time of the intervention, T int represents its duration, and A refers to the total number of new individuals entering the susceptible class during the intervention period. This piecewise definition allows us to model scenarios where external agents, such as governments or organizations, introduce surges of new participants into the discourse at strategically chosen times. By incorporating this time-dependent function, our model captures the transient and often abrupt changes observed in public opinion dynamics, providing a nuanced understanding of how targeted interventions influence the spread and evolution of opinions over time. In this study, we focus specifically on the time interval where A is active, analyzing the effects of the intervention period. In our simulations, we focus on the case where u ( t ) = A is constant, corresponding to a regime of sustained baseline engagement, in order to analyze the steady-state behavior of the system.

Engagement surges often arise from external factors such as major news events, viral media cycles, or coordinated platform-wide promotions. These triggers can cause abrupt increases in user participation, temporarily altering the trajectory of public discourse. Our model captures such surges through a time-dependent influx function, allowing for the analysis of how transient or sustained engagement influences long-term opinion dynamics.

A fundamental requirement in modeling such systems is ensuring that all state variables remain non-negative and bounded over time. This guarantees that the solutions remain physically meaningful, as negative values would not correspond to realistic interpretations of population sizes. The following theorem demonstrates that the total population remains constrained within a positively invariant region.

Theorem 1

Let μ denote the minimum of the coefficients μ 1 , μ 2 and μ 3 . Then, the closed region Ω = S , E , I p , I n , I m ∈ R + 5 | S + E + I p + I n + I m ≤ A / μ is positively invariant set for the model given in (1).

Proof. To analyze the behavior of the system in (Equation 1), we sum the equations and define the total population as N ( t ) = S ( t ) + E ( t ) + I p ( t ) + I n ( t ) + I m ( t ) , resulting in N ′ t = S ′ t + E ′ t + I p ′ t + I n ′ t + I m ′ t . By introducing μ = min { μ 1 , μ 2 , μ 3 } , we derive the inequality N ′ t ≤ A − μ N t , which clearly indicates that N ′ ( t ) becomes negative whenever N ( t ) > A / μ .

Solving this inequality leads to the conclusion that N t ≤ A μ + e − μ t N 0 − A μ , providing an upper bound for the total population as lim t → ∞ N t = A μ . Furthermore, for ( S , E , I p , I n , I m ) ∈ R + 5 , the following holds S ′ | S = 0 = A > 0 , E ′ | E = 0 = λ S I p + I n + I m ≥ 0 , I p ′ | I p = 0 = α 1 E + δ 1 I m ≥ 0 , I n ′ | I n = 0 = α 2 E + δ 2 I m ≥ 0 , I m ′ | I m = 0 = α 3 E ≥ 0 . Therefore, Ω constitutes a positively invariant set for the model (1), guaranteeing that no solution trajectory can leave through the boundary of Ω .

Analyzing the equilibrium points of the system provides insight into its long-term behavior. Equilibrium states correspond to points where the system remains unchanged over time, meaning all derivatives are set to zero. By solving the resulting algebraic equations, we can determine steady-state solutions that describe possible stable configurations of the system.

Solving for I m in terms of E yields the following relation I m = α 3 E δ 1 + δ 2 + μ 3 . (2) By adding the third and fourth equations in (1) and setting the result to zero, we obtain I p + I n = α 1 + α 2 + α 3 δ 1 + δ 2 δ 1 + δ 2 + μ 3 E μ 3 . (3) On the other hand, substituting (Equations 2, 3) into the expression E ′ = 0 , we obtain the following − α 1 + α 2 + α 3 + μ 2 + α 1 + α 2 + α 3 μ 3 λ S E = 0 . (4) This equation reveals two possible solutions, E = 0 and E ≠ 0 . For the former case, Equation 2 yields I m = 0 , and Equation 3 gives I p + I n = 0 . The nonnegativity constraints on I p and I n require I p = I n = 0 . Consequently, from S ′ = 0 , we find S = A / μ 1 . This result identifies the first equilibrium point, P 1 ( A / μ 1 , 0,0,0,0 ) , which corresponds to the dissemination-free equilibrium.

For a nonzero E , we solve (Equation 4) to determine S as follows S * = μ 3 α 1 + α 2 + α 3 + μ 2 λ α 1 + α 2 + α 3 (5) which is positive for all parameter combinations.

By substituting (Equations 2, 3 and 5) into S ′ = 0 and solving for E , we find E * = A α 1 + α 2 + α 3 + μ 2 − μ 1 μ 3 λ α 1 + α 2 + α 3 . The condition for E * to be positive is α 1 + α 2 + α 3 A λ − μ 1 μ 3 > μ 1 μ 2 μ 3 . (6) Then, (Equation 2) gives I m * = α 3 δ 1 + δ 2 + μ 3 E * . (7) Similarly, by substituting (Equations 3, 7) into I p ′ = 0 , we obtain the following quadratic polynomial I p 2 + k I p − l = 0 where k = μ 3 ω + α 3 δ 1 + δ 2 + μ 3 − α 1 + α 2 + α 3 μ 3 E * , l = α 1 + α 3 δ 1 δ 1 + δ 2 + μ 3 E * ω . For positive I p , the solution to this quadratic equation is I p * = − k 2 + 1 2 k 2 + 4 l , (8) which always exists since l is positive. Then, by using (Equations 8, 3), we find I n * = μ 3 ω − k − I p * , which remains positive for all parameter combinations. Thus, there exists the endemic equilibrium point P 2 ( S * , E * , I * p , I * n , I * m ) when the condition given in (Equation 6) holds.

The basic reproduction number, R 0 , plays a crucial role in analyzing the dynamics of dissemination in online public opinion systems. If R 0 > 1 , the dissemination persists, whereas it is eliminated for R 0 < 1 . This threshold mirrors its significance in such models, emphasizing the need for effective strategies to control the spread of online public opinion by lowering R 0 below one when it is necessary to limit its impact.

Using the next-generation matrix method and following the notation in [7], R 0 can be determined as the spectral radius of F V − 1 where the matrices F and V are defined as follows F = 0 λ A / μ 1 λ A / μ 1 λ A / μ 1 0 0 0 0 0 0 0 0 0 0 0 0 , V = − α 1 − α 2 − α 3 − μ 2 0 0 0 α 1 − μ 3 0 δ 1 α 2 0 − μ 3 δ 2 α 3 0 0 − δ 1 − δ 2 − μ 3 . Then, we obtain the basic reproduction number as R 0 = A λ μ 1 μ 3 1 − μ 2 α 1 + α 2 + α 3 + μ 2 . (9)

The dissemination-free equilibrium represents a state where the system reaches stability in the absence of active dissemination. The following theorem provides the conditions for local asymptotic stability.

Theorem 2

For the model defined by (1), the dissemination-free equilibrium is locally asymptotically stable when R 0 < 1 . However, if R 0 > 1 , the dissemination-free equilibrium becomes unstable.

Proof. The Jacobian matrix of the nonlinear system (1) is as follows J = − μ 1 − λ I p + I n + I m 0 − λ S − λ S − λ S λ I p + I n + I m − α 1 − α 2 − α 3 − μ 2 λ S λ S λ S 0 α 1 ω I n − μ 3 ω I p δ 1 0 α 2 − ω I n − μ 3 − ω I p δ 2 0 α 3 0 0 − δ 1 − δ 2 − μ 3 , (10) and the corresponding matrix at the dissemination-free equilibrium is given by J P 1 = − μ 1 0 − λ A / μ 1 − λ A / μ 1 − λ A / μ 1 0 − α 1 − α 2 − α 3 − μ 2 λ A / μ 1 λ A / μ 1 λ A / μ 1 0 α 1 − μ 3 0 δ 1 0 α 2 0 − μ 3 δ 2 0 α 3 0 0 − δ 1 − δ 2 − μ 3 . (11) The characteristic equation corresponding to (Equation 11) is ξ + μ 1 ξ + μ 3 ξ + δ 1 + δ 2 + μ 3 A 0 ξ 2 + A 1 ξ + A 2 = 0 where ξ represents the eigenvalue of J ( P 1 ) , and A 0 = μ 1 , A 1 = μ 1 α 1 + α 2 + α 3 + μ 2 + μ 3 , A 2 = − A λ α 1 + α 2 + α 3 + μ 1 μ 3 α 1 + α 2 + α 3 + μ 2 . If we use the basic reproduction number in (Equation 9), then we can express A 2 as A 2 = μ 1 μ 3 α 1 + α 2 + α 3 + μ 2 1 − R 0 . Therefore, all eigenvalues of the characteristic equation will have negative real parts if R 0 < 1 , and hence the model given by (1) is locally asymptotically stable at the dissemination-free equilibrium. On the other hand, the model is unstable at this equilibrium if R 0 > 1 .

Theorem 3

The dissemination-free equilibrium, P 1 , of the model defined by (1) is globally asymptotically stable within Ω whenever R 0 < 1 .

Proof. We define the following linear Lyapunov function V t = α 1 + α 2 + α 3 E t + α 1 + α 2 + α 3 + μ 2 I p + I n + I m .

Considering (1) together with (5), the derivative of the Lyapunov function with respect to t simplifies to V ′ t = α 1 + α 2 + α 3 E ′ t + α 1 + α 2 + α 3 + μ 2 I p ′ t + I n ′ t + I m ′ t = μ 3 α 1 + α 2 + α 3 + μ 2 α 1 + α 2 + α 3 α 1 + α 2 + α 3 + μ 2 λ S μ 3 − 1 I p + I n + I m ≤ μ 3 α 1 + α 2 + α 3 + μ 2 α 1 + α 2 + α 3 α 1 + α 2 + α 3 + μ 2 A λ μ 1 μ 3 − 1 I p + I n + I m .

Equation 9 indicates that the final inequality can be expressed as V ′ t ≤ μ 3 α 1 + α 2 + α 3 + μ 2 R 0 − 1 I p + I n + I m . For R 0 < 1 , the condition V ′ = 0 holds only if I p = I n = I m = 0 . Therefore, the maximum invariant set within { ( S , E , I p , I n , I m ) ∈ R + 5 | V ′ = 0 } is { P 1 } . By applying LaSalle’s invariance principle, it follows that the dissemination-free equilibrium, P 1 , is globally asymptotically stable when R 0 < 1 .

In Figure 2, the system is depicted for the parameter values A = 0.9 , λ = 0.1 , μ 1 = 0.7 , μ 2 = 0.4 , μ 3 = 0.8 , α 1 = 0.1 , α 2 = 0.2 , α 3 = 0.3 , ω = 0.2 , δ 1 = 0.03 and δ 1 = 0.04 , with six different initial conditions (5,5,2,5,5), (3,3,5,2,2), (2,4,3.8,3.5,3), (1,2,0.2,4,4), (0.2,1,1.6,1,1) and (4,0.4,1,0.5,0.5). These parameter settings correspond to R 0 = 0.0964286 < 1 , where the system settles at the dissemination-free equilibrium (1.28571,0,0,0,0). The figure illustrates the dynamic behavior of the system, showing that, regardless of the initial conditions, all state variables converge to the dissemination-free equilibrium.

The endemic equilibrium represents a steady-state solution where dissemination persists in the system at a constant level. Analyzing the stability of this equilibrium provides insight into the long-term behavior of the system, helping to identify conditions under which dissemination persists or diminishes over time.

Theorem 4

For the model defined by (1), the endemic equilibrium, P 2 , is locally asymptotically stable when the basic reproduction number R 0 > 1 .

Proof. We define the following coefficients B 0 = δ 1 + δ 2 + μ 3 B 1 = δ 1 + δ 2 + μ 3 A λ α 1 + α 2 + α 3 + μ 3 α 1 + α 2 + α 3 + μ 2 α 1 + α 2 + α 3 + μ 2 + μ 3 α 1 + α 2 + α 3 + μ 2 μ 3 , B 2 = A λ α 1 + α 2 + α 3 δ 1 + δ 2 + μ 3 α 1 + α 2 + α 3 + μ 2 + μ 3 α 1 + α 2 + α 3 + μ 2 μ 3 , B 3 = δ 1 + δ 2 + μ 3 A λ α 1 + α 2 + α 3 − μ 1 α 1 + α 2 + α 3 + μ 2 μ 3 . Then, the characteristic polynomial of the Jacobian matrix (Equation 10) corresponding to P 2 is ξ + δ 1 + δ 2 + μ 3 ξ + 2 ω k 2 + 4 l B 0 ξ 3 + B 1 ξ 2 + B 2 ξ + B 3 = 0 .

It is evident that B 0 , B 1 and B 2 are all positive. Furthermore, the condition in (11) for the existence of the endemic equilibrium, guarantees that B 3 and B 1 B 2 − B 0 B 3 remain also positive. Consequently, using the Routh-Hurwitz criterion, we determine that P 2 , when it exists, is locally asymptotically stable.

The inequality in (Equation 6) can be reformulated in terms of R 0 defined in (Equation 9), yielding the relation α 1 + α 2 + α 3 R 0 − 1 > 0 . This reformulation highlights that the endemic equilibrium is asymptotically stable for R 0 > 1 .

Figure 3 illustrates the surface corresponding to R 0 = 1 for a fixed μ 2 = 0.8 , with the x -axis representing α 1 + α 2 + α 3 , the y -axis representing λ / μ 3 , and the z -axis representing μ 1 / A . Parameter values below this surface indicate that the dissemination-free equilibrium is asymptotically stable, whereas values above the surface signify the asymptotic stability of the endemic equilibrium.

Theorem 5

The endemic equilibrium, P 2 , of the model defined by (1) is globally asymptotically stable within Ω whenever R 0 > 1 .

Proof. We consider a nonlinear Lyapunov function of the Goh–Volterra type, structured as V x = ∑ i = 1 n c i x i − x i * − x i * ⁡ ln x i x i * , which is particularly well-suited for systems with nonlinear interactions, such as those observed in biological or ecological models. Here, x i * represents the equilibrium value of the i -th state variable, and c i > 0 are positive constants. These functions are non-negative, achieving their minimum value at the equilibrium x i * .

Let us define the following Lyapunov function of this type for the model given by (1) V P = S − S * − S * ⁡ ln S S * + E − E * − E * ⁡ ln E E * + C I − I * − I * ⁡ ln I I * , (12) where P = ( S , E , I ) , I = I p + I n + I m , and C is a positive parameter that will later be determined to satisfy the necessary conditions, particularly those related to the derivative of the Lyapunov function.

By differentiating (Equation 12) and substituting the expressions for the derivatives defined in (1), we obtain V ′ P = S ′ − S * S ′ S + E ′ − E * E ′ E + C I ′ − I * I ′ I = 1 − S * S A − λ S I − μ 1 S + 1 − E * E λ S I − α + μ 2 E + C 1 − I * I α E − μ 3 I (13) where α = α 1 + α 2 + α 3 . At P 2 , solving for A from the first equation of system (1) yields A = λ S * I * + μ 1 S * . Substituting this into (Equation 13) and simplifying, we can group the remaining terms as follows V ′ P = λ S * I * + 2 μ 1 S * − μ 1 S − λ I * + μ 1 S * 2 S − λ S I E * E + α + μ 2 E * − C α E I − μ 3 I * + W P where W ( P ) = λ S * I − ( α + μ 2 ) E + C α E − C μ 3 I .

We set W ( P ) = 0 ; that is, λ S * I − α + μ 2 E + C α E − C μ 3 I = 0 . (14) A small deviation from the steady state, derived from (Equations 1 and 14), leads to C = λ μ 3 S * , α + μ 2 = λ S * I * E * and α = μ 3 I * E * . Substituting these values in (Equation 3), we express the derivative of the Lyapunov function as V ′ P = λ S * I * 3 − S * S − I * I E E * − S S * I I * E * E + μ 1 S * 2 − S * S − S S * . Since the arithmetic mean is greater than or equal to the geometric mean, we get 3 − S * S − I * I E E * − S S * I I * E * E ≤ 0 , 2 − S * S − S S * ≤ 0 , which implies that V ′ ( P ) ≤ 0 for R 0 > 1 . By LaSalle’s Invariance Principle, it follows that every solution of the system in (1) approaches the unique associated endemic equilibrium as t → ∞ , provided R 0 > 1 .

Figure 4 presents the system dynamics, utilizing the parameter values A = 4 , λ = 0.1 , μ 1 = 0.6 , μ 2 = 0.2 , μ 3 = 0.25 , α 1 = 0.1 , α 2 = 0.3 , α 3 = 0.2 , ω = 0.05 , δ 1 = 0.03 and δ 2 = 0.04 . The system is examined under the same six initial conditions as in Figure 3. With these parameters, the basic reproduction number satisfies R 0 = 2 > 1 , leading to an endemic equilibrium at (3.33,2.5,2.6652,2.7722,1.5625). According to Theorem 5, this equilibrium is globally asymptotically stable. The figure illustrates that, irrespective of the chosen initial conditions, all state variables ultimately settle at the endemic equilibrium.

At R 0 = 1 , the system in (1) undergoes a transcritical bifurcation, a phenomenon characterized by an equilibrium whose eigenvalue exhibits a real part that crosses zero. This transition marks a fundamental change in system stability. Using the parameter values A , α 1 , α 2 , α 3 , μ 2 and μ 3 as defined in Figures 4, 5 visualizes the system’s behavior in the ( R 0 , I p + I n + I m ) plane.

The graphical representation highlights the following distinct equilibrium branches in the system.

• The blue curve represents the stable branch of the dissemination-free equilibrium ( P 1 ) , where the system remains free from opinion dissemination.