Bifurcation Analysis for an OSN Model with Two Delays

: In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as P ∗ , and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks.


Introduction
The emergence of online social networks (OSNs) has significantly reshaped the landscape of information dissemination and interpersonal connectivity over the last two decades.Platforms like Facebook, Twitter, and Instagram have revolutionized how individuals exchange ideas and interact, profoundly influencing daily life.OSNs serve as virtual spaces where users can present themselves, engage with others, and forge connections irrespective of geographical boundaries.Their widespread adoption, particularly among tech-savvy generations, has had far-reaching implications across various domains, such as education, elections, and information dissemination.Understanding the intricate ways in which OSNs influence societal, political, and economic realms, as well as individual behaviors, has become increasingly imperative.
In the realm of OSN modeling, the total population N(t) at time t is often partitioned into three distinct sub-classes representing key populations within OSN dynamics: potential users, active users, and individuals opposed to OSNs, denoted by x(t), y(t), and z(t), respectively.Cannarella and Spechler [2] introduced the "infectious recovery" SIR-type model to analyze user adoption and abandonment of OSNs, later extended in ordinary, fractional, and stochastic differential equation models as given in [3,5,6].Graef et al. [5] ex-  Motivated by existing research and the nuanced complexities of OSNs, Wang and Wang [27] proposed a dynamic mathematical model capturing unique characteristics such as users' varying interests and the impact of time delays.Their model accounts for the transition of potential users to active ones and the eventual abandonment of OSNs by active users due to disinterest or interaction with those opposed to OSNs.This interaction is described by a system of differential equations as follows: where the parameters A > 0 and B ≥ 0 represent the rates that newcomers come into the community as either potential online network users or as people who are never interested in OSNs.α > 0 denotes the contact rate between the potential and active OSN users; µ > 0 is the death rate for all people; η > 0 is the contact rate between active users and people who are opposed to OSNs; δ > 0 is the transferring rate describing the rate the active users lose their interest and become opposing to OSNs; and τ ≥ 0 is the time delay that represents the time for active users to starting abandoning the network.Wang and Wang [27] performed a detailed analysis for System (2), including local and global analysis for user free equilibrium (UFE) and UPE.Hopf bifurcation was also carried out using the delay τ as the bifurcating parameter.Conditions and critical values were found that guarantee the occurrence of Hopf bifurcation.
Building upon prior work, considering the fact that it will take some time for active users to disengage after interacting with non-users, we introduce the following refined model that accounts for this time delay.Our proposed system of equations incorporates a time delay ρ, representing the period for active users to abandon OSNs after contact with non-users.This addition of a new time delay can indeed make it more representative of realworld situations and more accurately representing real-world dynamics and improving the reliability of predictions and control strategies.Notably, our model encompasses previous formulations as special cases, offering a comprehensive framework to study the evolving dynamics of OSNs and Let R 0 be the basic reproduction number defined by The following results are established by Wang and Wang [27].
Theorem 1.Let R 0 be defined by (6).If R 0 ≤ 1, then System (3) has a unique user free equilibrium P 0 = (A/µ, 0, B/µ) and it exists for all parameter values.If R 0 > 1, then System (3) has two equilibria: P 0 and a unique user prevailing equilibrium P * = (x * , y * , z * ), where z * is the unique positive root of the equation f (z) = g(z), such that z * > B/µ, and x * and y * are given by and Theorem 2. Let R 0 be defined by (6) and assume that τ = ρ = 0.If R 0 < 1, P 0 is locally asymptotically stable; if R 0 = 1, P 0 is neutrally stable; and if R 0 > 1, P 0 becomes unstable, and P * emerges and it is locally asymptotically stable.
The following result was established by Ruan and Wei [28] and will be used in this research.
Lemma 1.Consider the following exponential polynomial: n ]e −λτ 1 In this research, we were interested in finding out what network user dynamics the new model presents, in particular, whether or not a Hopf bifurcation will occur for this new OSN model after adding a time delay.In doing so, we performed a Hopf bifurcation analysis for System (3) using two delays τ and ρ as bifurcating parameters.We investigated the Hopf bifurcations at the unique user prevailing equilibrium point when R 0 > 1. Stability regions were established in terms of two delays τ and ρ.Conditions and critical curves were obtained so that the Hopf bifurcation occurs as (τ, ρ), passing through the boundary of the stability regions.
The remainder of the manuscript is structured as follows: In Section 2, we delve into Hopf bifurcation analysis concerning the interplay of two delays.We explore the establishment of stability regions and identify critical values under scenarios where either one delay is absent, or both delays are concurrently present.Our investigation delves into the conditions conducive to Hopf bifurcations and delineates the associated implications.To augment our theoretical insights, we present numerical simulations aimed at illustrating the dynamics of the system under consideration.
Finally, Section 3 encapsulates our findings and conclusions drawn from the preceding analyses.We synthesize the key insights gleaned from our study and discuss their broader implications in understanding the dynamics of online social networks.

Hopf Bifurcation
From Wang and Wang [27], we know that the dynamics of System (3) is completely determined by the basic reproduction number R 0 when delays ρ = τ = 0.In particular, we know that when R 0 > 1, the unique user prevailing equilibrium P * is locally asymptotically stable.We are interested in the question of whether the delays ρ and τ could cause the stability of the UPE P * to switch as they increase.In this section, we study the occurrence of Hopf bifurcations using the delays ρ and τ as the bifurcation parameters.Note that when R 0 > 1 there is a unique UPE P * = (x * , y * , z * ).For this section, we always assume that R 0 > 1.
The characteristic equation of System (3) at the unique equilibrium P * when ρ, τ ≥ 0 is the determinant of the matrix where and x * , y * , and z * are given in Theorem 1.
One root of Equation ( 9) is λ = −µ < 0. The other roots are determined by the transcendental equation: We know that if R 0 > 1 and ρ = τ = 0, all roots of Equation ( 11) have negative real parts and P * is locally asymptotically stable.Our interest is to see whether or not the delays ρ and τ cause the stability of P * to switch as ρ and τ increase while R 0 remains larger than the unity.Due to Lemma 1, we need to investigate if a zero of Equation (11) appears on or crosses the imaginary axis as ρ and τ increases.Keep in mind that when R 0 > 1, z * > B/µ, see [27].
2.1.Hopf Bifurcation When ρ = 0 For the case that ρ = 0, the Hopf bifurcation analysis was carried out completely by Wang and Wang [27].For completeness, we only cite key definitions and results here.We refer readers to [27] for a detailed analysis.When ρ = 0, Equation (11) becomes where Now, let λ = ωi (ω > 0) be a root to Equation (12).Plug it into (12), then ω has to satisfy the following equation: Then, the above equation can be rewritten as: The following result is well known.
Lemma 2. For Equation ( 14), we have Plug a 1 , b 1 , c 1 , given in ( 13) and x * , and y * , given in ( 7) and ( 8), into a 2 and b 2 , and we have where We then have the following results; see Wang and Wang [27].
(I) When any of the following conditions is satisfied, Equation ( 14) has no positive roots.
Now assume that R 0 > 1 and Equation ( 14) has at least one positive root.Solving p from Equation ( 14) for the positive roots gives Note that if Equation ( 14) has a unique positive root, then it is p + .Let ω ± = p ± and define Hence, τ ± n > 0 and Equation ( 11) has a pair of purely imaginary roots ±iω ± when τ = τ ± n for n = 0, 1, 2, • • • .
Theorem 4. Assume that R 0 > 1 and let a 2 , b 2 , P 1 , P 3 , z * , ω + , τ + 0 be defined above.Assume that P 1 and P 3 have unique positive roots z 1 and z 2 , respectively.We then have the following results.

Hopf Bifurcation
When τ = 0 When τ = 0, Equation (11) becomes where Now, let λ = ωi (ω > 0) be a root to Equation (20).When plugged into (20), separating the real and imaginary parts gives Squaring both sides and adding them together yields Then, the above equation can be rewritten as: Plug a 3 , b 3 , c 3 and d 3 given in ( 21) and x * and y * given in ( 7) and ( 8) into a 4 and b 4 , calculations yield where Q 1 , Q 2 and Q 3 are polynomials of z, such that Applying the results of Lemma 2, we have the following results.
Now assume that R 0 > 1 and Equation ( 24) has at least one positive root.Solving q from Equation ( 24) for the positive roots gives ) .
then all roots of Equation ( 20) have negative real parts for all delay ρ ≥ 0. Therefore, P * is locally asymptotically stable for all ρ ≥ 0.
Squaring both sides and adding them together yields and Now, we study the existence of positive solutions to the equation when τ ≥ 0. First, note that if ω = 0, then we have Therefore, it follows that we also have A function ρ(τ) > 0 as a function of τ can be found using (38), such that the stability region S in the τρ-space can be identified.P * is locally asymptotically stable for all (τ, ρ) in the interior of S, and Hopf bifurcation occurs as (τ, ρ) passes across the boundary of S, where S = {(τ, ρ) : 0 ≤ τ ≤ τ + 0 , 0 ≤ ρ ≤ ρ(τ)}.See Figure 7 for the stability region S and Figure 8 for solutions to converge to P * when (τ, ρ) = (0.1, 0.1) is in the interior of S. Also see Figure 9 for Hopf bifurcations to occur and periodic solutions to appear when (τ, ρ) = (0.2, 0.0314633) is on the boundary of the stability region S, and Figure 10 for solutions blow out when (τ, ρ) moves out of the stability region S.In this case, both τ + 0 > 0 and ρ + 0 > 0 exist, and they are A function ρ(τ) > 0 as a function of τ can be found using (38) such that the stability region S in the τρ-space can be identified.P * is locally asymptotically stable for all (τ, ρ) in the interior of S, and Hopf bifurcation occurs as (τ, ρ) passes across the boundary of S, where See Figure 5 for the stability region S and Figure 6 for solutions to converge to P * when 275 (τ, ρ) = (0.1, 0.1) is in the interior of S, and Figure 7 for Hopf bifurcations to occur and 276 periodic solutions to appear when (τ, ρ) = (0.So, Equation ( 14) has no positive roots, and that implies that τ + 0 > 0 does not exist.But in this case, ρ + 0 > 0 exists, and ρ + 0 = 0.325204.A function of ρ(τ) > 0 as a function of τ can be found using (38) such that the stability region S in the τρ-space can be identified.P * is locally asymptotically stable for all (τ, ρ)

Discussion
In this paper, we introduced and explored a mathematical model for online social networks, wherein the population is categorized into three distinct sub-classes: potential network users, active users, and individuals opposed to networks.Diverging from existing literature, our model accounts for the presence of individuals who will never express interest in using online networks.Additionally, active online social network users may exhibit a tendency to lose interest and subsequently abandon the platform over time, with or without interacting with non-users.
Assuming that the basic reproduction number R 0 exceeds unity, we delved into an investigation of whether time delays affecting active users' abandonment of the network can induce a switch in the stability of the unique user prevailing equilibrium (UPE) denoted as P * .We established conditions ensuring the asymptotic stability of P * for all delays τ ≥ 0 and ρ ≥ 0, enabling individuals across all three sub-classes to settle into equilibrium over time.Furthermore, we identified stability regions and associated conditions under which Hopf bifurcations occur as the delays (τ, ρ) traverse the boundaries of these regions.Consequently, periodic solutions emerged, leading to oscillations in the populations of the three sub-classes.
To validate our theoretical findings, we conducted numerical simulations, providing empirical evidence to support the dynamics predicted by our model.Through this comprehensive analysis, we shed light on the complex dynamics inherent in online social networks and elucidate the role of time delays in shaping equilibrium states and oscillatory behavior.Our study contributes to a deeper understanding of the underlying mechanisms driving the evolution of online social networks, with implications for diverse fields including sociology, network science, and computational modeling.
the sum of the orders of the zeros of P in the open right half plane can change only if a zero appears on or crosses the imaginary axis.

< 0 .
The discussion above establishes the following stability and Hopf bifurcation results.
2, 0.0314633) is on the boundary of the 277 stability region S.