Stability and Bifurcation Analysis For An OSN Model with Delay

In this research, we propose and study an online social network mathematical model with delay based on two innovative assumptions: (1) newcomers are entering community as either potential online network users or that who are never interested in online network at constant rates, respectively; and (2) it takes a certain time for the active online network users to start abandoning the network. The basic reproduction R 0 , the user-free equilibrium(UFE) P 0 , and the user-prevailing equilibrium(UPE) P ∗ are identiﬁed. The analysis of local and global stability for those equilibria is carried out. For the UPE P ∗ , using the delay τ as the Hopf bifurcation parameter, the occurrence of Hopf bifurcation is investigated. The conditions are established that guarantee the Hopf bifurcation occurs as τ crosses the critical values. Numerical simulations are provided to illustrate the theoretical results.


Introduction
Online Social Networks (OSNs) have become increasingly important and imperative for people to receive and spread information in the most recent two decades.The creation of Facebook, Twitter, Instagram, and other online social media networks has made information exchanging and spreading easy and it has changed people's daily life dramatically.OSNs allow people to present, interact, and connect themselves virtually with others.The popularity of OSNs, especially in the new high technology-oriented generations, has radically changed so many things such as education, election, information sharing, etc.It has become evidently urgent that a better understanding of how OSNs can influence a society's political and economical environments, even just people's daily life, will benefit all interested parties involved.Mathematical models have been developed to serve to better understand the OSN dynamics and have provided a deep insight on how people's opinion/behaviour is affected by social networks, see, for example [2,3,5,6,7,4,8,11,9,10,1].Many efforts have been made in the development of models for studying OSN dynamics leveraging SIR/SEIR disease type models.Readers are referred to [12,13,14,15,16] and references therein for some classic results and advances on SIR/SEIR mathematical models.In this spirit, the total population, denoted by N (t) at time t, is divided into three sub-classes representing three major populations in the OSN dynamics.They are potential OSN users, active OSN users, and people who are opposed to OSNs, with sizes denoted by x(t), y(t), and z(t) at time t, respectively.Cannarella and Spechler [2] proposed the infectious recovery SIR type model      x = −αxy/N, y = αxy/N − ηyz/N, z = ηyz/N, to analyze user adoption and abandonment of OSNs.This model was further extended to ordinary, fractional, and stochastic differential equation models in [3,5,6] respectively.The optimal control problem for an ordinary differential equation model was also studied in [7].The reader is referred to [2,3,5,6,7] for the details of the model development and meanings of the parameters.Particularly, Graef et al. [5] studied the following OSN model with demography to examine the user adoption and abandonment dynamics of OSNs.Both local and global stability analyses were carried out.Motivated by the research mentioned above and some unique characteristics and the complexity of OSNs such as the fact that there are people who are never interested in online social networks and active online network users may lose their interest after some time, the investigators propose a new dynamic mathematical model to incorporate these characteristics to study the OSN dynamics.Potential users become active after contacting with active users.Active users become opposed to OSNs either after contacting with those people who are opposed to OSNs or lose their interest after a period of time.The interaction and dynamics between these three compartments can be described by the following system of differential equations where the parameters A > 0 and B ≥ 0 represent the rates that newcomers come into the community as either potential 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.It is notable that when B = τ = 0, System (2) covers System (1) as a special case.
In this article, we will perform a detailed analysis for System (2).First, after a basic reproduction number is identified, the existence of equilibrium points is established based on the basic reproduction number.Then the local and global stability when τ = 0 is studied and we show that when the basic reproduction number R 0 ≤ 1, the unique user-free equilibrium P 0 is globally asymptotically stable, whereas R 0 > 1, P 0 becomes unstable and the unique user-prevailing equilibrium P * is globally asymptotically stable.Next, using τ as the bifurcation parameter, we investigate the Hopf bifurcations at the unique user-prevailing equilibrium point when R 0 > 1. Conditions and critical values are obtained so that the Hopf bifurcation will occur as τ passes through the critical values under some given conditions.
We need the following result due to Ruan and Wei [17].
Lemma 1.1.Consider the exponential polynomial P (λ, e −λτ ) = p(λ) + q(λ)e −λτ where p and q are real polynomials such that deg(q) < deg(p) and τ ≥ 0. As τ varies, the total number of zeros of P (λ, e −λτ ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
The rest of this article is organized as follows.In Section 2, the reproduction number R 0 is identified and the existence of equilibrium points are established based on the reproduction number.Local and global stability of equilibrium points when τ = 0 is studied in Sections 3 and 4. The conditions for occurrence of Hopf bifurcation at UPE as the delay τ passes through critical numbers are investigated in Section 5. Numerical simulations are provided to demonstrate our theoretical results in Section 6.The manuscript ends up with a brief discussion in Section 7.

Equilibrium points and basic reproduction number
In this section, we derive a basic reproduction number R 0 for System (2).Then equilibrium points are determined based on R 0 .To find the equilibrium points of System (2), we need to solve the following system of equations It's easy to show that the system has a unique user-free equilibrium P 0 = (A/µ, 0, B/µ) and it exists for all parameter values.Now we will try to find user-prevailing equilibrium(UPE) point(s) P * = (x * , y * , z * ), where y * > 0.
From the second and third equations of System (3), we can express x and y in terms of z as and Rewrite the first equation of (3) as A = x(αy + µ) and plug x and y given in (4) and ( 5) into this equation, we arrive at the equation which is equivalent to and Obviously, z * is a positive root of the equation and x * and y * can be found using expressions (4) and (5).From (5), it's easy to see that for y * to be positive, it requires that z * > B/µ.
Let R 0 be the basic reproduction defined by which gives the expected number of active users directly generated by one active user in a population where all individuals are potential users to network.The derivation of R 0 can be performed by the next generation matrix method given by van den Driessche and Watmough [18].If we rewrite System (2) as At the user-free equilibrium P 0 , we have where It follows that (8).We then have the following results regarding z * in terms of R 0 .
Lemma 2.1.Let z = B/µ, f , g, and R 0 be defined by ( 6) -(8) respectively.Then Proof.Note that f is a linear function with f (0) = Aαδ and g is a quadratic function with g(0 The lemma follows immediately from this equation. Remark 2.2.Note that if R 0 < 1, then either the equation f (z) = g(z) has a unique solution z * < z or it has no solutions.
The discussion above results in the following result.

Local stability of equilibrium points with no delay
In this section, we study local stability of P 0 and P * when delay τ = 0.The Jacobian matrix of System (2) at an equilibrium P = (x, y, z) is where It is clear that λ 1 = −µ is a negative root of Equation (9).The other two roots of Equation ( 9) are determined by the equation At P 0 , Equation (10) becomes At P * , a and b are given by, in terms of z * , using (4) and ( 5) From the discussion above, we know that if R 0 > 1, then z * > B/µ, or µz * − B > 0. Therefore, we have a > 0, b > 0. The two roots of Equation (10) have negative real parts.We thus obtain the following local stability results.Theorem 3.1.Let R 0 be defined in (8).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.

Global stability of equilibrium points with no delay
In this section, we study the global stability of P 0 and P * when delay τ = 0. Note that P 0 = (A/µ, 0, B/µ) and P * = (x * , y * , z * ) where z * > 0 is the unique positive root of the equation f (z) = g(z), and x * and y * are given by ( 4) and (5).
A standard argument, see, for example [19,20], can be used to show that the first octant R 3 + = {(x, y, z) : x ≥ 0, y ≥ 0, z ≥ 0} of R 3 is positively invariant with respect to System (2), and the system has a unique solution with any initial value starting from inside R 3 + .The first equation of System (2) clearly shows that if x(0) ≤ A/µ, then x(t) ≤ A/µ for all t > 0. Furthermore, if we add the three equations of System (2) together, we can get We, therefore, proved that the set is a global attractor and it's positively invariant with respect to System (2).We will use • Ω to denote the interior of Ω.Now, we are ready to prove the global stability of P 0 when R 0 ≤ 1.
Next, we will study the global asymptotic stability of P * = (x * , y * , z * ) in The derivative of V 2 along the solution of System (2) can be calculated as Note that The above V 2 (x, y, z) can be written as Now, from the first equation of System (3), we have Using expressions for x * and y * in (4) and ( 5), we can get We finally obtain Proof.Using the Lyapunov function given in ( 13) assuming δ = 0, by ( 14), we end up with Obviously, V 2 (x, y, z) ≤ 0 for all (x, y, z) ∈ • Ω and V 2 (x, y, z) = 0 holds only for the set The maximum invariant set in M is the unique equilibrium P * .Again, by the asymptotic stability theorem [21,22,24,23], all solutions starting inside • Ω converge to P * .This fact and the local stability of the equilibrium P * established in Theorem 3.1 imply the global asymptotic stability of P * .Remark 4.3.We want to point out that we only proved that P * is globally asymptotically stable in • Ω when δ = 0 in Theorem 4.2.But numerical simulations show that P * is globally asymptotically stable in • Ω as long as it exists, i.e., if R 0 > 1. See Figure 1 for the convergence of the solutions to P * for different initial points when δ > 0.Here we choose A = 2, B = 0.1, µ = .2,α = 1, β = 1, η = 0.5, δ = 0.4.Calculation shows that R 0 = 11.7647> 1 and P * = (3.7809,0.3276, 6.3816).Solutions converge to P * regardless where the solution starts.Remark 4.4.From Sections 3 and 4, we can conclude that the dynamics of System ( 2) is completely determined by the basic reproduction R 0 when there is no delay.If R 0 ≤ 1, P 0 is the only equilibrium and all solutions converge to P 0 .That means that in this case, the number of active online network users will eventually approach to zero.On the other hand, while R 0 > 1, all solutions converge to P * .The active online network users will settle at the level of y * > 0 over time.

Hopf bifurcation
As we see from Remark 4.2, the dynamics of System (2) is completely determined by the basic reproduction R 0 when the delay τ = 0. We are interested in the question that if the delay τ could cause the stability switch of the UPE P * as it increases.In this section, we study the occurrence of Hopf bifurcations using the delay τ as the bifurcation parameter.Note that when R 0 > 1 there is a unique UPE P * = (x * , y * , z * ).For this section, we will always assume that R 0 > 1.
The characteristic equation of System (2) at the unique equilibrium P * when τ ≥ 0 is the determinant of the matrix where One root of Equation ( 15) is λ = −µ < 0. The other roots are determined by the transcendental equation From the discussion in Section 2, we know that if R 0 > 1 and τ = 0, P * is locally asymptotically stable.Our interest is to see whether or not the delay τ will cause the stability of P * to switch as τ increases while R 0 remains larger than the unity.Due to Lemma 1.1, we need to investigate if a zero of Equation ( 17) will appear on or cross the imaginary axis.Keep in mind that when R 0 > 1, z * > B/µ.
From (16 and use expressions given in (4 and ( 5), we can get since z * > B/µ.Therefore, λ = 0 is not a root of (17).Now, let λ = ωi (ω > 0) be a root to Equation (17).Plug it into (17), we get Separating the real and imaginary parts gives Squaring both sides and adding them together yields Then the above equation can be rewritten as The following result is well known.Plug a 1 , b 1 , c 1 given in ( 16) and x * and y * given in ( 4) and ( 5) into a 2 and b 2 , a tedious and long calculation gives where Note that P 1 (z) is a degree three polynomial with P 1 (B/µ) = (Bη + δµ) 2 > 0 and P 1 (z) → −∞ as z → ∞.By Descartes' Rule of Signs, P 1 has one or three positive roots.Let z 1 > B/µ be the first value of z such that P 1 (z 1 ) = 0. Obviously, P 2 (z * ) > 0, and µz * − B > 0 as z * > B/µ if R 0 > 1. P 3 (z) is also a degree three polynomial of z such that and P 3 (z) → ∞ as z → ∞.Thus, P 3 has one or three positive roots.Let z 2 > B/µ be the first value of z such that P 3 (z 2 ) = 0.If we assume that both P 1 and P 3 have one positive root, i.e., z 1 and z 2 are the only positive numbers such that P 1 (z 1 ) = 0 and P 3 (z 2 ) = 0. We then have the following cases in terms of z 1 , z 2 and z * along with the signs of a 2 and b 2 : By Lemma 5.1, we have the following results based on these cases.
Theorem 5.2.Let R 0 > 1, and let a 2 , b 2 , P 1 and P 3 be defined by (21), ( 22), (23), and (24).Assume that P 1 and P 3 have a unique positive root z 1 and z 2 , respectively.We then have: (I) When any of the following conditions is satisfied, Equation (20) has no positive roots.
Now assume that R 0 > 1 and Equation ( 20) has at lease one positive root.We study Hopf bifurcations of System (2) at P * .Solving p from Eq.( 20) for the positive roots gives ) .

It follows that
Re dλ dτ We thus obtained that Re dλ In this case, Equation ( 20) has a unique positive root.We can find ω + = 0.5320039 and τ + 0 = 2.93232, that means by Theorem 5.2 that P * is locally asymptotically stable for all τ < 2.93232, and when τ = 2.93232, Equation ( 17) has a pair of purely imaginary roots ±0.5320039i and all other roots have negative real parts.Hopf bifurcation occurs as τ passes cross τ = 2.93232.See Figure 2(a) for the graphs of P 1 and P 2 , 2(b) for solutions to converge to P * for τ = 2, and 2(c) for Hopf bifurcations to occur and periodic solutions to appear when τ = 2.93232.In this case, again Equation ( 20) has a unique positive root.We can find ω + = 0.629831 and τ + 0 = 2.28009, that means by Theorem 5.2 that P * is locally asymptotically stable for all τ < 2.28009, and when τ = 2.28009, Equation ( 17) has a pair of purely imaginary roots ±0.629831i and all other roots have negative real parts.Hopf bifurcation occurs as τ passes cross τ = 2.28009.See Figure 3(a) for the graphs of P 1 and P 2 , 3(b) for solutions to converge to P * for τ = 2, and 3(c) for Hopf bifurcations to occur and periodic solutions to appear when τ = 2.28009.

Discussion
In this paper, we propose and study an online social network mathematical model.We divide the population into three major sub-classes: potential network users, active users, and people who are opposed to networks.Unlike the studies in the literature, in our model we assume that some people coming into the community that will never be interested in using online networks and active online social network users may lose their interest after a period of time (a time delay for abandoning networks).After a basic reproduction number R 0 is identified, we show that if R 0 ≤ 1, the system has a unique user-free equilibrium P 0 and it exists for all parameter values, whereas if R 0 > 1, the system has two equilibria: P 0 and a unique userprevailing equilibrium P * .Local and global stability analysis for both P 0 and P * is carried out when no delay is presented.In this case, the dynamics of the system is completely determined by the reproduction number R 0 .If R 0 ≤ 1, P 0 is globally asymptotically stable meaning that the number of active online social network users will converge to zero over time.Therefore, the social network will die out, and there will be fewer and fewer people using network.On the other hand, if R 0 > 1, i.e., each active user will generate more than one new network users, then P * is globally asymptotically stable meaning that the number of active network users will settle at a certain number over time.The network users will be persistent.Assuming that R 0 > 1, we investigated whether or not the time delay for active users to abandon the network may cause the stability of P * to switch as the delay τ is presented and increases.Conditions are established that guarantee that P * will stay asymptotically stable for all delay τ ≥ 0, so people in all three sub-classes will settle at their corresponding numbers over time.We also found conditions and critical numbers such that under these conditions Hopf bifurcation occurs as the delay increases and crosses the critical numbers.Therefore, periodic solutions appear and the numbers of three sub-classes oscillate over time.