A MULTI-STAGE SIR MODEL FOR RUMOR SPREADING

. We propose a multi-stage structured rumor spreading model that consists of ignorant, new spreader, old spreader, and stiﬂer. We derive a mean ﬁeld equation to obtain the multi-stage structured model on homogeneous networks. Since rumors spread from a few people, we consider a large population by setting the number of initial spread to one in total population n and limit-ing n to ∞ . We investigate a threshold phenomenon of rumor outbreak in the sense of the large population limit by studying the driven multi-stage structured model. The main conclusion of this paper is that the proposed model has a threshold phenomenon in terms of a basic reproduction number which is similar to the SIR epidemic model. We present numerical simulations to show the developed theory numerically.


1.
Introduction. The purpose of this paper is to study the large population limit, or macroscopic limit, on a multi-stage rumor spreading model, I = −kλ 1 S 1 I − kλ 2 S 2 I, S 1 = kλ 1 S 1 I + kλ 2 S 2 I − δ 1 S 1 , where k is the average degree of the network. We assume that, at the beginning, there is one new spreader. Since each variable is the fraction of populations, for a large number n (n → ∞), initial data satisfy I(0) = I 0 n = n − 1 n , S 1 (0) = (S 0 1 ) n = 1 n , S 2 (0) = (S 0 2 ) n = 0, R(0) = R 0 n = 0. (2) In this model, the population group is divided into four groups: ignorants (I), new spreaders (S 1 ), old spreaders (S 2 ), and stiflers (R). What distinguishes old spreaders from new spreaders here is the relative time that has passed since the infection. In other words, our model is structured by age of infection. The system has been obtained from the following rules: (1) Ignorants contact both spreaders. Some of them accept the rumor and become new spreaders and then spread the rumor to other ignorants. We assume that the acceptance rates of ignorant from new spreader and old spreader are λ 1 and λ 2 , respectively. (2) A new spreader becomes an old spreader with transition rate δ 1 and does not become a stifler directly. (3) When old spreaders lose their interest in the rumor, they become stiflers. We assume that if an old spreader contacts other spreaders and stiflers, then he or she loses interest in spreading. For simplicity, we assume the contact rates are all the same at σ. Additionally, we assume that an old spreader becomes a stifler with transition rate δ 2 .
The entire population is divided into three groups: ignorants (I), spreaders (S), stiflers (R). Ignorant group is people who have never heard of rumors and spreader group is people who spread rumor. Stifler group represents people who are aware of the rumor but do not spread it. To derive a mean field equation, we assume that ignorants I contact a spreader, and then one ignorant accepts the rumor with some probability. Since among the communicators, their respective influence and propagation power are different [6,4,3], we additionally assume that spreaders have a multi-stage structure, i.e., two types of spreaders. S 1 is the fraction of populations for relatively new spreaders who has recently heard rumors and S 2 is the fraction of populations for old spreaders who has been exposed to rumors relatively long ago. We assume that ignorants who meet new or old spreaders become spreaders. As time goes on, new spreaders become old spreaders with some probability, and old spreaders lose interest in the rumor and become stiflers. Moreover, if spreaders contact other spreaders or stiflers, then they become stiflers. See Figure 1.
Various theoretical and experimental aspects of rumor diffusion have been recently reported [4,5,20,35]. For example, authors in [4] considered the case that an ignorant can spread the rumor. In the real world, it is hard to distinguish ignorant and spreader precisely, and spreaders have various kinds of spreading rate. Spreaders spread rumors aggressively when they first hear rumors, and spreaders will spread the rumors less aggressively if they get used to it over time. From this simple observation, we consider an SIR type rumor spreading model with multistage structure on a homogeneous network. Figure 1. The rumor spreading process of the multi-stage SIR model Rumor spreading is a dynamic process that occurs often due to the characteristics of human society and is closely related to the phenomenon of information homogenization [2] in the macroscopic viewpoint. Ranging from global rumor to regional rumor in small village units, there are various forms of rumor spreading phenomena, and they have various effects on human society [22]. Rumors have had a great impact on human society for a long time; for instance, rumor has been used as a means of war, governance, and propaganda [17]. Rumors are now being transmitted more rapidly [11] and more extensively [24,29] due to the development of media and social network service; therefore, they are expected to exert more influence, and their role and essence are getting more attention from researchers.
Based on Daley and Kendall [7,8], a lot of researchers have tried to build mathematical models of rumor spreading [21,28]. Researchers have been trying to understand rumor spreading qualitatively. In [33,34], the author observed the existence of a critical threshold for a rumor spreading model numerically in small-world networks. In [23], the authors derived a mean-field equation for rumor spreading in complex heterogeneous networks. In [37], the authors considered an SIR type rumor spreading model with a forgetting mechanism. See also [26], for a spreading model in telecommunication networks where the structure of the network affects the model equations through the largest eigenvalue instead of the average degree.
As most scientific and theoretical studies, researchers have studied rumor spreading phenomena through simplified models. Research on rumor spreading has been also able to successfully derive mathematical and physical properties through such simplification, and it was possible to describe and study various derivative phenomena based on the modeling. As the research progresses, many authors provide rumor spreading models that are more realistic. See [30,32,36]. In this paper, we study a rumor model with multi-stage structure from this perspective. As mentioned earlier, the rumor spreading phenomenon in the real world has complex dynamics and various structures including multi-stage structure, which would have been omitted in the simplification process. Therefore, we try to understand in this paper how multi-stage structure affects rumor spreading models.
In general, an multi-stage structured model enables a more realistic representation for some dynamics phenomena [27]. In SIR models, many authors have considered a variety of equations to represent epidemic phenomena with the multistage structure. Their models are based on a system of partial differential equations with SIR structure; for example, [13,15,18]. For some threshold phenomena on an age structured epidemic model, see [16]. In this paper, we consider a simple SIR model for rumor spreading with two stages. See [19] for a similar discrimination approach on the epidemic model and [25] for an age structured model for rumor spreading.
The paper is organized as follows. In Section 2, we present our main result and provide a reduction of the multi-stage structured model to a system of two equations for rumor sizes φ 1 and φ 2 . In Section 3, we provide a proof of the main theorem. In Section 4, we present numerical simulations to ensure our result. Finally, we summarize our results in Section 5.

2.
Main theorem for rumor outbreak and equation reduction with auxiliary variables. In this section, we define rumor outbreak and present our main theorem. We consider new variables to prove our main theorem and derive reduced equations from (1). The equation reduction allows us to use steady state analysis. However, since there are two kinds of spreaders, we cannot deduce one equation that is equivalent with (1). Motivated by [12,31], we define the rumor sizes φ 1 and φ 2 of S 1 and S 2 , respectively: From this notion, we can define a rumor outbreak state in the sense of the large population limit.
Definition 2.1. Let {I n , (S 1 ) n , (S 2 ) n , R n } be a sequence of solutions to system (1) subject to initial data I 0 n , (S 0 1 ) n and (S 0 2 ) n = (R 0 ) n = 0, respectively. We define the final size of the rumor: Definition 2.2. Let I 0 n > 0, (S 0 1 ) n > 0, (S 0 2 ) n and R 0 n be sequences satisfying I 0 n → 1 − , (S 0 1 ) n → 0 + as n → ∞ and (S 0 2 ) n = R 0 n = 0 for n ∈ N, 1 = I 0 n + (S 0 1 ) n . Then we say that a rumor outbreak occurs if φ e > 0, where For the multi-stage model (1), we define the corresponding basic reproduction number such as Next, we present threshold phenomena with respect to R 0 for the asymptotic behavior of the multi-stage model, the so called rumor outbreak state. For the computation of R 0 in general structured models, we refer to [1,10,9].
Remark 1. In the case without multi-stage structure, there is a threshold phenomenon of rumor spreading dynamics for the basic reproduction number kλ/δ, where δ is the forgetting rate and λ is the acceptance rate. If kλ/δ > 1, then rumor outbreak occurs; if kλ/δ ≤ 1, then rumor outbreak does not occur. For details, see [12,31]. Similar to a mono-stage case, we can observe a threshold phenomenon in the multi-stage structured model.

Remark 2.
For the next generation matrix, we get For the infected compartments S 1 and S 2 , the Jacobian matrices for F and V at the disease free equilibrium are given by Therefore, This basic reproduction number coincides with the number which we proposed. However, this basic reproduction number obtaining from the next generation matrix explains the linear stability of the disease free equilibrium. As we can seen in [10], there are five assumptions for the global stability. However, the model (1) does not satisfy the assumptions in [10]. In particular, the fifth assumption is that 'if F is set to zero, then all eigenvalues of Df (x 0 ) have negative real parts', where f = F − V and x 0 is the disease free equilibrium. In our model, if we set F = 0, then the eigenvalues of Df (x 0 ) are −δ 1 , −δ 2 and 0 with multiplicity 2. Therefore, we investigate the asymptotic stability of the system (1) by employing the following method.
The following is the main theorem of this paper.
rumor outbreak occurs if and only if R 0 > 1.
Remark 3. Similar two-stage rumor spreading model was considered in [14]. Authors in [14] studied the locally asymptotic stability of equilibrium by using Routh-Hurwitz criteria and the global stability of internal equilibrium of their model by Lasalle's invariance principle. In this paper, they found five equilibrium depending on inflow and outflow rates. However, we assume that there is no inflow and outflow in our model. Thus, for any 0 ≤ I e ≤ 0, (I, S 1 , S 2 , R) = (I e , 0, 0, 1 − I e ) is equilibrium point of our model. Moreover, the stability of equilibrium points is not our scope. Here, we consider the thermodynamic limit or large population limit of rumor spreading model.

SUN-HO CHOI, HYOWON SEO AND MINHA YOO
From integrating the above, it follows that this implies We integrate the second equation in (1) to obtain By (4), the above result yields Note that, by the definitions of φ 1 and φ 2 , The change of variables leads us to By the initial data condition in (2) andφ 1 (t) = S 1 (t), we can geṫ Next we derive a equation forφ 2 . We integrate the third equation (1) to obtain By (3), S 1 (τ ) + S 2 (τ ) + R(τ ) = 1 − I(τ ), and this implies We use (4) to yield We note thatφ 1 (t) = S 1 (t) andφ 2 (t) = S 2 (t). Therefore, both are increasing functions, and for a fixed By the change of variables, we havė In summary, by (5) and (6), φ 1 and φ 2 satisfy the following system of differential equations:φ where Simultaneously, by (5) and (7), φ 1 and φ 2 also satisfẏ where 3. Steady state analysis and threshold phenomena. In this section, we prove the main theorem by using the reduced equations. We proceed to the steady state analysis based on (8) and (9). Note that on an infinite time scale (t → ∞), formally the following holds: and (S 1 , S 2 ) = (0, 0) is the unique steady state solution to the original system (1). From this simple observation, we can expect that (φ 1 (t), φ 2 (t)) converges to a solution to (8) and (9).

Proposition 1.
Let h(x) be the function defined in (12). We assume that Then we have From simple computations, it follows that By the definition of h,

SUN-HO CHOI, HYOWON SEO AND MINHA YOO
Since y ≥ 0, Φ 1 (y) ≥ and 0 < I 0 < 1, Therefore, By (13), we have Thus, Since h(0) is a negative real number, lim x→ 1 Moreover, by (14), This implies that Remark 4. We note that it is difficult to obtain an upper bound of Φ 1 (y) in h (x). Therefore, we cannot use this framework to obtain an lower bound of solution to h(x) = 0 for other case R 0 > 1.

3.2.
Case II (R 0 > 1): Since we cannot apply the framework which used in the previous subsection to this case, we need another auxiliary equation based on (9) to derive an estimate for φ 1 (t) for the case (R 0 > 1). Although this framework seems more complicated to obtain the desired estimate for φ 1 (t), we can calculate the fixed lower bound of lim t→∞ φ 1 (t) which is independent of initial data I 0 .
This yields Therefore, the inverseḡ −1 (x) exists on x ≥ 0. As Case I, we consider Proposition 2. Leth(x) be the function defined in (18) and we assume that Then there is a positive constant x 1 > 0 which is independent of initial data I 0 satisfying Here, x 0 > 0 is any positive solution tō h(x 0 ) = 0, Proof. Note that
In order to obtain an upper bound ofh (x), we need upper bounds of N (x) and D(x). We now assume that 0 ≤ x ≤ x 1 , where By (17), Here we used the assumption 0 < I 0 < 1 and the following simple inequality: Similarly, By (19) and (20), If we assume that , on x ∈ [0, x 1 ].
Sinceh(0) < 0, we can conclude that Therefore, the positive constant x 1 > 0 is independent of initial data I 0 , S 0 and the following holds: where x 0 is any zero satisfyingh (x 0 ) = 0.
3.3. The proof of the main theorem: In this subsection, we provide the proof of the main theorem. Assume that For fixed n ∈ N, let I 0 n be initial data of ignorant I n (t) that is the solution to (1). Then {φ n 1 (t), φ n 2 (t)} is the corresponding solution to the following equation: . By the definition of φ n 1 and φ n 2 : φ n 1 (t) and φ n 2 (t) are increasing functions with respect to t > 0 and φ n 1 (t) ≥ 0, φ n 2 (t) ≥ 0. From an elementary result of ordinary differential equations, it follows that φ n 1 (t) and φ n 2 (t) converge to x n 0 > 0 and y n 0 > 0, where {x n 0 , y n 0 } is a solution to Let {x n 0 , y n 0 } be any solution to (21). Then 0 = F Thus, For the remaining case, we assume that Similar to the previous case, {φ n 1 (t), φ n 2 (t)} is the solution to the following equation:φ . This also yields thath(x n 0 ) = 0. From Proposition 2, it follows that where x 1 > 0 is a positive constant which is independent of initial data I 0 n . Thus, 4. Numerical simulations. In this section, we provide numerical simulations to guarantee our main result. To carry out numerical simulations for several cases, we use the fourth order Runge-Kutta method and Matlab with time step size ∆t = 0.01. As we mentioned before, for a sufficiently large number n 1, there is a threshold phenomenon for long time asymptotics of the solution to (1). The equivalent condition for rumor outbreak is We assume that the average degree of the network k is 0.5, and the contact rate σ is 1. Additionally, we assume that n = 10 6 .
In Figure 3 (A), we take δ 1 = 0.1, δ 2 = 0.1, λ 1 = 3, and λ 2 = 2. Then, For the opposite case, we take δ 1 = 0.3, δ 2 = 0.3, λ 1 = 0.2, and λ 2 = 0.2. Then, In this case, rumor outbreak does not occur, and the results of simulation and theory are also consistent. Please see Figure 3 (B).  In the rest of this section, we will consider two phase diagrams to investigate the threshold phenomena on the asymptotic behavior of the solution (I, S 1 , S 2 , R).
• Case 1 (phase diagram I on (x, y) = (λ 1 , λ 2 ) with fixed δ 1 and δ 2 ): To clarify the threshold phenomena, we simulate several numerical experiments. We fix the values of δ 1 and δ 2 and change the vales of λ 1 and λ 2 . We choose δ 1 = 0.  Figure 4, we can check that the yellow area expresses the rumor non-outbreak territory that overlaps the region R 0 ≤ 1 exactly. The rest of the graph expresses the rumor outbreak territory and is the region R 0 > 1. Moreover, we can see that, in the upper region of the straight line the limit values lim t→∞ I(t) are less than one, and as the distance from the line to the point (λ 1 , λ 2 ) increases, the values of lim t→∞ I(t) gradually decrease. Figure 5 is a graph of R(10000) with the same parameters δ 1 , δ 2 , λ 1 and λ 2 , as in Figure 4. We easily verify that the outbreak interface takes place through the line R 0 = 1. Figure 6 shows solutions (I, S 1 , S 2 , R) to (1) for several cases by choosing We can check that the solutions are getting closer to the trivial solution as the value R 0 approaches 1.
If we choose λ 1 = 0.2 and λ 2 = 0.2, then the interface between the region of outbreak and non-outbreak is located in Thus, equivalently, we have   Figure 7 shows that in the lower region of a part of the hyperbola, the limit values of lim t→∞ I(t) are less than 1. Similar to Case 1, the values lim t→∞ I(t) gradually decrease as the distance from the interface line to the point (δ 1 , δ 2 ) increases. 5. Discussion and conclusion. In this paper, we present a mathematically rigorous proof for threshold phenomena on a simple multi-stage structured rumor spreading model. On the derivation of the model, we divided the spreaders into two groups: new (S 1 ) and old (S 2 ) spreaders. Motivated by the steady state analysis with multiple variables, we rigorously prove that threshold phenomena of the rumor outbreak exist in the large population limit sense (see the phase diagram for R in Figure 9). The basic reproduction number R 0 = kλ 1 δ 1 + kλ 2 δ 2 of our model is an extension of the classical basic reproduction number for the SIR rumor spreading model without multi-stage structure [31]. If the basic reproduction number is greater than one, i.e., R 0 > 1, then rumor outbreak occurs. On the other 2370 SUN-HO CHOI, HYOWON SEO AND MINHA YOO Figure 9. Surface graph R on (x, y) = (δ 1 , δ 2 ) with λ 1 = λ 2 = 0.2 hand, if R 0 ≤ 1, then rumor outbreak does not occur. In fact, our model can be derived from a system of stage-structured partial differential equations; similarly we can derive a multi-stage structured model for rumor spreading. In the future, we will extend the result of this paper to multiple stages and the original system of partial differential equations.