Qualitative analysis of a rumor transmission model with multiple transmission pathways.

1. Introduction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

S (t), I (t), R (t), represent the people who don't know, the propagator and the immune people respectively in the new media environment. W (t) represents the number of people who spread rumors through traditional media; N (t) represents the total population in the new media environment; [beta]I represents the infection rate coefficient of the propagators in S(t)and I(t); [beta]W represents the infection rate coefficient between S(t)and the propagators through traditional media; a represents when propagators meet or propagators and immune people meet, the probability of propagator turning into immune people; [gamma] represents the probability of propagators in new media environment turning into propagators in traditional media environment. [xi] represents the probability of the propagators in traditional media environment being removed. [mu] represents the probability of every individual moving in or out of certain groups. The rumors spreading chart is as follows:

[FIGURE 1 OMITTED]

Suppose the parameters are positive, the initial condition is:

S(0) [greater than or equal to] 0, I (0) [greater than or equal to] 0, R(0) [greater than or equal to] 0, W (0) [greater than or equal to] 0. (2)

For convenience, making dimensionless transform s = S / N, i = I / N, r R N, w = W / N, [[beta].sub.1] =[gamma][[beta].sub.w]N / [xi], [[beta].sub.2] = [[beta].sub.I]N,[??] = [alpha]N, Still taking S, , R, W ,[alpha] as s, i, r, w, [??], so system(1) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Initial condition is,

S(0) [greater than or equal to] 0, I (0) [greater than or equal to] 0, R(0) [greater than or equal to] 0, W(0) [greater than or equal to] 0. (4)

The lemmas about the existence, non-negativity and bound of the solution is as following: Lemma 1 suppose [epsilon] > 0, define,

[OMEGA]=(S, I, R,W [greater than or equal to] 0: S + I + R = 1}, [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] ={(S, I, R, W)[member of][OMEGA]: W [less than or equal to] 1 + [epsilon]}.

Solutions that satisfies the initial condition(4)in system(3) are (S, I, R, W) which exist and remain non-negative in [0, +[infinity]), (S,I,R, W)[member of][OMEGA], and when there be [T.sub.[epsilon]]> 0that makes t > [T.sub.[epsilon]], we have (S, I, R, W) [member of] [[OMEGA].sub.W [less than or equal to] 1+[epsilon]].

Prove: It's easy to prove that the solutions which satisfy initial condition (4) in system(3) are (S(t), I(t), R(t), W(t)) which exist and remain non-negative in [0, +[infinity]).

Let N (t) = S (t) +1 (t) + R(t), then from system (3) we can have N (t) = [mu]- [mu]N (t), which means N = 1 is a fixed point and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since N (0) = 1, we have S (t) +1 (t) + R(t) = 1 , which means any solution to system (3) match (S, I, R, W)[member of][OMEGA]. Also, Since,

dS/dt[|.sub.S=0] [greater than or equal to] 0, dI/dt[|.sub.I=0] [greater than or equal to] 0, dR/dt[|.sub.R=0] [greater than or equal to] 0

Thus the solution which satisfies initial condition(4)in system(3)is constant in positive ways.

Since I(t) [less than or equal to] 1,from the third equation in system (3) we can know that when W(t) > 1, W(t) < ,thus to any solution satisfies W > 1, W(t) shall be a monotone decreasing, thus W (t) is limited. Finally, we have to prove that any solution which satisfies initial condition is constant in positive ways in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]]. It's obvious that we only have to prove that W(t) [less than or equal to] 1+[epsilon] succeed to any solutions. Suppose that the solution that satisfies initial condition (4) in system (3), (t) [less than or equal to] 1+ , don't succeed, so any solution which satisfies W(0) > 1+[epsilon]has W(t) > 1+[epsilon]. Take [T.sub.[epsilon]=(W(0)-(1 + [epsilon]))/ ([xi][epsilon]), we have W([T.sub.[epsilon]]) > 1+[epsilon]. But from system(3) we can know that W(t) <-[xi][epsilon], thus W([T.sub.[epsilon]]) < W(0) - [xi][epsilon][T.sub.[epsilon]] < 1 + [epsilon], contradict. Thus any solution which satisfies initial condition is constant in positive ways in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]]. Theorem prove finished.

2. Partial stability of the equilibrium point

In this section we study the existence and the partial stability of the equilibrium point in system (3). We have the following theorems concerned with the existence of the equilibrium point.

Theorem 1.1 (1) System(3) always has boundary equilibrium point(Rumor-free equilibrium, RFE) [E.sub.0] = (1,0,0, 0);

(2) When basic reproduction number [R.sub.0SIWR] = [alpha] + [[beta].sub.1] + [[beta].sub.2]/[alpha] + [mu] > 1, system (3) has the only positive equilibrium point (Rumor- endemic equilibrium, REE) [E.sup.+] = (S*, [mu] - [mu]S*/([[beta].sub.1] + [[beta].sub.2])S*, [mu]- [mu]S*/([[beta].sub.1] + [[beta].sub.2])S*, 1- S* - [mu]-[mu]S*/([[beta].sub.1] + [[beta].sub.2])S*), S* = 1/[R.sub.0SIW].

Prove Let the right side of the equation in system (3) be 0, then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

It's obvious that the boundary equilibrium point [E.sub.0] = (1,0,0,0)always exist.

When S [not equal to] 0, I [not equal to] 0, R [not equal to] 0, W [not equal to] 0, from the third equation in equation group (5) we have W * = I *. Take it into the first equation, then we have.

[mu]-([[bea].sub.1] +[[beta].sub.2]) S* I* -[mu]S* = 0 (6)

Add the first, second and forth equation in equation group (5) and we have

R* = 1 -S *-I* (7)

Put (6)and(7) into the second equation in system (5) and we have

S * = [alpha] + [mu]/[alpha]+[[beta].sub.1] + [[beta].sub.2]

From literature (Chen Lansun, Chenjian, 1993), we can calculate that the basic reproduction number is

[R.sub.oSIWR] = [alpha] + [[beta].sub.1] + [[beta].sub.2]/[alpha] + [mu]

When [R.sub.0SIWR] > 1, it's easy to calculate that I * = W* = [mu]- [mu]S*/([[beta].sub.1] + [[beta].sub.2])S*. Theorem prove finished.

Theorem 1.2 Let [R.sub.0SIR] , [R.sub.0SIWR] be the basic reproduction number of the SIR model and the SIWRmodel respectively, then [R.sub.0SIR] < [R.sub.0SIWR].

In fact, when [R.sub.0SIR] = [alpha] + [[beta].sub.2]/[alpha] + [mu], it's easy to prove that Rosir < [R.sub.0SIWR].

Note: When system doesn't contain W, system (3) degenerates into a SIR rumors spread model. Based on the practical meaning of the basic reproduction number (Describes the number of people who are infected in the average infection period when an infected person is involved in all the vulnerable populations. It's generally believed that only when R0> 1, the disease will spread in the region, thus we can make a series of precautions and methods to cure the epidemic by the value of [R.sub.0]), theoremi.2shows that it's not accurate to ignore the rumors spread between traditional media and new media in rumors spread model and get the basic reproduction number which is less than the actual situation. Thus will largely influence the various qualitative analyses and the making of prevention strategies based on the value of [R.sub.0].

As for epidemic dynamics, analyze the spread of the epidemic by studying the epidemic growth rate is a simple and effective method of analysis. Similarly, we can calculate the growth rate of the spread of rumors, first through calculating system (3)'s main characteristic value in Jacobian determinant of the boundary equilibrium point to get the initial growth of the spread of rumors.

r = [[beta].sub.2] - [mu] - [xi] + [square root of [([[beta].sub.2] - [mu]- [xi]).sup.2] + 4 [xi]([[beta].sub.1] + [[beta].sub.2] -[mu])]/2 =[??]-[xi][square root of [([??]+[xi]).sup.2]+4[xi] [[beta].sub.1]]/2

When [[beta].sub.1] [right arrow] 0, r [right arrow] [??], and r [greater than or equal to] [??], [??] = [[beta].sub.2]-[mu] refers to the initial growing rate of the rumors spread in SIR model.

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

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Thus, we can draw the following conclusion.

Theorem 1.3 When [R.sub.0SIWR] > 1, dr/d[[beta].sub.2] > dr/d[[beta].sub.1].

Note: Theorem demonstrates that when [R.sub.0SIWR] > 1, the growing rate is more sensitive in the spread between people in new media compared with the spread between people in new media and traditional media.

About the stability of the equilibrium point, we have following theorem.

Theoremi.4 When [R.sub.0SIWR] < 1, Eo is partly asymptotically stable; when [R.sub.0SIWR] > 1, [E.sub.0] is unstable.

Prove The characteristic equation of system (3) in the boundary equilibrium point [E.sub.0] is

[([lambda] + [mu]).sup.2] ([[lambda].sup.2] +([xi]- [[beta].sub.2]+[mu])[lambda] + [xi]([mu]- [[beta].sub.1] - [[beta].sub.2])) = 0 (8)

Apparently, the two characteristic roots of characteristic equation (8) are [[lambda].sub.1,2] = -[mu]< 0. From Routh-Hurwitz Stability Criterion , we can know that the stability of [E.sub.0] is determined by the signs of root in the following equation:

[[lambda].sup.2] +([xi]- [[beta].sub.2] +[mu])[lambda]+[xi]([mu]- [[beta].sub.1] -[[beta].sub.2]) = 0

And we have [[lambda].sub.3] + [[lamabda].sub.4] = -([xi] - [[beta].sub.2] + [mu]) , [[lambda].sub.3][[lambda].sub.4] = [xi]([mu]- [[beta].sub.1] -[[beta].sub.2]). Thus, when [R.sub.0SIWR] < 1, all the roots of the characteristic equation(8) has negative real parts, and [E.sub.0] is partly asymptotically stable; when [R.sub.0SIWR] > 1, [E.sub.0] is unstable. Theorem prove finished.

Theorem 1.5 When [R.sub.0SIWR] > 1, [E.sup.+] is partly asymptotically stable; when [R.sub.0SIWR] < 1, [E.sup.+] is unstable.

Prove System (3)'s characteristic equation at positive equilibrium point [E.sup.+] is

([lambda] + [mu])([[lambda].sup.3] + [[alpha].sub.1][[lambda].sup.2] + [[alpha].sub.2][lambda] + [[alpha].sub.3])=0 (9)

Among which

[[alpha].sub.1] = [xi] + [[beta].sub.1] S* +[mu]/S *

[[alpha].sub.2] = ([xi] + [[beta].sub.1] S*)[mu]/S* +([alpha]+ [[beta].sub.1] + [[beta].sub.2]) [[beta].sub.2] ([mu]- [mu]S*) / ([[beta].sub.1]+ [[beta].sub.2])

[[alpha].sub.3] = ([alpha]+ [[beta].sub.1] + [[beta].sub.2])[xi]([mu]-[mu]S*)

Apparently, the characteristic root of characteristic equation (9) is [[lambda].sub.1] =-[mu]< 0, thus the stability of [E.sup.+] is determined by the signs of root in the following equation:

[[lambda].sup.3] + [[alpha].sub.1][[lambda].sup.2] + [[alpha].sub.2][lambda] + [[alpha].sub.3] = 0

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[alpha].sub.3] = ([alpha] + [[beta].sub.1] + [[beta].sub.2])[xi]([mu]- [mu]S*) > 0, thus the sign of [H.sub.3] is the same as the sign of [H.sub.2].

When [R.sub.0SIWR] >1,

[f.sub.min] ([xi])[??]([alpha]+[mu]) [[beta].sub.1](2[[beta].sub.1] + 2[[beta].sub.2]-[alpha]- [[beta].sub.1] --[[beta].sub.2]/[alpha] + [[beta].sub.1] + [[beta].sub.2] + [mu]([alpha]+ [[beta].sub.1] + [[beta].sub.2]) ([[beta].sub.1] + [[beta].sub.2])/[([alpha] + [mu]).sup.2] [[beta].sub.1)> 0,

Thus, [H.sub.2] > 0.

From Routh-Hurwitz Stability Criterion we can know that when [R.sub.0SIWR] > 1, E + is partly asymptotically stable; when [R.sub.0SIWR] < 1, E + is unstable. Theorem prove finished.

3. Overall Asymptotic Stability of the Equilibrium Point

In this section, we will further study the overall asymptotic stability of the equilibrium point in system(3).

Theorem 2.1 When [R.sub.0SIWR] < 1, the equilibrium point [E.sub.0] in system(3) concerned with [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] is overall asymptotically stable; when [R.sub.0SIWR] = 1, [E.sub.0] concerned with [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] is overall attracted.

Prove Here we use the Lyapunov-LaSalle invariance principle to prove that [E.sub.0] is overall attracted concerned with [[OMEGA].sub.W [less than or equal to] 1+[epsilon]]. Consider the non-negative function on the compact set.

V = (S -1--ln S) +I + R + [[beta].sub.1] + [[beta].sub.2]/[xi] W (10)

Apparently, V is constant in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]], and along with system(3), the derivative of the solution satisfies,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When [R.sub.0SIWR] [less than or equal to] 1, V [|.sub.(3)] [less than or equal to] 0,thus, V is a Lyapunov function of system(3) in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]]. Define the subset of [[OMEGA].sub.W [less than or equal to] 1+[epsilon]], E is

E = {(S, I, W,R)|(S ,I ,W, R) [member of] [[OMEGA].sub.W [less than or equal to] 1+[epsilon]], V = 0}

Meanwhile, let M be the biggest fixed subset of system (3) in E. From the invariance of M and system (3), it's easy to prove that E = [E.sub.0].

Thus, from the Lyapunov-LaSalle invariance principle we can know that [E.sub.0] is overall attracted. And from Theorem 1.4, when [R.sub.0SIWR] < 1, [E.sub.0] is partly asymptotically stable. Therefore, when [R.sub.0SIWR] < 1, the equilibrium point [E.sub.0] in system(3) concerned with [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] is overall asymptotically stable; when [R.sub.0SIWR] = 1, [E.sub.0] concerned with CW<1+e is overall attracted. Theorem prove finished.

Theorem 2.2 When [R.sub.0SIWR] > 1, the positive equilibrium point E + of system (3) is overall asymptotically stable.

Prove Consider the non-negative function on the compact set.

[[OMEGA].sub.W [less than or equal to] 1+[epsilon]]

V = (S--S* -lnS) + (I-I *lnI)+ [[beta].sub.1]S*/[xi](W--W *lnW) + (R--R *lnR)

Apparently, V is constant in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]], and along with system(3), the derivative of the solution satisfies,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Among which

g(R) = I * R * (I + R)R +1 (I * +R *)R *-(I * +R *)I * [R.sup.2]--I[R.sup.*2] (I + R) = -[(I * R--R * I).sup.2] [less than or equal to] 0

And since

[mu]-[[beta].sub.1] S*I* = [mu]([[beta].sub.1] + [[beta].sub.2]) + [mu][[beta].sub.1] S*/[[beta].sub.1] + [[beta].sub.2] [greater than or equal to] 0, S*/S + S/S* - 2 [greater than or equal to] 0

And

I/W + S/S* W/I + S*/S - 3 [greater than or equal to] 0

Thus, V [|.sub.(3)] [less than or equal to] 0,therefore, V is a Lyapunov function of system(3) in [[OMEGA].sub.W [less than or equal to] 1+[epsilon]].

Define the subset of [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] be E'

E' = {(S,I,W,R)I(S,I,W,R) [member of] [[OMEGA].sub.W [less than or equal to] 1+[epsilon]] [??] = 0}

Meanwhile, let M' be the biggest fixed subset of system(3) in E'.

Thus,

E' = {(S, I,R,M) I (S,I, R, M) [member of] [OMEGA], S = S*, I = I*, R = R*, W = W *} .

From the invariance of M' and system (3), it's easy to prove that E' = [E.sup.+]. Thus, from the Lyapunov-LaSalle invariance principle we can know that E + is overall attracted. And from Theorem 1.5, when [R.sub.0SIWR] > 1, [E.sup.+] is partly asymptotically stable. Therefore, when [R.sub.0SIWR] > 1, [E.sup.+] is overall asymptotically stable.

4. Numerical Simulation Analyses and Discussion

In this section, according to the practical meaning of system (3), we run numerical simulation in Matlab software and discuss the influence to rumors spreading via various ways of transmission. In the following chart, the red line represents the rail line of SIR model, while the blue line represents the rail line of SIWR model.

Choose parameter [mu] = 0.1, [[beta].sub.1] = 0.3, [[beta].sub.2] = 0.3, [alpha] = 0.25. Choose [xi] = 0.3 from SIWR model, the initial value is (S(0),I(0),W(0),R(0)) = (0.9,0.1,0,0), correspondingly, in SIR model, [xi] = 0, initial value is (S(0),I(0),R(0)) = (0.9,0.1,0). By calculating we can know that the basic reproduction number of SIWR model [R.sub.0SIWR] = [alpha] + [[beta].sub.1] + [[beta].sub.2]/[alpha] + [mu] = 2.2 > 1, and there exists positive equilibrium point [E.sup.+.sub.SIWR] = (0.454,0.171,0.171,0.374); the basic reproduction number of SIR model SIR model [R.sub.0SIR] = [alpha] + [[beta].sub.2]/[alpha]+[mu] = 1.6 > 1, and there exists a positive equilibriumpoint [E.sup.+.sub.SIR] = (0.625,0.15,0.225) .Apparently, [R.sub.0SIR] < [R.sub.0SIWR] succeed (theorem2). Line graphi shows the trends of population density in two different models.

[FIGURE 2 OMITTED]

From line graph1 we can see that, with the spreading of rumors, the density of the propagators in SIR and SIWR models both have instantaneous growth point, then the density of propagators begins to stabilize after reaching a peak value. The density of the unknown people keeps decreasing while the density of the immune people keeps increasing until they reach a stable status respectively. However, by comparing the two models we find that the densities of unknown people in SIR model is bigger than that of SIWR model, while the density of the propagators and the immune people in SIR model are smaller that that of SIWR model. This fully demonstrates that compared with SIWR model, SIR model would underestimate the scale of rumors spreading. From the overall trend of each rail line, when the basic reproduction number is bigger than 1, the rumors will be spread, which means the positive equilibrium point is globally asymptotically stable. (Theorem 2.2) Therefore, in the prevention and control of the spread of rumors, the rational use of the basic reproduction number can provide an important reference for the effective prevention of the proliferation in the spread of rumors.

5. Conclusion

1. This essay established the rumor propagation model (SIWR rumor propagation model) with various transmission routes and obtained the basic reproductive number [R.sub.0SIWR]. When [R.sub.0SIWR] < l, the boundary equilibrium point (Rumor-free equilibrium, RFE) Eo of system (3) is globally asymptotically stable. When [R.sub.0SIWR] > l, the positive equilibrium point (Rumor-endemic equilibrium, REE) [E.sup.+] of system (3) is globally asymptotically stable.

2. By the comparison of the basic reproduction number of SIWR rumor propagation model and SIR rumor propagation model, it is discovered that the basic reproductive number of SIWR rumor propagation model is greater than the basic reproductive number SIR rumor propagation model, which means that SIR model would underestimate the basic reproduction number.

3. The numerical results show that, compared with SIWR model, SIR model underestimates the scale of the spread of rumors. Thus, the diversity of communication channels has a significant impact on the scale of the spread of rumor. This also confirms that the network platform of new media isn't the only way to spread rumors in the era of the new media network, and other forms of distribution still play a very important role. To prevent and control the spread of rumors, government should take into account both the traditional media and the new media, and make reasonable and feasible control measures such as using multi-channel ways to distribute the refutal of rumors--from traditional media to new media, from the press conference to government micro-blogging, etc.

Recebido/Submission: 10/9/2015

Aceitacao/Acceptance: 27/11/2015

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Chen Hua (1)

chenghua@163.com

Xi'an University of Science and Technology, 710054, Shaanxi, Xi'an, China

DOI: 10.17013/risti.18B.326-338
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