## Abstract

Mobile Opportunistic Networks (MONs) are effective solutions to uphold communications in the situations where traditional communication networks are unavailable. In MONs, messages can be disseminated among mobile nodes in an epidemic and delay-tolerant manner. However, MONs can be abused to disseminate misinformation causing undesirable effects in the general public, such as panic and misunderstanding. To deal with this issue, we first propose a formal model to formulate the process of misinformation propagation in MONs, considering human psychological behaviors. Secondly, we explore a general framework to describe the random node mobility, and derive a new contact rate between nodes, which is closely related to mobility properties of nodes. Thirdly, we propose a novel approach based on vaccination and treatment strategies for inhibiting misinformation propagation in human MONs. Moreover, a novel pulse control model of misinformation propagation is developed. Finally, through the derivation and stability analysis of a misinformation-free period solution of the proposed model, we obtain a threshold upon which misinformation dies out in a human MON. The extensive simulation results validate our theoretical analysis.

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## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 61373083, 61402273); the Fundamental Research Funds for the Central Universities of China (Grant No. GK201401002); the Program of Shaanxi Science and Technology Innovation Team of China (Grant No. 2014KTC-18); and the Natural Science Basis Research Plan in Shaanxi Province of China under Grant No. 2014JQ8353.

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## Appendices

### Appendix A Proof of Theorem 1

We first consider the existence of the misinformation-free solution of system (6), and then prove the globally asymptotically stability of this solution. From the second equation of system (7), considering the fact that the initial value of *r*(*t*) in a pulse period (*k*
*τ*,(*k*+1)*τ*] is *r*(*k*
*τ*
^{+}), we have

From equations (8) and (12), we have

Thus, *s*((*k*+1)*τ*)=1−*r*(*k*
*τ*
^{+})*e*
^{−(η + δ)τ} and *r*((*k*+1)*τ*) = *r*(*k*
*τ*
^{+})*e*
^{−(η + δ)τ}. Furthermore, *s*((*k*+1)*τ*
^{+})=(1−*σ*
_{1})[1−*r*(*k*
*τ*
^{+})*e*
^{−(η + δ)τ}] and *r*((*k*+1)*τ*
^{+}) = *r*(*k*
*τ*
^{+})*e*
^{−(η + δ)τ} + *σ*
_{1}[1−*r*(*k*
*τ*
^{+})*e*
^{−(η + δ)τ}]. Now, let *y*
_{
k
} = *r*(*k*
*τ*
^{+}). Then, we have

For equation (14), let *y*
_{
k+1} = *y*
_{
k
} = *y*. Then, we have the only fixed point of equation (14) \(\widetilde {y}=\frac {\sigma _{1}}{1-(1-\sigma _{1})e^{-(\eta +\delta )\tau }}\). Thus, there exists the only *τ*-period solution of system (7) \((\widetilde {s}(t), \widetilde {r}(t))\), where \(\widetilde {r}(t)=\widetilde {y}e^{-(\eta +\delta )(t-k\tau )}, \widetilde {s}(t)=1-\widetilde {r}(t)=1-\widetilde {y}e^{-(\eta +\delta )(t-k\tau )}\), and *k*
*τ*<*t*≤(*k*+1)*t*. Moreover, the solution \((\widetilde {s}(t), \widetilde {r}(t))\) is globally asymptotically stable [34].

### Appendix B Proof of Theorem 2

###
*Proof*

We rewrite matrix *J* as follows:

where

and

□

Matrix *W* is rewritten as

where \(W_{1} = \left (\begin {array}{cc} {1 - \sigma _{1} } & 0 \\ {\sigma _{1} } & 1 \end {array}\right )\), \(W_{2} = \left (\begin {array}{cc} 1 & 0 \\ 0 & {1 - \sigma _{2} } \end {array}\right )\), and \(B_{2} = \left (\begin {array}{cc} 0 & 0 \\ 0 & {\sigma _{2} } \end {array} \right )\). Obviously, *W* has four eigenvalues *χ*
_{1}, *χ*
_{2}, *χ*
_{3}, and *χ*
_{4}. According to the Fluoquet theory [34], the necessary and sufficient conditions that the misinformation-free *τ*-period solution of system (6) is locally asymptotically stable are |*χ*
_{1}|<1, |*χ*
_{2}|<1, |*χ*
_{3}|<1, and |*χ*
_{4}|<1. To check the above conditions, we integrate equation (10) in pulse period [0,*τ*] to obtain \(\Phi (\tau ) = e^{{\int }_{0}^{\tau } {J(t)dt} }\). Then we have

where \(e^{{\int }_{0}^{\tau } {A_{1} dt} } = \left (\begin {array}{cc} {e^{- \delta \tau } } & {\eta \tau e^{- \delta \tau } } \\ 0 & {e^{- (\delta + \eta )\tau } } \end {array}\right )\overset {\Delta }{=} \Phi _{11}\). As a result, we have

Hence, *χ*
_{1} and *χ*
_{2} are the eigenvalues of *W*
_{1}Φ_{11}, and *χ*
_{3} and *χ*
_{4} are the eigenvalues of *W*
_{2}Φ_{22}. In fact, *χ*
_{1} and *χ*
_{2} are the solutions of the following equation

where *U* is a unit matrix. We have

From equation (15), we obtain the following equation

Through solving equation (16), we obtain

where *C*=(1−*σ*
_{1} + *σ*
_{1}
*η*
*τ* + *e*
^{−ητ})^{2}−4(1−*σ*
_{1})*e*
^{−ητ}. Obviously, we have |*χ*
_{1}|<1 and |*χ*
_{2}|<1. Next, let *ψ*= max{|*χ*
_{3}|,|*χ*
_{4}|}. Then, we say that the misinformation-free *τ*-period solution of system (6), denoted as \((\widetilde {s}(t),0,0,\widetilde {r}(t))\), is locally asymptotically stable when *ψ*<1.

### Appendix C Proof of Theorem 4

###
*Proof*

From the first equation and the fourth equation of system (6), we obtain

Let \(X = W_{2} e^{{\int }_{0}^{\tau } {Zdt} }\), where \(Z = \left (\begin {array}{ll} - \lambda + \theta + \delta & \alpha \mu (\tilde s(t) + \varepsilon _{1} )\\ \lambda & \beta \mu (\tilde s(t) + \varepsilon _{1} ) - \varepsilon - \delta \end {array}\right )\), and *ε*
_{1} is a sufficiently small positive number. Obviously, there exist two eigenvalues of matrix *X* which are *χ*
_{5} and *χ*
_{6}. Let *χ*= max{|*χ*
_{5}|,|*χ*
_{6}|}. From the condition of the locally asymptotically stability of solution \((\widetilde {s}(t),0,0,\widetilde {r}(t))\) to system (6) |*ψ*|<1, we can find a sufficiently small *ε*
_{1} such that *χ*<1. Now, we construct the following differential pulse system

From Theorem 1, we see that there exists a globally asymptotically stable *τ*-period solution of system (18), denoted as \((\widetilde {x}(t), \widetilde {y}(t))\), and \((\widetilde {x}(t), \widetilde {y}(t))\rightarrow (\widetilde {s}(t), \widetilde {r}(t))\) when *t*→*∞*. Thus, for any *ε*
_{1}>0, *t*
_{1}>0 such that \(\widetilde {y}(t)>\widetilde {r}(t)-\varepsilon _{1}\) when *t*>*t*
_{1}. According to the comparison theorem of the pulse differential equation, we have

□

For simplicity, we assume that equation (19) holds for any time *t*. From the second equation of system (6) and equation (4), we have \(\frac {dl(t)}{dt}=\alpha \mu s(t)i(t)-(\lambda +\theta +\delta )l(t)=\alpha \mu (1-l(t)-i(t)-r(t))i(t)-(\lambda +\theta +\delta )l(t) \leq \alpha \mu (1-l(t)-i(t)-\widetilde {r}(t)+\varepsilon _{1})i(t)-(\lambda +\theta +\delta )l(t)\leq \alpha \mu (\widetilde {s}(t)+\varepsilon _{1})i(t)-(\lambda +\theta +\delta )l(t)\). Similarly, from the third equation of system (6), we have \(\frac {di(t)}{dt}\leq \lambda l(t)+\beta \mu (\widetilde {s}(t)+\varepsilon _{1})i(t)-(\varepsilon +\delta )i(t)\). Next, we construct the following system

We can rewrite equation (20) as

Let \(G(t) = \left (\begin {array}{c} u(t) \\v(t) \end {array}\right )\). Then, we can rewrite equation (21) as

Solving equation (22), we have \(G(t) = G(0)e^{{{\int }_{0}^{t}} {Z(t)dt} }\) when 0<*t*≤*τ*. Thus, for any time *t*∈(*k*
*τ*,(*k*+1)*τ*], we have \(G(t) = G(0)X^{k} e^{{\int }_{k\tau }^{t} {Z(t)dt}}\), where *k* is a non-negative integer. As *X*<1, we have *G*(*t*)→0 when *t*→*∞*. This implies that we have (*l*(*t*),*i*(*t*))→(0,0) when *t*→*∞*.

In the following, we prove that \(r(t)\rightarrow \widetilde {r}(t)\) and \(s(t)\rightarrow \widetilde {s}(t)\) as *t*→*∞*. We have, for any 0<*ε*
_{2}<(*η* + *δ*)*γ*
*μ*, *t*
_{2}>0 such that 0<*l*(*t*)<*ε*
_{2} and 0<*i*(*t*)<*ε*
_{2} when *t*>*t*
_{2}. Without loss of generality, we assume 0<*l*(*t*)<*ε*
_{2} and 0<*i*(*t*)<*ε*
_{2} for any time *t*>0. Then, we have \(-(\eta +\delta )r(t)<\frac {r(t)}{dt}<\varepsilon \varepsilon _{2}+\theta \varepsilon _{2}-\gamma \mu (1-2\varepsilon _{2}-r(t)) \varepsilon _{2}-\eta r(t)-\delta r(t)=\varepsilon \varepsilon _{2}+\theta \varepsilon _{2}-\gamma \mu (1-2\varepsilon _{2}) \varepsilon _{2}-(\eta +\delta -\gamma \mu \varepsilon _{2})r(t)=(\varepsilon +\theta -\gamma \mu )\varepsilon _{2}+2{\gamma \mu \varepsilon _{2}^{2}}-(\eta +\delta -\gamma \mu \varepsilon _{2})r(t) \triangleq \varepsilon _{3}-(\eta +\delta -\gamma \mu \varepsilon _{2})r(t)\), where \(\varepsilon _{3}=(\varepsilon +\theta -\gamma \mu )\varepsilon _{2}+2{\gamma \mu \varepsilon _{2}^{2}}\). Next, we construct the following system

Obviously, system (23) has a globally asymptotically stable *τ*-period solution:

where \(\widetilde {z}^{\ast }=\frac {\frac {(1-\sigma _{1})\varepsilon _{3}}{F}(1-e^{-F\tau })+(1-2\varepsilon _{2})\sigma _{1}+\varepsilon _{2}\sigma _{2}} {1-(1-\sigma _{1})e^{-F\tau }}\), and *F* = *η* + *δ*−*γ*
*μ*
*ε*
_{2}. Thus, for any *ε*
_{4}>0, there exists a time *t*
_{3} such that \(\widetilde {r}^(t)-\varepsilon _{4}<r(t)<\widetilde {z}^(t)+\varepsilon _{4}\) when *t*>*t*
_{3}. When *ε*
_{2}→0 and *t*→*∞*, we have \(\widetilde {r}(t)-\varepsilon _{4}<r(t)<\widetilde {r}(t)+\varepsilon _{4}\). Thus, we obtain \(r(t)\rightarrow \widetilde {r}(t)\). Furthermore, \(s(t)\rightarrow \widetilde {s}(t)\) when *t*→*∞*. This implies that the misinformation-free *τ*-period solution of system (6) \((\widetilde {s}(t), 0, 0, \widetilde {r}(t))\) is globally attractive.

As the misinformation-free *τ*-period solution of system (6) is both locally asymptotically stable and globally attractive, it is globally asymptotically stable.

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Wang, X., Lin, Y., Zhao, Y. *et al.* A novel approach for inhibiting misinformation propagation in human mobile opportunistic networks.
*Peer-to-Peer Netw. Appl.* **10, **377–394 (2017). https://doi.org/10.1007/s12083-016-0438-3

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### Keywords

- Mobile opportunistic network
- Misinformation propagation
- Contact rate
- Mobility property
- Pulse control model
- Stability analysis