A novel approach for inhibiting misinformation propagation in human mobile opportunistic networks

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.

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

$$ r(t)=r(k\tau^{+})e^{-(\eta+\delta)(t-k\tau)}, k\tau<t\leq (k+1)t. $$

(12)

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

$$ s(t)=1-r(k\tau^{+})e^{-(\eta+\delta)(t-k\tau)}, k\tau<t\leq (k+1)t. $$

(13)

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

$$ y_{k+1}=y_{k}e^{-(\eta+\delta)\tau}+\sigma_{1}\left[1-y_{k}e^{-(\eta+\delta)\tau}\right]. $$

(14)

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:

$$J = \left( \begin{array}{cc} {A_{1} } & {B_{1} } \\ 0 & {A_{2} } \end{array}\right)\Phi (\tau ), $$

where

$$A_{1} = \left( \begin{array}{cc} { - \delta } & \eta \\ 0 & { - (\eta + \delta )} \end{array}\right), $$

$$B_{1}=\left( \begin{array}{cc} 0 & { - (\alpha + \beta + \gamma )\mu\tilde s(t)} \\ \theta & {\varepsilon + \gamma \mu\tilde s(t)} \end{array}\right), $$

and

$$A_{2} = \left( \begin{array}{cc} { - (\lambda + \theta + \delta )} & {\alpha \mu\tilde s(t)} \\ \lambda & {\beta \mu\tilde s(t) - \varepsilon - \delta } \end{array}\right). $$

□

Matrix W is rewritten as

$$J = \left( \begin{array}{cc} {W_{1} } & {B_{2} } \\ 0 & {W_{2} } \end{array}\right)\Phi (\tau ), $$

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

$$\Phi (\tau ) = \left( \begin{array}{cc} {e^{{\int}_{0}^{\tau} {A_{1} dt} } } & {e^{{\int}_{0}^{\tau} {A_{1} dt} }e^{{\int}_{0}^{\tau} {B_{1} dt} } } \\ 0 & {e^{{\int}_{0}^{\tau} {A_{2} dt} } } \end{array}\right)\overset{\Delta}{= } \left( \begin{array}{cc} {\Phi_{11} } & {\Phi_{12} } \\ 0 & {\Phi_{22} } \end{array}\right), $$

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

$$\begin{array}{@{}rcl@{}} &&W = \left( \begin{array}{cc} {W_{1} } & {B_{2} } \\ 0 & {W_{2} } \end{array}\right)\left( \begin{array}{cc} {\Phi_{11} } & {\Phi_{12} } \\ 0 & {\Phi_{22} } \end{array}\right)\\ &&= \left( \begin{array}{cc} {W_{1} \Phi_{11} } & {W_{1} \Phi_{12} + B_{2} \Phi_{22} } \\ 0 & {W_{2} \Phi_{22} } \end{array}\right). \end{array} $$

Hence, χ ₁ and χ ₂ are the eigenvalues of W ₁Φ₁₁, and χ ₃ and χ ₄ are the eigenvalues of W ₂Φ₂₂. In fact, χ ₁ and χ ₂ are the solutions of the following equation

$$ |\chi U-W_{1}\Phi_{11}|=0, $$

(15)

where U is a unit matrix. We have

$$\begin{array}{@{}rcl@{}} &&W_{1} \Phi_{11} = \left( \begin{array}{cc} {1 - \sigma_{1} } & 0 \\ {\sigma_{1} } & 1 \end{array}\right)\left( \begin{array}{cc} {e^{- \delta \tau } } & {\eta \tau e^{- \delta \tau } } \\ 0 & {e^{- (\delta + \eta )\tau } } \end{array}\right)\\ &&\qquad= e^{- \delta \tau } \left( \begin{array}{cc} 1 - \sigma_{1} & (1 - \sigma_{1} )\eta \tau\\ \sigma_{1} & \sigma_{1} \eta \tau + e^{\eta \tau } \end{array}\right). \end{array} $$

From equation (15), we obtain the following equation

$$ \chi^{2}-(1-\sigma_{1}+\sigma_{1}\eta\tau+e^{-\eta\tau})e^{-\delta\tau}\chi+(1-\sigma_{1})e^{-(\eta\tau+2\delta\tau)}=0. $$

(16)

Through solving equation (16), we obtain

$$ \chi_{1,2}=\frac{(1-\sigma_{1}+\sigma_{1}\eta\tau+e^{-\eta\tau}\pm \sqrt{C})e^{-\delta\tau}}{2}, $$

(17)

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

$$\left\{ \begin{array}{l} \frac{ds(t)}{dt} > \delta + \eta r(t) - \delta s(t)\\ \frac{dr(t)}{dt} > - (\eta + \delta )r(t) \end{array}. \right. $$

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

$$ \left\{ \begin{array}{l} \left. \begin{array}{l} \frac{dx(t)}{dt} = \delta + \eta y - \delta x(t) \\ \frac{dy(t)}{dt} = - (\eta + \delta )y(t) \end{array} \right\} t \ne k\tau ,\\ \left. \begin{array}{l} x(t^ + ) = (1 - \sigma_{1} )x(t) \\ y(t^ + ) = y(t) + \sigma_{1} x(t) \end{array} \right\}t = k\tau , \\ x(0^ + ) = s(0^ + ) = s_{0} , \\ y(0^ + ) = r(0^ + ) = r_{0} . \end{array} \right. $$

(18)

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

$$ r(t)\geq y(t)>\widetilde{r}(t)-\varepsilon_{1}. $$

(19)

□

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

$$ \left\{ \begin{array}{l} \left. \begin{array}{l} \frac{du(t)}{dt} = \alpha\mu (\tilde s(t) + \varepsilon_{1} )v(t) - (\lambda + \theta + \delta )u(t) \\ \frac{dv(t)}{dt} = \lambda u(t) + (\beta \mu(\tilde s(t) + \varepsilon_{1} ) - \varepsilon - \delta )v(t) \end{array} \right\}t \ne k\tau , \\ \left. \begin{array}{l} u(t^ + ) = u(t) \\ v(t^ + ) = (1 - \sigma_{2} )v(t) \end{array} \right\}t = k\tau , \\ u(0^ + ) = l(0^ + ) = l_{0} , \\ v(0^ + ) = i(0^ + ) = i_{0} . \end{array} \right. $$

(20)

We can rewrite equation (20) as

$$ \left\{ \begin{array}{l} \frac{d}{dt}\left( \begin{array}{c} {u(t)} \\ {v(t)} \end{array}\right) = Z\left( \begin{array}{c} {u(t)} \\ {v(t)} \end{array}\right),t \ne k\tau , \\ \left( \begin{array}{c} {u(t^ + )} \\ {v(t^ + )} \end{array}\right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & {1 - \sigma_{2} } \end{array}\right)\left( \begin{array}{c} u(t) \\ v(t) \end{array}\right),t = k\tau. \end{array} \right. $$

(21)

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

$$ \left\{ \begin{array}{l} \frac{dG(t)}{dt} = ZG(t),t \ne k\tau , \\ G(t^ + ) = W_{2} G(t),t = k\tau , \\ G(0) = \left( \begin{array}{c} l_{0} \\ i_{0} \end{array}\right). \end{array} \right. $$

(22)

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

$$ \left\{\begin{array}{l} \frac{dz(t)}{dt} = \varepsilon_{3} - (\eta + \delta - \gamma \mu\varepsilon_{2} )z(t),t \ne k\tau ,\\ z(t^ + ) = (1 - \sigma_{1} )z(t) + (1 - 2\varepsilon_{2} )\sigma_{1} + \varepsilon_{2} \sigma_{2} ,t = k\tau , \\ z(0^ + ) = r_{0}. \end{array} \right. $$

(23)

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

$$\widetilde{z}(t)=\frac{\varepsilon}{\eta+\delta-\gamma\mu\varepsilon_{2}}+\left( \widetilde{z}^{\ast}-\frac{\varepsilon_{2}} {\eta+\delta-\gamma\mu\varepsilon_{2}}\right)e^{-(\eta+\delta-\gamma\mu\varepsilon_{2})(t-k\tau)}, $$

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.