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For the sake of simplicity, we set [THETA](t) = [[summation].sub.k']P(k' | k)[S.sub.k],(t) = (1/(k)) [[summation].sub.k'], k'P(k')[S.sub.k'](t) representing the probability that a randomly selected neighbor of a given node is a spreader while [PHI](t) = [[summation].sub.k']P(k' | k)([R.sub.k],(t)) = (1/(k)) [[summation].sub.k'], k'P(k')[R.sub.k'](t) denoting the probability that a randomly selected neighbor of a given node is a stifler. Then, system (1) can be rewritten as
Equation (1) indicates that the density of ignorants varies over time t, which consists of three parts: the density of ignorants varies over time t caused by ignorants contacting spreaders and ignorants becoming spreaders at the rate [lambda]; the density of ignorants varies over time t caused by ignorants contacting spreaders and ignorants becoming stiflers at the rate y due to refusal mechanism; the density of ignorants varies over time t caused by ignorants contacting spreaders and ignorants becoming stiflers at the rate [beta] due to refutation mechanism.
Finally, when the system reaches a stable state, there are no spreaders and only ignorants and stiflers remain.
Calculated by the Runge-Kutta method, Figure 4 vividly shows how the density of spreaders and stiflers changes over time steps on homogeneous network before and after the refutation mechanism is considered, respectively.
Figure 6 shows how the densities of stiflers change over time for different refuting rates.
We then derived mean-field equations to describe changes in the proportions of spreaders, ignorants, and stiflers. The classical SIR rumor-spread model and Nekovee's rumor-spread model with a forgetting mechanism are special cases of the model.