Abstract
We propose to model the phenomenon of the spread of a rumor in social networks in this paper. From an existing SIR model, we manipulate a new one that is based on the model of cholera in order to take into account professional pages that specialize in spreading rumors. In the second part, we introduce a control strategy to fight against the diffusion of the rumor. Our main objective is to characterize the three optimal controls that minimize the number of spreader users, fake pages, and the corresponding costs. For that matter, using the maximum principle of Pontryagin, we prove the existence and we give characterization of our controls. Numerical simulations are given to concretize our approach.
Highlights
In 3, we present an optimal control problem for the proposed model where we give some results concerning the existence of the optimal control, and we characterize the optimal controls using the Pontryagin maximum principle in discrete time
In order to demonstrate the effectiveness of the model we have proposed, we will present a numerical simulation with the following figure so that we can see how well the model adapts to reality
Initial values are approximate data that we suggested after studying and researching some statistics about the users of social networks; the values are attached in the table
Summary
If I(0) ≥ 0, Sp(0) ≥ 0, St(0) ≥ 0, and F(0) ≥ 0, the solutions I(t), Sp(t), St(t), and F(t) of system (1) are positive for all t ≥ 0. We prove that I(t) ≥ 0, Sp(t) ≥ 0, St(t) ≥ 0, and F(t) ≥ 0
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