Optimal control of a delayed rumor propagation model with saturated control functions and $$L^1$$
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ORIGINAL ARTICLE
Optimal control of a delayed rumor propagation model with saturated control functions and L1‑type objectives Abdelhadi Abta1 · Hassan Laarabi2 · Mostafa Rachik2 · Hamad Talibi Alaoui3 · Salahaddine Boutayeb1 Received: 11 February 2020 / Revised: 7 July 2020 / Accepted: 7 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Rumor is an important form of social interaction, and its spreading has a significant impact on human lives. The optimal control theory is an important tool to better manage the spread of rumors. Most of the literature on rumor propagation models deals with quadratic cost functions relative to the control variable. In this paper, we have considered a time-delay rumor propagation model with saturated control functions and an objective function of L1-type linear with respect to the control variables. In the general case, the introduction of the delay in the dynamic systems represents the time lag between the action on the system and the response of the system to this action. The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation. Keywords Rumor propagation · Delay differential equations · Optimal control
1 Introduction Rumors are part of our everyday life, and its spreading has a significant impact on human lives. Rumor generally refers to any subject that can be propagated from person to another. As it is well known, rumor spreading, as an important topic in psychology and sociology, has a significant influence on a variety of human affairs, for example, causing damaging and perhaps even deadly consequences or shaping the public opinion of a society or a market (Kosfeld 2005; Nekovee et al. 2007). The influence of rumors is too great to be ignored; however, the prevention and control of rumors are also very difficult (Wang et al. 2014). Therefore, studying the features of rumor * Abdelhadi Abta [email protected] 1
Department of Mathematics and Computer Science, Poly‑disciplinary Faculty, Cadi Ayyad University, P.O Box 4162, Safi, Morocco
2
Analysis, Modeling and Simulation Laboratory, Faculty of Sciences Ben M’Sik, Hassan II University, Av Driss El Harti, P.O Box 7955, Sidi Othmane, Casablanca, Morocco
3
Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, P.O Box 20, El Jadida, Morocco
propagation and then mastering their propagation mechanisms are very important to reduce in an optimal way the effect of rumor spreading (see, e.g., Huo and Ma 2018; Kandhway and Kuri 2014a, b; Lin et al. 2015 and references cited therein). Up to now, several rumor propagation models have been well studied. The most popular transmission model of rumors, conceptually similar to the susce
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