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Analyzing the dynamics of a rumor transmission model with incubation. (English) Zbl 1244.91079
Summary: We consider a rumor transmission model with incubation that incorporates constant recruitment and has infectious force in the latent period and infected period. By carrying out a global analysis of the model and studying the stability of the rumor-free equilibrium and the rumor-endemic equilibrium, we use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. It shows that either the number of rumor infective individuals tends to zero as time evolves or the rumor persists. We prove that the transcritical bifurcation occurs at $R_0$ crosses the bifurcation threshold $R_0 = 1$ by projecting the flow onto the extended center manifold. Since the rumor endemic level at the equilibrium is a continuous function of $R_0$, as a consequence for successful eradication of the rumor, one should simply reduce $R_0$ continuously below the threshold value $1$. Finally, the obtained results are numerically validated and then discussed from both the mathematical and the sociological perspectives.

MSC:
91D30Social networks
92D30Epidemiology
37N40Dynamical systems in optimization and economics
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