MATHEMATICAL ANALYSIS OF RUMOUR SPREADING MODEL IN SOCIAL NETWORKS

Abstract

The rapid spread of rumours through online social networks significantly influences public perception, decision-making, and societal stability. This study extends the traditional Susceptible-Infectious-Hibernator-Recovered (SIHR) rumour transmission model by incorporating external triggers (y) and delayed remembering mechanisms governed by a time delay (r ). The extended model captures the complex interplay between forgetting, memorydriven re-engagement, and external triggers of rumour activity. The system of delay differential equations is solved using the Fourth-Order Runge-Kutta (RK4) method implemented in Python and the Differential Transform Method (DTM) in Maple for comparative numerical analysis. A comprehensive stability analysis is conducted to identify equilibrium states and evaluate the longterm behaviour of the system under varying parameter conditions. Simulation results highlight the critical roles of remembering mechanisms, forgetting rates, and external stimuli in determining whether a rumour dies out or persists. The time-delay-dependent reproduction number is derived to identify bifurcation points and thresholds for instability. The fmdings offer practical insights into how memory and external triggers influence rumour spread, informing the design of timely and effective strategies for misinformation control in contexts such as crisis response, political communication, and public health.