Ali Abu Safyan, Javeed Shumaila, Faiz Zeshan, Baleanu Dumitru
Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy.
Department of Mathematics, COMSATS University Islamabad, Islamabad Campus, Islamabad, Pakistan.
PLoS One. 2024 Mar 12;19(3):e0293807. doi: 10.1371/journal.pone.0293807. eCollection 2024.
We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model's sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.
在本研究中，我们构建了一个社交媒体成瘾与抑郁（SMAD）的数学模型。对社交媒体成瘾与抑郁无病平衡点（SMADDFEP）、社交媒体成瘾与抑郁地方病平衡点（SMADEEP）以及基本再生数（R0）等关键方面进行了定性分析。结果表明，如果R0 < 1，SMADDFEP是局部渐近稳定的。利用卡斯蒂略 - 查韦斯定理建立了SMADDFEP的全局渐近稳定性。另一方面，如果R0 > 1，根据李雅普诺夫定理，唯一的地方病平衡点（SMADEEP）是局部渐近稳定的，并且根据中心流形定理，该模型在R0 = 1处呈现前向分岔。为检验该模型的敏感性，我们计算了归一化前向敏感性指数，并进行了偏秩相关系数（PRCC）分析，以描述参数对SMAD的影响。使用四阶龙格 - 库塔（RK - 4）格式获得的数值结果表明，成瘾个体数量的增加会导致抑郁个体数量的增加。
