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. 2024 Mar 12;19(3):e0293807.
doi: 10.1371/journal.pone.0293807. eCollection 2024.

Mathematical modelling, analysis and numerical simulation of social media addiction and depression

Abu Safyan Ali  1   2 , Shumaila Javeed  2   3   4 , Zeshan Faiz  2 , Dumitru Baleanu  3
Affiliations

Mathematical modelling, analysis and numerical simulation of social media addiction and depression

Abu Safyan Ali et al. PLoS One. .

Abstract

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.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Compartmental diagram of SMAD model.
Fig 2
Fig 2. Variation of different parameters on the R0.
Fig 3
Fig 3. Sensitivity Index Of Reproduction Number R0.
Fig 4
Fig 4. PRCC values representing the sensitivities of the model outputs with respect to R0.
Fig 5
Fig 5. Solution of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq0 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq0 with R0 < 1.
Fig 6
Fig 6. Seperate plots of all classes of solution of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq0 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq0 with R0 < 1.
Fig 7
Fig 7. Separate surface plots of solution of all classes of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq0 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq0 with R0 < 1.
Fig 8
Fig 8. Solution of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq1 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq1 with R0 > 1.
Fig 9
Fig 9. Separate plots of solution of all classes of social of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq1 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq1 with R0 > 1.
Fig 10
Fig 10. Comparison of addicted and depresses individuals of social media addiction and depression (SMAD) model at R0 < 1 and R0 > 1.
Fig 11
Fig 11. Separate surface plots of solution of all classes of social of social media addiction and depression model (SMAD) obtained by RK-4 scheme for Eq1 with IC’s and step size h = 0.001.
Numerical scheme converge to true equilibrium point Eq1 with R0 > 1.
Fig 12
Fig 12. Numerical results of social media addiction and depression (SMAD) at different values of Φ = (0.5, 0.7, 0.9).
Fig 13
Fig 13. Numerical results of socail media addiction and depression (SMAD) at different values of τ = (0.05, 0.09, 0.13).
Fig 14
Fig 14. Numerical results of social media addiction and depression (SMAD) at different values of ϕ = (0.6, 0.8, 1).
Fig 15
Fig 15. Numerical results of social media addiction and depression (SMAD) at different values of α = (0.3, 0.5, 0.7).
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