A Discrete-Time Compartmental Model of Problematic Pornography Use Dynamics with Clinical Intervention, Analysed Using Discrete Physics-Informed Neural Networks (D-PINNs)

Abstract

Continuous-time compartmental models, formulated as systems of ordinary differential equations, are the dominant mathematical framework for describing the population-level
dynamics of behavioural-health conditions such as problematic pornography use. In practice, however, the longitudinal data available to clinicians and researchers, periodic clinical-screening waves, survey administrations, or treatment-program intake records, are inherently discrete, sampled at fixed reporting intervals rather than continuously. This paper develops and analyses a discrete-time counterpart to a continuous compartmental model of problematic pornography use dynamics under clinical intervention, formulated directly as a system of nonlinear difference equations rather than as a discretisation applied after the fact. The model partitions a population into four interacting groups, evaluated at discrete reporting indices n = 0,1,2,...: susceptible individuals (Sn), individuals exhibiting problematic or compulsive use patterns (Pn), individuals engaged in clinical treatment (Cn), and recovered individuals (Rn). We establish the positivity and boundedness of solutions, locate the problematic-use-free and persistent-use equilibria, which coincide algebraically with those of the underlying continuous system, and derive the same basic reproduction number R0 via the Next Generation Matrix method. Local stability is then analysed using the discrete-time Jacobian and the Jury (discrete Routh–Hurwitz) stability criterion, requiring all eigenvalues to lie strictly within the unit circle, a materially different and generally more restrictive condition than the continuous requirement of negative real part. We show analytically that the discrete and continuous eigenvalues are related by a unit shift, and derive an explicit, computable sufficient condition under which continuous-time stability at R0 < 1 carries over to the discrete-time system. To complement this analysis, we implement a discrete Physics-Informed Neural Network (D-PINN) that enforces the governing difference equations directly as a residual loss, without requiring automatic differentiation, and use it to solve the forward and inverse problems from synthetic discretely sampled observational data. The trained network reproduces the forward trajectories with normalized root-mean-square errors below 0.55 across all four compartments, and recovers three of the four unknown parameters with relative error under 10%, while the clinical-intervention-efficacy parameter is markedly more weakly identifiable under discrete, sparser sampling than in the continuous-time formulation, a finding we discuss in detail.