A delayed SIAMR prostate cancer model with vertical transmission and treatment: stability analysis and numerical simulation

Vollständiger Abstract

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​In this paper, we propose and analyse a delayed-compartmental mathematical model of prostate cancer dynamics that incorporates vertical transmission and therapeutic intervention. The population is stratified into five compartments: susceptible (S), infected/virally activated (I), asymptomatic infectious under treatment (A), malignant untreated (M), and recovered (R). A discrete time delay τ > 0 is introduced in the force-of-infection term to model the age-dependent onset of susceptibility in middle-aged adult males. Vertical transmission of viral etiology at a rate qΛ ensures that a fraction of newborns enter the infected compartment directly, rendering the disease persistent regardless of the basic reproduction number R₀. We rigorously establish: (i) well-posedness and positive invariance of solutions; (ii) the explicit formula R₀ = β(1−q)Λ / [μ(α+μ)]; (iii) local and global asymptotic stability of the disease-free equilibrium E₀ when R₀ < 1; (iv) existence of a unique endemic equilibrium E* that persists for all q > 0; and (v) existence of a critical delay threshold τ_c beyond which a Hopf bifurcation occurs, generating stable limit cycles. Numerical simulations using a delayed Euler scheme confirm all theoretical findings and illustrate distinct qualitative behaviours across parameter regimes.