Dynamics of an SEIDW epidemic model for primary amebic meningoencephalitis threshold analysis, bifurcation, and optimal environmental control.

Primary amebic meningoencephalitis (PAM), caused by the thermophilic amoeba Naegleria fowleri, is a rare but almost invariably fatal neurological infection. Unlike classical communicable diseases, PAM transmission is exclusively environmentally driven, with no human-to-human transmission. This paper develops and rigorously analyzes a novel susceptible-exposed-infected-dead-water (SEIDW) epidemic model that explicitly couples human infection dynamics with an environmental reservoir of N. fowleri. We establish the fundamental properties of the model including positivity, boundedness, and invariance of solutions. Unlike classical SIR-type models where the basic reproduction number R e derives from human-to-human transmission cycles, the one-directional transmission structure of PAM (environment → human only, with no human-to-environment feedback) renders the standard next-generation matrix approach inapplicable. We instead derive the environmental reproduction number R e = r W / ( δ W + κ C w ) via Jacobian eigenvalue analysis and demonstrate that it serves as a sharp threshold parameter: when R e < 1 , the disease-free equilibrium is globally asymptotically stable; when R e > 1 , a unique endemic equilibrium exists and is globally asymptotically stable under biologically plausible conditions. We conduct a complete bifurcation analysis revealing that the model undergoes a forward transcritical bifurcation at R e = 1 . Sensitivity analysis reveals that R e depends exclusively on environmental parameters (r W , δ W , κC w ), while human parameters (β, σ, α) affect disease burden but not persistence-a finding with profound public health implications. We extend the model to include time-dependent optimal control strategies representing behavioral interventions, environmental sanitation, and personal protective measures. Pontryagin's maximum principle yields characterization of optimal controls, and numerical simulations demonstrate that integrated environmental management is substantially more effective than clinical interventions alone. PRCC and variance-based sensitivity analysis via Latin Hypercube Sampling confirms model robustness under parameter variability. This mathematical framework provides rigorous foundations for designing evidence-based PAM prevention strategies.