Transmission dynamics of yellow fever with toxic infected population

This study develops a deterministic two-sex compartmental model to investigate the population-level effects of continuous sterile-male releases on Anopheles mosquitoes. The model distinguishes aquatic stages, unmated females, fertile-mated females, sterile-mated females, wild males, and released sterile males, while incorporating density-dependent aquatic recruitment, sex-specific adult emergence, natural mortality, and mating competition between wild and sterile males. We establish the nonnegativity and boundedness of solutions and characterise the wild-mosquito-free and nontrivial wild-mosquito equilibria. The wild-mosquito-free equilibrium is locally asymptotically stable because viable reproduction requires the joint presence of unmated females and wild males, generating a mate-finding Allee effect near extinction. Accordingly, population persistence is characterised by an explicit release-dependent threshold rather than a conventional infectious-disease reproduction number. When the intrinsic demographic condition $\mathcal{Q}>1$ holds, two nontrivial wild-mosquito equilibria may coexist below the critical release rate. Numerical eigenvalue calculations at sampled baseline release levels identify the lower branch as unstable and the upper branch as locally asymptotically stable. A center-manifold/Lyapunov–Schmidt argument establishes a nondegenerate saddle–node bifurcation of the full six-dimensional system at the critical continuous release rate. Numerical simulations are consistent with bistability under subcritical releases and show suppression of the selected established wild population under the sustained supercritical scenarios examined here; these observations are not asserted as a global convergence theorem. For the illustrative baseline parameter set, the critical release rate is approximately 1850 sterile males per day. One-at-a-time and Latin-hypercube/partial rank correlation analyses identify recruitment, maturation, adult survival, sterile-male mortality, and mating competitiveness as major determinants of the release requirement and post-intervention mosquito abundance. Although the numerical threshold is not an operational release recommendation, the framework provides an analytically transparent basis for evaluating continuous sterile-insect interventions and identifying the parameters requiring reliable field estimation.