This article presents a mathematical model designed to simulate the impact of targeted interventions aimed at preventing HIV transmission among female sex workers (FSWs) and their clients, while also analyzing their effects on the health of the general population. The compartmental model distinguishes between high-risk populations (FSWs and their clients) and low-risk populations (sexually active men and women in the general population), and links prevention efforts in high-risk groups to the evolution of the epidemic in the low-risk population. The fundamental properties of the model, such as the positivity of solutions and the boundedness of the system, have been verified, and the basic reproduction number R<SUB>0</SUB> has been calculated. Finally, the stability of the model was studied using Varga’s theorem and the Lyapunov method. Simulation results show that targeted prevention among FSWs and their clients reduces HIV incidence in the general population. This framework provides a valuable tool for guiding policymakers in the design of effective strategies to combat the epidemic, especially relevant in the context of suspension of USAID funding.

A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2265;</mo><mn>4</mn></mrow></semantics></math>, the residual set <math><semantics><mrow><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>&#x007B;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2212;</mo><mi>p</mi><mo>&#x2223;</mo></mrow></semantics></math> <math><semantics><mrow><mi>p</mi><mo>&#x003C;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>,</mo><mi>p</mi><mo>&#x2208;</mo><mi>&#x2119;</mi><mo>&#x007D;</mo></mrow></semantics></math> is defined, and the universal residual set <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub><mo>=</mo><msub><mrow><msup><mstyle mathsize='140%' displaystyle='true'><mo>&#x222A;</mo></mstyle><mtext>&#x200B;</mtext></msup></mrow><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow></msub><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math> is constructed. It is shown that <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub></mrow></semantics></math> contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies <math><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x2265;</mo><mn>2</mn></mrow></semantics></math> for all <math><semantics><mrow><mi>E</mi><mo>&#x2265;</mo><mn>8</mn></mrow></semantics></math>, and that the cumulative partition count satisfies <math><semantics><mrow><munder><mstyle mathsize='140%' displaystyle='true'><mo>&#x2211;</mo></mstyle><mrow><mi>E</mi><mo>&#x2264;</mo><mi>N</mi></mrow></munder><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x226B;</mo><mfrac><mrow><msup><mi>N</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mtext>log</mtext></mrow><mn>4</mn></msup><mi>N</mi></mrow></mfrac></mrow></semantics></math>. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi></mrow></semantics></math> is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>7</mn></mrow></semantics></math> can be expressed as <math><semantics><mrow><mi>n</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn>2</mn><mi>q</mi></mrow></semantics></math>, where <math><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>&#x2208;</mo><mi>&#x2119;</mi></mrow></semantics></math>. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>9</mn></mrow></semantics></math>. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence.