SEIR Epidemic + Vaccination

SEIR Model with Vaccination: The SEIR model tracks Susceptible→Exposed→Infectious→Recovered transitions. Vaccination moves S→V (vaccinated) at rate v·ε per susceptible per day. The herd immunity threshold H=1−1/R₀: when the vaccinated fraction exceeds H, R_eff=R₀·(1−v·ε)<1 and the outbreak cannot sustain itself. With R₀=2.5, H=60%. The network view shows stochastic individual-based transmission on a random contact graph. Force of infection β=R₀·γ. Exposed individuals are infectious after incubation 1/σ days; recovered gain immunity. Real pathogens: COVID-19 R₀≈2–5, measles R₀≈12–18, influenza R₀≈1.2–1.4.