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An epidemiologist models a disease with a basic reproduction number R₀ = 2.5. In a population of 100,000 with 1 initial case, how many total cases occur after 4 generations of transmission, assuming no immunity?
Generation 0: 1
Generation 1: 1 × 2.5 = 2.5
Generation 2: 2.5 × 2.5 = 6.25
Generation 3: 6.25 × 2.5 = 15.625
Generation 4: 15.625 × 2.5 = 39.0625
Total cases = 1 + 2.5 + 6.25 + 15.625 + 39.0625 = 64.4375 ≈ 64.44
But since we count cumulative infections including initial, and model is geometric:
Sum = (rⁿ⁺¹ - 1)/(r - 1) = (2.5⁵ - 1)/(2.5 - 1) = (97.65625 - 1)/1.5 = 96.65625/1.5 ≈ 64.4375
#### 64.44