A patient’s gut at baseline contains Pathogen R = 1.0 × 10³ cells (highly resistant) and Commensal S = 1.0 × 10⁷ cells (largely susceptible). Under antibiotic therapy, R replicates with net growth r_R = 0.7 day−1 and S has net growth r_S = 0.0 day−1 while therapy continues. Assuming exponential growth for R and no net growth for S during 5 days of therapy, calculate R(t) after 5 days and determine whether R would exceed S in absolute number. Show the calculation steps and final decision; use N(t) = N0·e^rt.

• A. R(5) ≈ 3.3 × 10⁴ cells; R remains far below S (1.0 × 10⁷), so R will not exceed S after 5 days under the given rates.
• B. R(5) ≈ 1.0 × 10⁸ cells; R will exceed and dominate S because resistant organisms always outgrow commensals rapidly under therapy.
• C. R(5) ≈ 1.0 × 10³ cells; R does not grow at all during therapy so it cannot outcompete S under any circumstances.
• D. R(5) ≈ 3.3 × 10⁶ cells; R will approximately equal S and there is a high probability of superinfection within 5 days.

Answer: (A)