An epidemiologist models the spread of a virus in a city of 2 million people. Initially, 0.1% are infected. Each day, the number of infected individuals increases by 15% of the current infected count, while 0.5% of infected recover. After 3 days, how many people are infected (rounded to the nearest whole number)? - Coaching Toolbox

Understanding the Context

Each day, two key factors influence the count:

• Infection growth: The number of infected individuals increases by 15% of the current infected count.
  
• Recovery: Simultaneously, 0.5% of infected individuals recover and move to the recovered group.

Let’s simulate day-by-day to determine total infected after 3 days, with values rounded to the nearest whole number.

Day-by-Day Breakdown

Key Insights

Day 0 (Initial):

• Infected at start: 2,000

Day 1:

• Infection growth:
  Increase = 2,000 × 0.15 = 300
  New infected before recovery = 2,000 + 300 = 2,300
  
• Recoveries:
  Recoveries = 2,000 × 0.005 = 10
  
• Total infected after Day 1:
  2,300 – 10 = 2,290

Day 2:

• Infection growth:
  Increase = 2,290 × 0.15 = 343.5 ≈ 344
  Total before recovery = 2,290 + 344 = 2,634
  
• Recoveries:
  Recoveries = 2,290 × 0.005 = 11.45 ≈ 11
  
• Total infected after Day 2:
  2,634 – 11 = 2,623

Day 3:

• Infection growth:
  Increase = 2,623 × 0.15 = 393.45 ≈ 393
  Total before recovery = 2,623 + 393 = 3,016
  
• Recoveries:
  Recoveries = 2,623 × 0.005 = 13.115 ≈ 13
  
• Total infected after Day 3:
  3,016 – 13 = 3,003

Final Thoughts

Final Result

After 3 days, the number of infected individuals is approximately 3,003, rounded to the nearest whole number.

Using the Formula for Insight

This process mirrors exponential growth with a net daily multiplier:
Each day, infected individuals grow by 15% (×1.15), while recoveries reduce the count by 0.5% (×0.995). The net daily factor is:
1.15 × 0.995 ≈ 1.14425

Using exponential modeling:
Infected after 3 days = 2,000 × (1.14425)³ ≈ 2,000 × 1.503 ≈ 3,006
Approximation confirms the iterative result: rounding differences explain the slight variance due to discrete daily adjustments.

In a city of 2 million, starting with a modest 2,000 infected and applying 15% daily growth offset by 0.5% recovery yields approximately 3,003 infected individuals after 3 days. This model highlights how quickly early outbreaks can escalate—even with partial recovery—emphasizing the importance of timely intervention and surveillance.