Total Infections in a 4-Week Outbreak: Understanding the Progression Using Exponential Growth

When modeling infectious disease spread, one key question is: how many total people will be infected over the first four weeks? This article explains a fundamental calculation: the sum of infections over time using exponential growth, specifically the formula:

Total Infections = Sum from k=0 to 3 of (2.5)^k plus initial

Understanding the Context

This recurring model helps public health analysts estimate early-stage transmission dynamics and plan interventions effectively.

What Does the Formula Represent?

The expression sum from k=0 to 3 of (2.5)^k computes new infections week by week, where each term represents the number of new infections during week k, starting with week 0 (the initial case). Multiplying this sum by the initial number of infections gives the total infections across four weeks.

Total infections = sum from k=0 to 3 of (2.5)^k for new infections in week k, plus initial

Key Insights

Breaking Down the Weekly Infections

Using a growth factor of 2.5, the daily exponential spread model projects:

  • Week 0 (Initial):
    New infections = (2.5)^0 = 1
    Assumed: 1 initial infected individual

  • Week 1:
    New infections = (2.5)^1 = 2.5

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But question implies multiple.
Insight: Perhaps "each holds half of the previous" starts after the first, and the first is 729, second is 364.5, but the *total* to be poured is more? But it’s only 729.
Another possibility: The initial 729 ml is split into portions where each portion is half the size of the container that holds it, and containers are nested with halving capacity. But we are to find how many containers are used to hold the entire volume if each subsequent container holds half the previous.

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Final Thoughts

  • Week 2:
    New infections = (2.5)^2 = 6.25

  • Week 3:
    New infections = (2.5)^3 = 15.625

Each value reflects compounded spread—each generation of infections fuels the next, consistent with a reproduction number R ≈ 2.5.

Calculating the Total Infections

We sum the week-by-week infections:

Total infections (weeks 0–3) = (2.5)^0 + (2.5)^1 + (2.5)^2 + (2.5)^3
= 1 + 2.5 + 6.25 + 15.625
= 25.375

If multiplied by the initial case (1), the total new infections across four weeks is 25.375. This continuous model approximates cumulative exposure in early outbreak phases.

Why This Model Matters