Total infections = sum from k=0 to 3 of (2.5)^k for new infections in week k, plus initial - High Altitude Science
Total Infections in a 4-Week Outbreak: Understanding the Progression Using Exponential Growth
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.
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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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.
