A research article estimates that a certain virus spreads such that each infected person infects 1.5 others every week. Starting with 4 infected individuals, how many are infected after 4 weeks?

["Estimating Virus Spread: A Research-Based Model Showing Exponential Growth", "Understanding how a virus spreads is critical for public health planning and intervention. Recent research highlights the concept of basic reproduction number R₀, which measures how many people, on average, one infected person transmits the virus to under ideal conditions. A compelling study estimates that a particular virus spreads with an R₀ of 1.5 — meaning each infected person infects 1.5 new individuals each week.", "This article explores how exponential growth models predict the number of infections over time, using real-world data inspired by such research. Specifically, we examine the scenario starting with 4 initial infections and project how many people are infected after 4 weeks.", "---", "### Horizontal Transmission and Exponential Growth", "In epidemiology, when R₀ > 1, a virus can trigger exponential growth in cases. Each infected individual passes the virus to 1.5 new people per week. Since transmission is typically stepwise (weekly), the number of new infections builds on prior ones.", "Using discrete weekly intervals, the total number of infected individuals grows geometrically:", "[\n\ ext{Total infected after } n \ ext{ weeks} = \ ext{Initial infected} \ imes (R₀)^n\n]", "However, this formula assumes only current infectious individuals transmit, so the total number of new infections per week increases multiplicatively.", "But to calculate the cumulative infections, including everyone ever infected, we model:", "- Week 0: 4 infected (initial)\n- Each infected individual in week k spreads the virus to 1.5 individuals in week k+1", "Thus, new infections per week follow a geometric progression with ratio 1.5.", "Let’s compute week-by-week:", "---", "### Weekly Infection Progression", "- Week 0 (Start): 4 infected\n- Week 1:\n New infections = ( 4 \ imes 1.5 = 6 )\n Total infected = ( 4 + 6 = 10 )", "- Week 2:\n New infections = ( 6 \ imes 1.5 = 9 )\n Total infected = ( 10 + 9 = 19 )", "- Week 3:\n New infections = ( 9 \ imes 1.5 = 13.5 \approx 13.5 ) (keep decimal for precision)\n Total infected = ( 19 + 13.5 = 32.5 )", "- Week 4:\n New infections = ( 13.5 \ imes 1.5 = 20.25 )\n Total infected = ( 32.5 + 20.25 = 52.75 )", "---", "### Final Calculation – Cumulative Infections After 4 Weeks", "Using the full geometric growth for cumulative infections:", "The total number infected after 4 weeks follows the geometric series:", "[\n\ ext{Total} = a \cdot \frac{r^{n+1} - 1}{r - 1}\n]", "Where:\n- ( a = 4 ) (initial infections)\n- ( r = 1.5 ) (multiplier per week)\n- ( n = 4 ) weeks (counting from week 0 to week 4 inclusive)", "[\n\ ext{Total} = 4 \cdot \frac{1.5^5 - 1}{1.5 - 1} = 4 \cdot \frac{7.59375 - 1}{0.5} = 4 \cdot \frac{6.59375}{0.5} = 4 \cdot 13.1875 = 52.75\n]", "---", "### Interpretation", "After 4 weeks, the cumulative number of infected people is approximately 52.75, but since people are whole individuals, we interpret this as 53 people infected when rounded to the nearest whole number.", "However, to maintain scientific precision and align with research estimates, the exact estimate is 52.75, indicating rapid exponential spread driven by a reproduction number of 1.5.", "---", "### Public Health Implications", "Even with modest R₀ of 1.5, unchecked spread leads to explosive case growth. Public health strategies—such as isolation, contact tracing, vaccination, and behavioral changes—are essential to reduce effective reproduction number and flatten the curve.", "---", "Conclusion", "Using a research-backed model of viral transmission, starting from just 4 infected individuals, the virus spreads so that each person infects 1.5 others weekly. After 4 weeks, this exponential trend results in approximately 53 people infected, demonstrating how relatively small initial clusters can escalate into widespread outbreaks within limited time.", "Key takeaway: Early intervention drastically influences outcomes in such exponential epidemics.", "Keywords: virus spread, R₀ 1.5, exponential growth, cumulative infections, public health modeling, infection projection, epidemiological research, infectious disease spread, weekly transmission model."]