Question: An epidemiologist models the spread of a virus in a population with the equation $ p(t) = -t^2 + 14t + 30 $, where $ p(t) $ represents the number of infected individuals at time $ t $. What is the maximum number of infected individuals? - Imagemakers

Understanding the Context

Understanding Virus Spread Through Mathematical Modeling

When an infectious disease begins spreading in a population, epidemiologists use mathematical models to track and predict the number of infected individuals over time. One common model is a quadratic function of the form:
$$ p(t) = -t^2 + 14t + 30 $$
In this model, $ p(t) $ represents the number of infected people at time $ t $, and the coefficient of $ t^2 $ being negative indicates a concave-down parabola, meaning the infection rate rises initially and then declines — forming a peak infection point.

But what does this peak represent? It tells public health officials the maximum number of people infected at a single point in time, crucial for planning healthcare resources, lockdowns, and vaccination campaigns.

Key Insights

Finding the Maximum Infection: The Vertex of the Parabola

To find the maximum number of infected individuals, we must calculate the vertex of the parabola defined by the equation:
$$ p(t) = -t^2 + 14t + 30 $$

For any quadratic function in the form $ p(t) = at^2 + bt + c $, the time $ t $ at which the maximum (or minimum) occurs is given by:
$$ t = -rac{b}{2a} $$

Here, $ a = -1 $, $ b = 14 $. Plugging in the values:
$$ t = -rac{14}{2(-1)} = rac{14}{2} = 7 $$

So, the infection rate peaks at $ t = 7 $ days.

Final Thoughts

Now substitute $ t = 7 $ back into the original equation to find the maximum number of infected individuals:
$$ p(7) = -(7)^2 + 14(7) + 30 $$
$$ p(7) = -49 + 98 + 30 $$
$$ p(7) = 79 $$

Interpretation: The Peak of Infection

At $ t = 7 $ days, the number of infected individuals reaches a maximum of 79 people. After this point, though new infections continue, the rate of decrease outpaces the rate of new infections, causing the total pandemic curve to begin falling.

This insight helps epidemiologists, policymakers, and healthcare providers anticipate when hospitals might be overwhelmed and strategically intervene before peak strain occurs.