Question

Health An epidemic was spreading such that \(t\) weeks after its outbreak it had infected \(N_{1}(t)=0.1 t^{2}+0.5 t+150, \quad 0 \leq t \leq 50\) people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model \(N_{2}(t)=-0.2 t^{2}+6 t+200\)

Short answer

The rate of infection before the vaccine was introduced was \(N_{1}'(t)=0.2t + 0.5\) and after the vaccine was introduced, it was \(N_{2}'(t)=-0.4t + 6\). To predict the number of infections after a period, use the respective polynomials and don't forget to adjust the time scale.

Step-by-step solution

Understanding the models used

Let's break down the two models. The first polynomial, \(N_{1}(t)=0.1 t^{2}+0.5 t+150\), governs the infection before the vaccine. Here, \(t\) is the time in weeks since the outbreak began, while \(N_{1}(t)\) is the number of infected people. The second polynomial, \(N_{2}(t)=-0.2 t^{2}+6 t+200\), models the number of infections post-vaccine. \(t\) is also measured from the time of the outbreak, not from the time the vaccine was administered.

Finding the infection rates before the vaccine

By differentiating the first polynomial, \(N_{1}'(t)=0.2t + 0.5\), you get the rate of infection before the vaccine. This shows the number of new infections per week before the vaccine.

Finding the infection rates after the vaccine

Similarly, differentiate the second polynomial to get \(N_{2}'(t)=-0.4t + 6\). This is the rate of infection per week after the vaccine. Notice that the rate is negative, implying the number of infected people was falling.

Predicting situation after given period

Suppose we want to know the number of people infected 40 weeks after the outbreak. We know that 25 weeks after the outbreak, the infection model changed. So for the first 25 weeks, we use the first equation \(N_{1}(t)\) and for the next 15 weeks, we use \(N_{2}(t-25)\), remembering to adjust the \(t\) value since \(N_{2}(t)\) measures time from the outbreak, not from when the vaccine was introduced. Thus, \(N_{1}(25) + N_{2}(15)\) will give us the total number of infections after 40 weeks.

Key concepts

Infection Rate Calculation

Understanding the rate at which infections spread during an epidemic is crucial for effectively managing public health responses. Our example uses polynomial functions to model the number of infections over time, with two different functions representing the periods before and after the introduction of a vaccine.

In the periods defined, infection numbers are represented by the equations: for the pre-vaccine period, it is given as \(N_{1}(t)=0.1 t^{2}+0.5 t+150\), and for the post-vaccine period, as \(N_{2}(t)=-0.2 t^{2}+6 t+200\). Calculating the infection rate involves finding the derivative of the corresponding polynomial.

The derivative of a polynomial gives us the rate of change of the quantity with respect to time—in this case, the infection rate. For the first polynomial, differentiating yields \(N_{1}'(t)=0.2t + 0.5\), indicating the increasing rate of new infections per week before the vaccine. After the vaccine is introduced, the infection rate is represented by the derivative of the second polynomial, \(N_{2}'(t)=-0.4t + 6\), showing a decreasing rate of new infections, which is a good indicator of the vaccine's effectiveness over time.

Differentiation of Polynomials

Differentiation is a method used in calculus to find the rate at which a quantity changes. In our epidemic model, differentiation helps determine the infection rate by finding the derivative of the polynomial function that models the number of infected people as a function of time.

Consider the pre-vaccine infection model, \(N_{1}(t)=0.1 t^{2}+0.5 t+150\). Differentiating this polynomial with respect to time \(t\) involves applying basic rules of differentiation: the power rule, the constant multiple rule, and the addition rule. Applying these rules, we calculate the first derivative as \(N_{1}'(t)=2 \times 0.1t + 0.5 = 0.2t + 0.5\).

For the post-vaccine model, \(N_{2}(t)=-0.2 t^{2}+6 t+200\), the differentiation process is similar. Taking the derivative gives us \(N_{2}'(t)=2 \times -0.2t + 6 = -0.4t + 6\). These derivatives allow us to analyze the rate at which the epidemic spreads before and after the vaccine implementation. They are essential for planning public health interventions and understanding the epidemic's dynamics.

Epidemic Modeling

Epidemic modeling is an invaluable tool for predicting the spread of infectious diseases and evaluating control strategies. Polynomial models, like the ones represented in our exercise, provide simplified versions of reality that can still offer valuable insights.

In constructing a model, we assume factors such as transmission rates, recovery rates, and, as highlighted in this exercise, the impact of vaccination. These models can be highly complex, but polynomial models serve as a more accessible starting point. By analyzing the coefficients and the degree of the polynomials, we can interpret the basic behavior of the epidemic over time.

To predict the situation after a given period, such as 40 weeks, we piece together the data from pre-vaccine and post-vaccine models, which provides a composite view of the epidemic's progression. This bridging of models allows for the adjustment to new circumstances, like the implementation of a vaccine, and offers a more complete analysis of the epidemic's potential course.