Question

the total number of reported cases of an illness in a large city days after the start of an outbreak is modeled by thefunction that is a solution to the logistic differential equation . if there are 5reported cases of the illness initially, what is the limiting value for the total number of reported cases of the illnessas increases?

Answer 1

The limiting value for the total number of reported cases as time increases is known as the carrying capacity 'K' of the environment, a value at which logistic growth levels off. Without specific data, the numerical value of 'K' cannot be determined, but it represents the maximum sustainable number of cases in the logistic growth model.

Step-by-step explanation:

The question pertains to the concept of logistic growth and the carrying capacity in the context of modeling the spread of an illness in a population using differential equations. In logistic growth, an initial exponential phase of the growth is followed by a slow down as the number of cases approach the carrying capacity of the environment. This carrying capacity, often denoted as 'K', represents the maximum number of individuals that can be sustained by the available resources without harming the environment. The logistic growth model predicts that once the population reaches this level, growth will plateau or level off.

Given that there are initially 5 reported cases and assuming the model follows the logistic curve smoothly without overshoots or technological interference, the limiting value for the total number of reported cases will be the carrying capacity 'K'. The information provided does not specify a numerical value for 'K', so we cannot provide a specific number. However, if we had data points from various weeks, as suggested in the example figures, we could estimate 'K' by analyzing the trend of the reported cases over time and determining the point at which they level off.

Answer 2

The logistic differential equation models the spread of an illness, and as time increases, the total number of reported cases approaches the limiting value of 1400.

The correct answer is option B.

The given differential equation dy/dt = (1/5600) * y * (1400 - y) models the spread of an illness in a large city over time. To find the limiting value as time (t) increases, we examine the behavior of the solution y(t).

The logistic differential equation has two equilibrium solutions: y = 0 and y = 1400. When y = 0, the derivative dy/dt is zero, indicating no growth. Similarly, when y = 1400, the derivative is also zero, representing the limiting value where the spread of the illness reaches saturation.

Given the initial condition y(0) = 5, the solution y(t) approaches the limiting value of 1400 as time t increases. Therefore, the answer is:

B. 1400

The question probable may be:

The total number of reported cases of an illness in a large city t days after of an outbreak is modeled by the function y= F(t) that is a solution to the logistic differential equation dy/dt = 1/5600 y(1400-y). If there are 5 reported cases of the illness initially, what is the limiting value for the total number of reported cases of illness as t increases.

A. 700

B. 1400

C. 2800

D. 5600