Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.3.40c

Chapter 9, Problem 9.3.40c

{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation
dP/dt=kP(1−P/A),P0=P_0,
where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

c. For a fixed value of K and A, describe the long-term behavior of the solutions, for any P0 with 0<P0<A.

Verified step by step guidance

Identify the logistic differential equation given: \[\frac{dP}{dt} = kP\left(1 - \frac{P}{A}\right),\] where \(k > 0\) is the infection rate, \(A\) is the total population, and \(P(t)\) is the number of infected individuals at time \(t\).

Recognize that the logistic equation has two equilibrium (steady-state) solutions where \[\frac{dP}{dt} = 0.\] These occur when either \(P = 0\) or \(P = A\).

Analyze the stability of these equilibria by considering the sign of \[\frac{dP}{dt}\] near these points: For \(0 < P < A\(, the term \[1 - \frac{P}{A}\] is positive, so \[\frac{dP}{dt} > 0,\] meaning \)P\) increases; for \(P > A\(, \[1 - \frac{P}{A} < 0,\] so \[\frac{dP}{dt} < 0,\] meaning \)P\) decreases.

From this, conclude that \(P = 0\) is an unstable equilibrium (since any small positive initial infection grows), and \(P = A\) is a stable equilibrium (since the infection level tends to approach \(A\) over time).

Therefore, for any initial infected population \(P_0\) with \(0 < P_0 < A\), the solution \(P(t)\) will increase over time and approach the total population \(A\) as \(t \to \infty\), modeling the spread of the infection until everyone is infected.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logistic Differential Equation

The logistic differential equation models population growth with a limiting factor, expressed as dP/dt = kP(1 - P/A). Here, P(t) represents the population at time t, k is the growth rate, and A is the carrying capacity or maximum population. The term (1 - P/A) slows growth as P approaches A, reflecting resource limitations or saturation effects.

Equilibrium Solutions and Stability

Equilibrium solutions occur where the rate of change is zero (dP/dt = 0), meaning the population remains constant. For the logistic equation, equilibria are at P = 0 and P = A. Stability analysis shows that P = 0 is unstable (small infections grow), while P = A is stable, indicating the population tends to this value over time.

Long-term Behavior of Solutions

For initial values 0 < P0 < A, the solution P(t) increases over time and approaches the stable equilibrium A as t → ∞. This means the number of infected individuals will grow until it saturates at the total population size A, modeling the spread of the epidemic until everyone is infected in the absence of recovery or intervention.

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Related Practice

Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1.

y′(t) = 2y + 4

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Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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Textbook Question

46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t)+km(t)=I, where m(t) is the mass of the drug in the blood at time t≥0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I=10mg/hr and k=0.05 hr^−1.

c. What is the equilibrium solution?

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Textbook Question

{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).

c. Find the solution of this separable equation assuming v(0)=0 and 0<v<g/b.

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Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions.

Textbook Question

{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.

c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).