Chapter 11

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}0 & -2 \\ 1 & -3\end{array}\right]\)

Some diseases are lethal; not every individual infected by the disease will recover; some will die. Assume that in one unit of time a fraction m of infected individuals will die (m is called the mortality rate). We will assume that the habitat this population lives in is at its carrying capacity. If no individuals die then no reproduction occurs. If individuals die, then resources are freed up and more individuals will be born: one birth for every death that occurs. That is, the number of individuals born in one unit of time is equal to the number of individuals who die in that unit of time. In Problems 38–41 you will analyze models for lethal diseases. In Problems 38–40 you should assume that infants are initially uninfected by the disease but are also not immune to it, so new individuals added to the population are all in the susceptible class. In this problem we will determine the stability of equilibria in an SIRS model that includes mortality. Consider a population of size \(N=100\). The SIRS model with mortality for this population is: $$ \begin{aligned} \frac{d S}{d t} &=-\frac{1}{200} S I+\frac{1}{10} R+\frac{1}{3} I \\ \frac{d I}{d t} &=\frac{1}{200} S I-\frac{2}{3} I \\ \frac{d R}{d t} &=\frac{1}{3} I-\frac{1}{10} R \end{aligned} $$ (a) What is the mortality rate for this disease? (b) Rewrite the system of differential equations as a part of differential equations with \(S\) and \(I\) as dependent variables. (c) Find all equilibria lying within the domain of the system. (d) By linearizing the differential equations around the equilibria that you discovered in (c), classify each of the equilibria (e.g." as stable node, spiral, or saddle).

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}1 & -2 \\ 1 & -3\end{array}\right]\)

Some diseases are lethal; not every individual infected by the disease will recover; some will die. Assume that in one unit of time a fraction m of infected individuals will die (m is called the mortality rate). We will assume that the habitat this population lives in is at its carrying capacity. If no individuals die then no reproduction occurs. If individuals die, then resources are freed up and more individuals will be born: one birth for every death that occurs. That is, the number of individuals born in one unit of time is equal to the number of individuals who die in that unit of time. In Problems 38–41 you will analyze models for lethal diseases. In Problems 38–40 you should assume that infants are initially uninfected by the disease but are also not immune to it, so new individuals added to the population are all in the susceptible class. In this problem we will determine the stability of equilibria in an SIRS model that includes mortality. Consider a population of size \(N=250\). Assuming a mortality rate \(m=1 / 4\), our SIRS model becomes: $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{500} S I+\frac{1}{5} R+\frac{1}{4} I \\ \frac{d I}{d t}=\frac{1}{500} S I-\frac{1}{3} I \\ \frac{d R}{d t}=\frac{1}{12} I-\frac{1}{5} R \end{array} $$ (a) Write the system of differential equations as a part of differential equations with \(S\) and \(I\) as dependent variables. (b) Find all equilibria lying within the domain for this model. (c) By linearizing the differential equations around the equilibria that you discovered in (c), classify each of the equilibria (e.g., as stable node, spiral, or saddle).

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{rr}-2 & -3 \\ 1 & 3\end{array}\right]\)

Some diseases are lethal; not every individual infected by the disease will recover; some will die. Assume that in one unit of time a fraction m of infected individuals will die (m is called the mortality rate). We will assume that the habitat this population lives in is at its carrying capacity. If no individuals die then no reproduction occurs. If individuals die, then resources are freed up and more individuals will be born: one birth for every death that occurs. That is, the number of individuals born in one unit of time is equal to the number of individuals who die in that unit of time. In Problems 38–41 you will analyze models for lethal diseases. In Problems 38–40 you should assume that infants are initially uninfected by the disease but are also not immune to it, so new individuals added to the population are all in the susceptible class. Assume that all individuals in the population are equally likely to be parents to the \(m I\) offspring added to the population in each unit of time. If an offspring is born to an infected parent it will be born infected (i.e., into the infectious class). Similarly, offspring born to susceptible parents are susceptible, and offspring born to recovered parents are recovered. Derive an SIRS model to describe the spread of this disease. There is no need to analyze your model.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be real, distinct, and nonzero. Analyze the stability of the equilibrium \((0,0)\), and classify the equilib\mathrm{\\{} r i u m ~ a c c o r d i n g ~ t o ~ w h e t h e r ~ i t ~ i s ~ a ~ s i n k , ~ a ~ s o u r c e , ~ o r ~ a ~ s a d d l e ~ point. \(A=\left[\begin{array}{ll}4 & -1 \\ 5 & -1\end{array}\right]\)

A particular infectious disease confers lifelong immunity to any individual who recovers from the disease. The population size is \(N=200 .\) Assume that the spread of the disease can be described by an SIR model: $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{100} S I \\ \frac{d I}{d t}=\frac{1}{100} S I-6 I \\ \frac{d R}{d t}=6 I \end{array} $$ Assuming that \(R(0)=0\) initially and \(I(0)=5\), calculate a bound on the maximum number of individuals who will catch the disease.

In Problems 43-56, we consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}2 & -1 \\ 3 & 0\end{array}\right]\)

A particular infectious disease confers lifelong immunity to any individual who recovers from the disease. The population size is \(N=100\). Assume that the spread of the disease can be described by an SIR model: $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{300} S I \\ \frac{d I}{d t}=\frac{1}{300} S I-\frac{1}{9} I \\ \frac{d R}{d t}=\frac{1}{9} I \end{array} $$ (a) Assuming that \(R(0)=0\) initially and \(I(0)=5\), calculate a bound on the maximum number of individuals who will catch the disease. (b) Assume that a vaccination program means that half of the population start out immune to the disease, i.e., \(R(0)=50\). Assume also that there are initially 5 infected individuals (i.e., \(I(0)=5\) ). Recalculate the maximum bound on the number of individuals who will eventually catch the disease.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}-1 & -5 \\ 4 & -3\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}-2 & 4 \\ -3 & -2\end{array}\right]\)

In some diseases (such as herpes simplex), an individual may apparently recover from the disease, and in fact gain immunity to it, but the disease continues to be harbored in the person’s body, breaking out some time after they recover from the initial infection. To model this process we will modify our SIRS model as follows: Since there is no loss of immunity, a = 0. However, in each unit of time a fraction r (r is a constant called the rate of relapse) of the individuals from the recovered class become infected with the disease. You will analyze models for relapsing infections. Assume that the following model can be used to represent the spread of a relapsing infection in a population of size \(N=100\) and with relapse rate \(r=\frac{1}{100}:\) $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{50} S I \\ \frac{d I}{d t}=\frac{1}{50} S I-\frac{1}{10} I+\frac{1}{100} R \\ \frac{d R}{d t}=\frac{1}{10} I-\frac{1}{100} R \end{array} $$ (a) What is the domain for this differential equation system? (b) Find all of the possible equilibria for this system of differential equations. (c) Use the fact that \(S+I+R=100\) to eliminate \(S\) from the system, and to write it as a pair of differential equations with \(I\) and \(R\) as dependent variables. (d) By linearizing the differential equation system near each of the equilibria that you discovered in part (b), classify these equilibria (e.g., as a stable node, spiral, or saddle).

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}1 & -2 \\ 2 & 1\end{array}\right]\)

In some diseases (such as herpes simplex), an individual may apparently recover from the disease, and in fact gain immunity to it, but the disease continues to be harbored in the person’s body, breaking out some time after they recover from the initial infection. To model this process we will modify our SIRS model as follows: Since there is no loss of immunity, a = 0. However, in each unit of time a fraction r (r is a constant called the rate of relapse) of the individuals from the recovered class become infected with the disease. You will analyze models for relapsing infections. Assume that the following model can be used to represent the spread of the infection in a population of size \(N=250\) and with relapse rate \(r=\frac{1}{50}:\) $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{100} S I \\ \frac{d I}{d t}=\frac{1}{100} S I-\frac{1}{5} I+\frac{1}{50} R \\ \frac{d R}{d t}=\frac{1}{5} I-\frac{1}{50} R \end{array} $$ (a) What is the domain for this differential equation system? (b) Find all of the possible equilibria for this system of differential equations. (c) Use the fact that \(S+I+R=250\) to eliminate \(S\) from the system and to write it as a pair of differential equations with \(I\) and \(R\) as dependent variables. (d) By linearizing the differential equation system near each of the equilibria that you discovered in part (b), classify these equilibria (e.g., as a stable node, spiral, or saddle). (e) We will now sketch the direction of flow for the system in the \(I R\) -plane. (i) First sketch the \(\frac{d I}{d t}=0\) and \(\frac{d R}{d t}=0\) isoclines. (ii) Add arrows to show the directions of the vector field on the isoclines that you drew in part (i). Then add arrows showing the direction of the vector field in the regions between isoclines.

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}0 & -1 \\ 1 & -1\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}2 & -4 \\ 2 & -3\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}4 & 5 \\ -3 & -3\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}1 & -4 \\ 1 & -1\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}-1 & 1 \\ -3 & 1\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}3 & -2 \\ 1 & 3\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}0 & -3 \\ 2 & 2\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{rr}1 & 2 \\ -5 & -3\end{array}\right]\)

We consider differential equations of the form \(\frac{d \mathbf{x}}{d t}=A \mathbf{x}(t)\) where $$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ The eigenvalues of A will be complex conjugates. Analyze the stability of the equilibrium \((0,0)\), and classify the equilibrium according to whether it is a stable spiral, an unstable spiral, or a center. \(A=\left[\begin{array}{ll}2 & -3 \\ 3 & -2\end{array}\right]\)

In Problems 57-66, we consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-1 & -2 \\ 1 & 3\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{ll}-2 & 2 \\ -4 & 3\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-1 & -1 \\ 5 & -3\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{ll}2 & 2 \\ 2 & 1\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{ll}1 & 3 \\ 2 & 3\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{ll}-1 & 5 \\ -3 & 1\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-2 & 3 \\ 1 & -4\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}-2 & -7 \\ 1 & 2\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}1 & 2 \\ -1 & -1\end{array}\right]\)

We consider differential equations of the form \(\frac{d x}{d t}=A x(t)\) where \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Analyze the stability of the equilibrium \(\mathbf{( 0 , 0 ) , \text { and classify the }}\) equilibrium. \(A=\left[\begin{array}{rr}2 & 2 \\ 3 & -2\end{array}\right]\)

The following system has two distinct real eigenvalues, but one eigenvalue is equal to 0 : $$ \frac{d \mathbf{x}}{d t}=\left[\begin{array}{ll} 4 & 2 \\ 2 & 1 \end{array}\right] \mathbf{x}(t), x(t)=\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ (a) Find both eigenvalues and the associated eigenvectors. (b) From the general solution of (11.27) find \(x_{1}(t)\) and \(x_{2}(t)\). (c) The vector field is shown in Figure \(11.38\). Sketch the lines corresponding to the eigenvectors. Compute \(d x_{2} / d x_{1}\), and conclude that all direction vectors are parallel to the line of eigenvectors corresponding to the nonzero eigenvalue. Describe in words how solutions starting at different points behave.