Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

It may be that recovered individuals do not have life time immunity; they become susceptible after a period \(p,\) and one may write $$ \begin{array}{l} s^{\prime}(t)=\alpha r(t-p)-\beta \times s(t) \times i(t) \\ i^{\prime}(t)=\beta \times s(t) \times i(t)-\gamma i(t) \\ r^{\prime}(t)=\gamma i(t)-\alpha r(t-p) \end{array} $$ This system is considerably more complex than Equations \(18.61,\) and is simplified by letting \(p=0 .\) $$ \begin{aligned} s^{\prime}(t) &=\alpha r(t)-\beta \times s(t) \times i(t) \\ i^{\prime}(t) &=\beta \times s(t) \times i(t)-\gamma i(t) \\ r^{\prime}(t) &=\gamma i(t)-\alpha r(t) \end{aligned} $$ This system involves three functions and three equations and is beyond our exposition. However, you may be able to analyze it. a. Show that \(\left(s_{0}, 0,0\right)\) is an equilibrium point, for any \(s_{0}\). b. Guess or compute the Jacobian matrix at \(\left(s_{0}, 0,0\right)\). c. The characteristic values of the Jacobian matrix at \(\left(s_{0}, 0,0\right)\) are \(\beta s_{0}-\gamma\) and \(-\alpha .\) What is the criterion for an epidemic (the number of infected will increase when a small number, \(\epsilon,\) of infected individuals enter a population of \(s_{0}\) susceptible). d. Solve the equations $$ \begin{array}{ll} x^{\prime}(t)=-\beta s_{0} y(t)+\alpha z(t) & x(0)=s_{0} \\ y^{\prime}(t)=\left(\beta s_{0}-\gamma\right) y(t) & y(0)=\epsilon \\ z^{\prime}(t)=\gamma y(t)-\alpha z(t) & z(0)=0 \end{array} $$ Solve first for \(y(t)\), then for \(z(t)\) and then for \(x(t)\).

Step-by-Step Solution

Answer

An epidemic occurs if \(\beta s_0 - \gamma > 0\). Solve \(y(t) = \epsilon e^{(\beta s_0 - \gamma)t}\), include in \(z(t), x(t)\).

Step 1: Finding Equilibrium Point

To show that \((s_0, 0, 0)\) is an equilibrium point, we need to check if the derivatives \(s'(t), i'(t), r'(t)\) are all zero at \((s_0, 0, 0)\). Substituting \(s(t) = s_0\), \(i(t) = 0\), and \(r(t) = 0\) into the equations:\[ s'(t) = \alpha \cdot 0 - \beta \cdot s_0 \cdot 0 = 0 \]\[ i'(t) = \beta \cdot s_0 \cdot 0 - \gamma \cdot 0 = 0 \]\[ r'(t) = \gamma \cdot 0 - \alpha \cdot 0 = 0 \]All derivatives are zero indicating that \((s_0, 0, 0)\) is an equilibrium point.

Step 2: Compute Jacobian Matrix

The Jacobian matrix at a given point provides information about the system dynamics. Here, we compute the Jacobian at \((s_0, 0, 0)\). First, we identify partial derivatives:1. \(\frac{\partial s'}{\partial s} = -\beta i(t)\)2. \(\frac{\partial s'}{\partial i} = -\beta s(t)\)3. \(\frac{\partial s'}{\partial r} = \alpha\)4. \(\frac{\partial i'}{\partial s} = \beta i(t)\)5. \(\frac{\partial i'}{\partial i} = \beta s(t) - \gamma\)6. \(\frac{\partial i'}{\partial r} = 0\)7. \(\frac{\partial r'}{\partial s} = 0\)8. \(\frac{\partial r'}{\partial i} = \gamma\)9. \(\frac{\partial r'}{\partial r} = -\alpha\)Substitute \((s_0, 0, 0)\):\[\begin{pmatrix}0 & -\beta s_0 & \alpha \0 & \beta s_0 - \gamma & 0 \0 & \gamma & -\alpha\end{pmatrix}\]

Step 3: Characteristic Values of the Jacobian Matrix

The characteristic values or eigenvalues of the Jacobian matrix give insights into the stability of the system.For the diagonal form:- Given eigenvalues are \(\beta s_0 - \gamma\) for the upper left 2x2 block and \(-\alpha\) for the remaining element.- If \(\beta s_0 - \gamma > 0\), the system has instability leading to an epidemic.

Step 4: Criterion for an Epidemic

The epidemic occurs if the number of infected increases, i.e., if \(\beta s_0 - \gamma > 0\), leading to instability or exponential growth of infected individuals in the population.

Step 5: Solve for \(y(t)\)

Start by solving the equation for \(y(t)\):\[y'(t) = (\beta s_0 - \gamma) y(t), \quad y(0) = \epsilon\]This is a simple first-order linear differential equation with the solution:\[y(t) = \epsilon e^{(\beta s_0 - \gamma)t}\]

Step 6: Solve for \(z(t)\)

Use the solution for \(y(t)\) to solve for \(z(t)\):\[z'(t) = \gamma y(t) - \alpha z(t)\]Substitute \(y(t) = \epsilon e^{(\beta s_0 - \gamma)t}\) into the equation:\[z'(t) = \gamma \epsilon e^{(\beta s_0 - \gamma)t} - \alpha z(t)\]This can be solved using integrating factors or similar methods; the analytical form will vary accordingly.

Step 7: Solve for \(x(t)\)

Finally, solve for \(x(t)\):\[x'(t) = -\beta s_0 y(t) + \alpha z(t)\]Now substitute the known \(y(t) = \epsilon e^{(\beta s_0 - \gamma)t}\) and the derived expression for \(z(t)\) to find \(x(t)\). This involves integrating the expressions accurately to find how \(s(t) = x(t)\) changes over time.

Key Concepts

Epidemiology Models

In understanding disease dynamics, epidemiology models play a crucial role. They help researchers and scientists predict how diseases spread and develop strategies to control them. One common model used is the SIR model, which divides a population into susceptible (S), infected (I), and recovered (R) compartments. In this exercise, we consider an extension where recovered individuals might lose immunity and become susceptible again.
This complexity is introduced by replacing traditional equations with delay differential equations, accounting for the time factor, \(p\). However, to simplify the model, the delay is set to zero, changing the system dynamics.
The model equations are:

• \(s'(t) = \alpha r(t) - \beta \cdot s(t) \cdot i(t)\)
• \(i'(t) = \beta \cdot s(t) \cdot i(t) - \gamma i(t)\)
• \(r'(t) = \gamma i(t) - \alpha r(t)\)

By analyzing these equations, we gain insights into how an infection can move through a population under varying conditions.

Equilibrium Point

In the context of differential equations, an equilibrium point is a state where the system remains constant over time if left undisturbed. For our model, the point \((s_0, 0, 0)\) represents a situation where everyone is susceptible but no one is infected or recovered.
To confirm this as an equilibrium point, we substitute \(s(t) = s_0\), \(i(t) = 0\), and \(r(t) = 0\) into the model's differential equations.

• \(s'(t) = \alpha \cdot 0 - \beta \cdot s_0 \cdot 0 = 0\)
• \(i'(t) = \beta \cdot s_0 \cdot 0 - \gamma \cdot 0 = 0\)
• \(r'(t) = \gamma \cdot 0 - \alpha \cdot 0 = 0\)

Since all derivatives are zero, the point remains stable, making it an equilibrium. The system does not change unless disturbed, such as introducing a small number of infected individuals.

Jacobian Matrix

The Jacobian matrix helps analyze the system's stability near an equilibrium point by capturing partial derivatives of each function with respect to each variable. It’s a crucial tool in determining how small changes in the population affect the overall dynamics.
For the equilibrium point \((s_0, 0, 0)\), we compute partial derivatives to construct the Jacobian:

• \(\frac{\partial s'}{\partial s} = -\beta i(t)\)
• \(\frac{\partial s'}{\partial i} = -\beta s(t)\)
• \(\frac{\partial s'}{\partial r} = \alpha\)
• \(\frac{\partial i'}{\partial s} = \beta i(t)\)
• \(\frac{\partial i'}{\partial i} = \beta s(t) - \gamma\)
• \(\frac{\partial i'}{\partial r} = 0\)
• \(\frac{\partial r'}{\partial s} = 0\)
• \(\frac{\partial r'}{\partial i} = \gamma\)
• \(\frac{\partial r'}{\partial r} = -\alpha\)

After substitution, the Jacobian simplifies to:\[\begin{pmatrix}0 & -\beta s_0 & \alpha \0 & \beta s_0 - \gamma & 0 \0 & \gamma & -\alpha\end{pmatrix}\]
Analyzing this matrix reveals the nature and stability of the equilibrium.

Stability Analysis

Stability analysis uses the Jacobian matrix to determine the behavior of an equilibrium point. By examining its eigenvalues, we can predict if small perturbations grow or dissipate over time.
For our system, the Jacobian's characteristic equations yield eigenvalues: \(\beta s_0 - \gamma\) and \(-\alpha\). These values provide a criterion for stability.

• If \(\beta s_0 - \gamma > 0\), the eigenvalue is positive, indicating instability at the equilibrium. This means that introducing a small number of infected individuals will lead to an increase in infections, resulting in an epidemic.
• If \(\beta s_0 - \gamma \leq 0\), the equilibrium is stable, and infections do not increase.

This analysis is crucial in epidemiology to predict outbreaks and create strategies to mitigate the spread of disease.