Problem 35 - Calculus for Biology and Medicine

Question

The spread of a disease through a population of 250 individuals is represented by the following SIRS model; $$ \begin{array}{l} \frac{d S}{d t}=R-\frac{1}{50} S I \\ \frac{d I}{d t}=\frac{1}{50} S I-\frac{1}{10} I \\ \frac{d R}{d t}=\frac{1}{10} I-R \end{array} $$ In this problem we will sketch the directions of the solution in the SI-plane. (a) Eliminate \(R\) to rewrite the equation system as a system of differential equations in the dependent variables \(S(t)\) and \(I(t)\). (b) Draw the zero isoclines of your system from part (a). (c) Find all of the equilibria for this model and classify them (e.g., as stable nodes, unstable nodes, or saddles) by analyzing the linearized system. (d) Add to your plot from part (b) arrows showing the direction of the vector field on the isoclines, and in the regions between the isoclines.

Step-by-Step Solution

Answer

R is eliminated to focus on S and I, obtaining equilibria at (5,0), and sketching reveals the direction field.

Step 1: Eliminate Variable R

To eliminate R, use the fact that the total population is constant, i.e., \( S + I + R = 250 \). From this, we can express R as \( R = 250 - S - I \). Substitute this into the equation for \( \frac{dS}{dt} \): \( \frac{dS}{dt} = (250 - S - I) - \frac{1}{50}SI \).

Step 2: Rewrite Equations in Terms of S and I

Substitute \( R = 250 - S - I \) back into the system:\[ \begin{align*} \frac{dS}{dt} &= (250 - S - I) - \frac{1}{50} SI, \ \frac{dI}{dt} &= \frac{1}{50} SI - \frac{1}{10} I. \end{align*} \]

Step 3: Determine Zero Isoclines

The zero isoclines occur when the derivatives are zero. For \( \frac{dS}{dt} = 0 \), we get \( (250 - S - I) = \frac{1}{50} SI \). For \( \frac{dI}{dt} = 0 \), we derive \( \frac{1}{50} SI = \frac{1}{10}I \), simplifying to \( S = 5 \).

Step 4: Find Equilibria

Solve for equilibrium points by setting both derivatives to zero. Solving \( \frac{dS}{dt} = 0 \) and \( \frac{dI}{dt} = 0 \) yields solutions like \((S, I) = (5,0)\). Analyze by substituting these back to confirm that they are equilibria.

Step 5: Analyze Stability of Equilibria

Using linearization, consider the Jacobian matrix derived from the system. For \( (S, I) = (5,0) \), evaluate the Jacobian. The eigenvalues of the Jacobian at this point determine stability (e.g., if all real parts are negative, it is a stable node).

Step 6: Sketch the Direction Field

Plot isoclines \((S = 250 - I, S = 5)\). Use calculated equilibria and analyze stability to add arrows indicating vector field direction along isoclines and intermediate regions.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They play a vital role in modeling real-world situations where quantities change over time. In the context of the SIRS model, differential equations are used to represent the rate of change of susceptible (\( \frac{dS}{dt} \)), infected (\( \frac{dI}{dt} \)), and recovered (\( \frac{dR}{dt} \)) populations.

• The equation for susceptible individuals accounts for both the rate at which recovered individuals become susceptible again and the rate at which susceptible individuals become infected.
• Infected individuals have an equation that indicates the rate at which infections occur and the rate at which infected individuals recover.
• The recovered individuals' equation reflects the conversion from infected to recovered and from recovered back to susceptible.

The comprehensive interplay of these variables highlights the power of differential equations in predicting disease dynamics over time.

Equilibria Analysis

Equilibria analysis involves finding all the points in a model where the variables do not change over time. For the SIRS model, this means solving the equations \( \frac{dS}{dt} = 0 \) and \( \frac{dI}{dt} = 0 \) to find the equilibrium points.

• At these points, the disease spread is neither increasing nor decreasing, meaning the system is in a steady state.
• For example, an equilibrium occurs at \( (S, I) = (5, 0) \) when the population of susceptible individuals is 5, and there are no infected individuals.
• This analysis helps identify critical thresholds for disease eradication or persistence.

By understanding the equilibria, we gain insight into potential scenarios where the disease could be controlled.

Stability Analysis

Stability analysis is a crucial aspect of understanding equilibria in a mathematical model. It helps determine whether the system will return to equilibrium after a disturbance. Two main methods are used in stability analysis: linearization and analyzing the Jacobian matrix.

• Linearization involves approximating the system near an equilibrium point and analyzing the simplified, linear system.
• The Jacobian matrix, which consists of partial derivatives of the system's equations, helps determine the behavior around equilibria.
• By evaluating the eigenvalues of the Jacobian, we can classify equilibrium points. If all eigenvalues have negative real parts, the equilibrium is stable, meaning small disturbances die out over time.

For the SIRS model, if the equilibrium point at \((S, I) = (5,0)\) is stable, minor changes in infected or susceptible individuals will not cause significant deviations from the equilibrium.

Direction Field

A direction field, or vector field, represents the tendencies or directions of systems of differential equations graphically. In the SIRS model, direction fields allow us to visualize how the population of susceptible and infected individuals changes over time. This is achieved by plotting arrows indicating the direction of change at various points.

• Isoclines, where derivatives are zero, are particularly important as they represent constant solutions, making them key features in direction fields.
• For example, the isocline at \(S = 5\) indicates that at any time, the direction of change along this line involves no change in infected individuals.
• By sketching these lines and adding arrows, we can predict the trajectory and behavior of disease spread across different scenarios.

Direction fields provide a visual way to understand the complex interplay between variables in the differential equation system, offering intuitive insights into the dynamics of disease transmission.