Problem 32 - Biocalculus Calculus for the Life Sciences

Question

$$\begin{array}{c}{\text { The Kermack-McKendrick equations describe the out- }} \\ {\text { break of an infectious disease. Using } S \text { and } I \text { to denote the }} \\ {\text { number of susceptible and infected people in a population, }} \\ {\text { respectively, the equations are }} \\\ {S^{\prime}=-\beta S I \quad I^{\prime}=\beta S I-\mu I}\end{array}$$ $$\begin{array}{l}{\text { where } \beta \text { and } \mu \text { are positive constants representing the }} \\ {\text { transmission rate and rate of recovery. }} \\ {\text { (a) Verify that } \hat{I}=0, \text { along with any value of } S, \text { is an }} \\ {\text { equilibrium. }}\end{array}$$ $$\begin{array}{l}{\text { (b) Calculate the Jacobian matrix. }} \\ {\text { (c) Using your answer to part (b), determine how large } S} \\ {\text { must be to guarantee that the disease will spread when }} \\ {\text { rare. }}\end{array}$$

Step-by-Step Solution

Answer

(a) \( \hat{I} = 0 \), any \( S \) is equilibrium. (b) Jacobian: \[ J = \begin{bmatrix} -\beta I & -\beta S \\ \beta I & \beta S - \mu \end{bmatrix} \]. (c) \( S > \frac{\mu}{\beta} \) for spread.

Step 1: Define Equilibrium

An equilibrium point occurs when both derivatives \( S' \) and \( I' \) are equal to zero. We must find values for \( S \) and \( I \) such that these conditions hold true.

Step 2: Verify Equilibrium for \( \hat{I}=0 \)

Starting with the second equation \( I' = \beta S I - \mu I = 0 \), set \( I = 0 \) which satisfies this equation for any \( S \). Similarly, for the first equation \( S' = -\beta S I = 0 \), where \( I = 0 \), \( S' \) is also zero. Hence \( \hat{I} = 0 \) with any value of \( S \) is an equilibrium.

Step 3: Jacobian Matrix Calculation

Find the Jacobian matrix by calculating the partial derivatives of \( S' \) and \( I' \) with respect to \( S \) and \( I \): - \( \frac{\partial S'}{\partial S} = -\beta I \) - \( \frac{\partial S'}{\partial I} = -\beta S \) - \( \frac{\partial I'}{\partial S} = \beta I \) - \( \frac{\partial I'}{\partial I} = \beta S - \mu \) The Jacobian matrix \( J \) is therefore:\[ J = \begin{bmatrix} -\beta I & -\beta S \ \beta I & \beta S - \mu \end{bmatrix} \]

Step 4: Analyzing Stability at \( (S, I) = (S, 0) \)

Substitute \( (S, I) = (S, 0) \) into the Jacobian. The matrix becomes:\[ J = \begin{bmatrix} 0 & -\beta S \ 0 & \beta S - \mu \end{bmatrix} \]The eigenvalues \( \lambda_1 \) and \( \lambda_2 \) are the diagonal elements: \( \lambda_1 = 0 \) and \( \lambda_2 = \beta S - \mu \).

Step 5: Determine Conditions for Disease Spread

For the disease to spread when rare, \( \lambda_2 \) should be positive. This means:\[ \beta S - \mu > 0 \]Solving for \( S \), we find \( S > \frac{\mu}{\beta} \). Hence, \( S \) must be greater than the threshold \( \frac{\mu}{\beta} \) for the disease to spread.

Key Concepts

Infectious Disease Modeling

Infectious disease modeling is a crucial tool in understanding how diseases spread through populations. Models help scientists predict outbreaks, plan interventions, and allocate resources effectively. The goal is to simulate the dynamics of disease transmission, considering the interactions between susceptible individuals and those infected. In this exercise, we examine the Kermack-McKendrick Model, which is a foundational mathematical framework for infectious disease dynamics.

• "Susceptible" individuals (\(S\)) are those at risk of contracting the disease.
• "Infected" individuals (\(I\)) are currently experiencing the disease and can potentially transmit it to others.

The model uses differential equations to describe changes in these groups over time. Specifically, it considers the rates at which individuals become infected or recover. Understanding these dynamics is critical for managing public health responses to infectious diseases.

Equilibrium Analysis

Equilibrium analysis involves finding the conditions under which populations of susceptible and infected individuals remain constant over time. In the Kermack-McKendrick model, equilibrium is achieved when both derivatives \(S'\) and \(I'\) are zero.

• When \(I=0\), the population is in a disease-free state.
• Susceptibles can adopt any value since no new infections will occur.

This setup reveals that \(\hat{I}=0\), or zero infected individuals, is always an equilibrium regardless of \(S\). This equilibrium tells us that the disease won't spontaneously flare up unless new infections are introduced to the system.

Jacobian Matrix

The Jacobian matrix is a crucial mathematical tool used to understand the stability of equilibria within a system of differential equations. It provides insight into how small changes in state variables impact the system's behavior near equilibrium points. In our exercise, the Jacobian matrix for the Kermack-McKendrick model is derived by finding the partial derivatives of the differential equations:
\[ J = \begin{bmatrix} -\beta I & -\beta S \ \beta I & \beta S - \mu \end{bmatrix} \]
Analyzing the Jacobian helps us determine whether an equilibrium is stable or unstable.

• The elements associated with parameters \(\beta\) and \(\mu\) signal how transmission and recovery rates influence stability.
• Specifically, we look for \(\lambda_2 = \beta S - \mu\), to determine if perturbations lead to increases or decreases in infection rates.

Disease Spread Conditions

Understanding the conditions necessary for disease spread is vital for controlling epidemics. In our context, the disease will spread if the condition \(\beta S - \mu > 0\) is met. This inequality is derived from the eigenvalues of the Jacobian matrix. If \(\lambda_2\) is positive, indicating that \(S > \frac{\mu}{\beta}\), the infection can spread since each infected person, on average, passes the disease to more than one susceptible individual.

• \(\beta\) represents the effective contact rate, or how often an infected person encounters and successfully transmits the disease to a susceptible person.
• \(\mu\) is the recovery rate which denotes how quickly infected individuals recover and become non-infectious.

If the number of susceptible individuals \(S\) exceeds this threshold, public health measures may be required to prevent the outbreak from accelerating.