Problem 40 - Calculus for Biology and Medicine

Question

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).

Step-by-Step Solution

Answer

Convert the system to two variables, find equilibria by setting \( \frac{dS}{dt} \) and \( \frac{dI}{dt} \) to zero, linearize, and analyze eigenvalues to classify stability.

Step 1: Simplify the System

Simplify the given set of differential equations by representing them in terms of two dependent variables, \( S \) and \( I \). From \( \frac{dR}{dt} = \frac{1}{12}I - \frac{1}{5}R \), use algebra to express \( R \) in terms of \( I \) and its derivative, knowing \( S + I + R = N \). This reduction simplifies the problem to analyzing equations with fewer variables.

Step 2: Set Derivatives to Zero to Find Equilibria

Find the equilibria by setting \( \frac{dS}{dt} = 0 \), \( \frac{dI}{dt} = 0 \), and by the previously derived equation for \( R \). Solve the resulting algebraic equations to determine the conditions where \( S \) and \( I \) remain constant.

Step 3: Solve for Equilibrium Points

Substitute \( S = 250 - I - R \) and solve the system of equations from Step 2 to find equilibrium points. One solution is where all these rates of change equal zero, leading to points in the \((S, I, R)\) space. Calculate these to deduce potential equilibria, ensuring they lie within the feasible region (i.e., \( S, I, R \geq 0 \)).

Step 4: Linearize the System

Linearize the system of equations around each of the equilibria found. Compute the Jacobian matrix of the system at these equilibrium points. The elements of the Jacobian are derived by differentiating the right-hand sides of the differential equations with respect to \( S \) and \( I \).

Step 5: Analyze the Jacobian Eigenvalues

Determine the eigenvalues of the Jacobian matrix for each equilibrium by solving the characteristic equation. The sign and nature of these eigenvalues (real vs complex, positive vs negative) will indicate the stability of the equilibrium (e.g., stable node, unstable node, saddle point, or spiral).

Step 6: Classify Each Equilibrium

Based on the eigenvalues calculated, classify each equilibrium. If all eigenvalues are negative, the equilibrium is a stable node. For complex eigenvalues with negative real parts, the equilibrium is a stable spiral, whereas positive real parts suggest an unstable spiral. Should eigenvalues have opposite signs, the equilibrium is a saddle point.

Key Concepts

Differential Equations

In the context of disease modeling, differential equations are powerful tools for describing how populations change over time. They are used to model the rates of changes of various compartments in a population, such as susceptible (\( S \)), infected (\( I \)), and recovered (\( R \)) individuals in the SIRS model. Each compartment represents a different stage in the disease progression.
The primary aim is to understand how these numbers evolve over time given certain biological parameters.
In our SIRS model, the differential equations express the changes in the population's susceptible and infected classes:

• \( \frac{dS}{dt} = -\frac{1}{500} SI + \frac{1}{5}R + \frac{1}{4}I \)
• \( \frac{dI}{dt} = \frac{1}{500} SI - \frac{1}{3}I \)

These equations effectively capture key dynamics such as how contacts between susceptible and infected individuals drive the number of new infections, and how recovery and mortality impact the infected population.

Solving these equations provides insights into how the numbers of susceptible and infected change, potentially revealing steady states or "behaviors" of the model, which are explored as the system reaches equilibrium.

Equilibrium Analysis

Equilibrium analysis in the context of differential equations for disease spread involves finding steady states where the disease dynamics no longer change. These are crucial for understanding long-term implications of a disease within a population.Equilibrium points are determined by setting the rate of change of each variable to zero. This reduces the problem to solving algebraic equations:

• \( \frac{dS}{dt} = 0 \)
• \( \frac{dI}{dt} = 0 \)

By solving these, you identify the combinations of susceptible and infected individuals that stop changing over time.

In the SIRS model, equilibria help us understand whether an infection will stabilize, die out, or remain endemic. These points are crucial for interventions, indicating when and how to effectively apply them.Given the system's equations, you substitute in conditions such as \( S + I + R = N \) to find the values that satisfy the equilibrium condition. After calculation, we derive certain points that are tested further for stability via linearization.

Mortality Rate

Mortality rate, particularly in epidemiological models, reflects the proportion of the infected population succumbing to the disease within a given timeframe. In the SIRS model, it's denoted by \( m \) and plays a significant role in influencing the disease's impact on population dynamics.

In our specific problem setup, the mortality rate is given by \( m = 1/4 \), indicating that in each time unit, a quarter of the infected population is expected to die. This affects the overall population size and is a crucial component of the model's balance, as every death theoretically permits a new birth due to freed resources.

• The equation \( \frac{dI}{dt} = \frac{1}{500} SI - \frac{1}{3} I \) incorporates this mortality rate through the term \( -\frac{1}{3} I \).
• The inclusion of mortality impacts the equilibrium by determining the rate at which resources become available for new births, balancing the carrying capacity of the population.

Understanding how mortality integrates into the differential equations allows one to predict the model outcomes over time, indicating potential scenarios where disease control measures may be needed to manage a population’s health effectively.