Problem 39 - 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 \(N=100\). 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).

Step-by-Step Solution

Answer

The mortality rate is \( \frac{2}{3} \). Simplified, the system is \((S, I)=(0,0)\). The equilibrium \((0,0)\) needs classification via eigenvalue analysis.

Step 1: Identify the Mortality Rate

The mortality rate, \(m\), is derived from the term \(-\frac{2}{3} I\) in the \(\frac{dI}{dt}\) equation. This term represents the net death rate for the infected population. Therefore, the mortality rate is \(m = \frac{2}{3}\).

Step 2: Simplify the System Using S and I

To express the system with \(S\) and \(I\) only, note that \(N = S + I + R = 100\) and express \(R\) in terms of \(S\) and \(I\): \(R = 100 - S - I\). Substitute \(R\) in any term involving \(R\), resulting in two equations: \(\frac{dS}{dt} = -\frac{1}{200} SI + \frac{1}{10} (100 - S - I) + \frac{1}{3} I\), and \(\frac{dI}{dt} = \frac{1}{200} SI - \frac{2}{3} I\).

Step 3: Find Equilibria

Set the equations \(\frac{dS}{dt} = 0\) and \(\frac{dI}{dt} = 0\) and solve for \(S\) and \(I\). From \(\frac{dI}{dt} = 0\), \(I=0\) or \(S = \frac{400}{3}\). Substitute \(I = 0\) into \(\frac{dS}{dt}\) and calculate to check possible values of \(S\). One equilibrium is \((S, I) = (0, 0)\). If \(S = \frac{400}{3}\), \(I = 50\), \(R=100 - 50 - \frac{400}{3}\) is impossible. Thus, only \((S, I) = (0, 0)\) exists.

Step 4: Linearize and Classify the Equilibrium

Linearize the system around \((S, I) = (0, 0)\) by finding the Jacobian matrix of the system. Evaluate the Jacobian at the equilibrium point \((0, 0)\). Analyze the eigenvalues of the matrix to classify the equilibrium. Use the system derived into variables \(S\) and \(I\): \(J = \begin{bmatrix} \frac{\partial }{\partial S}\frac{dS}{dt} & \frac{\partial }{\partial I}\frac{dS}{dt} \ \frac{\partial }{\partial S}\frac{dI}{dt} & \frac{\partial }{\partial I}\frac{dI}{dt} \end{bmatrix}\). Evaluate the eigenvalues.

Key Concepts

Mortality Rate

The mortality rate is an essential concept in epidemiological models like the SIRS model. It represents the fraction of the infected population that dies in a given time period. In the context of the exercise, the term \(-\frac{2}{3} I\) from the differential equation for \(\frac{dI}{dt}\) helps us determine the mortality rate. Here, \(m = \frac{2}{3}\) implies that two-thirds of the infected individuals do not survive.
This rate affects the overall dynamics by removing individuals from the population, thus influencing susceptible, infected, and recovered compartments. Understanding the mortality rate is crucial for predicting the course of an outbreak and making necessary public health interventions. The mortality rate's integration into models helps assess a disease's severity and the required health system responses.

Differential Equations

Differential equations are vital in modeling the spread of infectious diseases. They allow us to describe how the number of susceptible, infected, and recovered individuals evolves over time. In our SIRS model, we use a system of differential equations:
\[ \begin{aligned} \frac{d S}{d t} &=-\frac{1}{200} S I+\frac{1}{10} R+\frac{1}{3} I, \ \frac{d I}{d t} &=\frac{1}{200} S I-\frac{2}{3} I, \ \frac{d R}{d t} &=\frac{1}{3} I-\frac{1}{10} R \end{aligned} \]
These equations use parameters like infection rates and mortality to simulate real-world disease spread.

This system describes the interaction between different compartments and how individuals move from being susceptible to infected and eventually recovering. By manipulating these equations, students can explore various scenarios and understand how changes in parameters affect the disease dynamics. Particularly, simplifying the system to use fewer variables, as was done in the step-by-step solution, helps emphasize key aspects of the disease's impact.

Equilibrium Stability

Equilibrium stability is a fundamental aspect when analyzing models like the SIRS with mortality. An equilibrium point is where the differential equations do not change over time, meaning the population dynamic has stabilized.
Finding equilibria involves setting equations like \(\frac{dS}{dt} = 0\) and \(\frac{dI}{dt} = 0\) and solving for \(S\) and \(I\). It's about determining states where the population ceases to evolve.

Not all equilibria are stable. To assess stability, we use linearization and analyze the Jacobian matrix at each equilibrium point. This involves calculating eigenvalues that inform us about the equilibrium's nature. At the equilibrium \((S, I) = (0, 0)\), these values determine whether a slight disturbance will return the system to equilibrium or divert it elsewhere. This classification helps in anticipating whether the disease will persist or fizzle out.