Problem 42 - Calculus for Biology and Medicine

Question

A particular infectious disease confers lifelong immunity to any individual who recovers from the disease. The population size is \(N=200 .\) Assume that the spread of the disease can be described by an SIR model: $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{100} S I \\ \frac{d I}{d t}=\frac{1}{100} S I-6 I \\ \frac{d R}{d t}=6 I \end{array} $$ Assuming that \(R(0)=0\) initially and \(I(0)=5\), calculate a bound on the maximum number of individuals who will catch the disease.

Step-by-Step Solution

Answer

The maximum number of individuals who will catch the disease is approximately 195.

Step 1: Understand the SIR Model Equations

The SIR model describes how a disease spreads through a population. The equations provided are for the susceptible ( S ), infected ( I ), and recovered ( R ) individuals over time. The derivatives describe rates of change: \( \frac{dS}{dt} = -\frac{1}{100}SI \), \( \frac{dI}{dt} = \frac{1}{100}SI - 6I \), and \( \frac{dR}{dt} = 6I \). Initially, \( N = 200 \), \( R(0) = 0 \), and \( I(0) = 5 \). Since the total population is constant, \( S(t) + I(t) + R(t) = N \).

Step 2: Find Initial Number of Susceptible Individuals

From the total population and initial values of \( I(0) \) and \( R(0) \), we can find \( S(0) \). Since \( N = 200 \) and \( I(0) = 5 \), we have \( S(0) = 200 - 5 - 0 = 195 \).

Step 3: Analyze the Dynamics of the System

The equation for \( \frac{dR}{dt} \) tells us that \( 6I \) individuals recover per unit time. The increase in \( R \) over time is directly tied to the decrease in \( S \) via \( \frac{dS}{dt} = - \frac{1}{100}SI \). The goal is to determine how \( R \) changes until \( I = 0 \).

Step 4: Calculate the Maximum Number of Individuals Infected

In an SIR model, \( R(\infty) \) or the final recovered number represents all who were ever infected. Given no new births are assumed, we need the total ever infected, \( R(\infty) - R(0) \) when \( I(t) = 0 \). The starting number of infected is 5, and all recover eventually since \( \frac{dR}{dt} = 6I \).

Step 5: Use Conservation of Population to Find Bound

Since \( S(t) + I(t) + R(t) = N \) and at the maximum spread \( I(t) + R(t) \) tends to approach \( N \), use \( R(\infty) = N - S(t) \) to find the bound when \( I(t) \) reduces to 0: \( R(\infty) = 200 - S(t) \). Assuming the worst-case scenario: \( S(t) \) minimizes along the trajectory without any events causing early immunity.

Step 6: Conclude with the Numerical Bound

Numerically simulate if required; however, the SIR dynamics imply \( R(\infty) = R(0) + \text{Maximum Infected} = 200 - \text{Minimum } S \). Assume maximum 195 initially gain infection immunity: the bound on max infected individuals becomes \( R(\infty) = 200 - 0 = R(\infty) \) based on the minimum remaining S equal to 0. Therefore, the maximum number of individuals who will ever hold infection rolls out around the initial susceptible count since \( R + I \) integrates to N.

Key Concepts

Infectious Disease

Understanding infectious diseases requires knowing how they spread among populations. The SIR model is a mathematical way to describe this spread. By breaking down the population into three groups: Susceptible ( S), Infected ( I), and Recovered ( R), we can track how the disease affects them over time. Infections happen when susceptible individuals come into contact with infected ones. As these interactions happen, some get sick, increasing the infected count, while others recover, moving into the recovered group.
The SIR model equations represent these rates of change in a structured way. For instance, the equation for \( \frac{dS}{dt} \) shows how quickly individuals are moving from susceptible to infected due to interactions. This allows us to simulate and predict the course of an outbreak in a given population.

Lifelong Immunity

Lifelong immunity plays a vital role in the dynamics of an infectious disease spread. It's the condition where once a person has recovered from an illness, they are protected against getting infected again. This concept is crucial because it reduces the pool of susceptible individuals over time, eventually slowing down and stopping the outbreak.
In the SIR model, individuals in the recovered category, R , are those who have gained lifelong immunity. They no longer participate in the chain of infection, effectively lowering the transmission potential of the disease. This aspect of immunity explains why the SIR model often results in the curve flattening out, indicating that the epidemic has reached its end as most individuals have moved into the recovered state.

Population Dynamics

Population dynamics in the context of infectious diseases refer to how the disease impacts different sections of the population over time. In our exercise, the dynamics are captured by observing the rate of changes in the numbers of S , I , and R. This is fundamental, as it helps predict how quickly a disease can spread and the eventual total people affected.

• The interaction between susceptible and infected individuals drives the dynamics of infection spread.
• Recoveries, as represented by \( \frac{dR}{dt} = 6I \) , show how quickly immunity spreads in the population.
• Population conservation ensures the total remains constant, \( S + I + R = N \) .

Understanding these dynamics helps anticipate medical needs and plan interventions more effectively.

Differential Equations

Differential equations provide a way to mathematically model how diseases propagate within a population. They define the rate of change of the susceptible, infected, and recovered groups over time. For instance, \( \frac{dI}{dt} = \frac{1}{100}SI - 6I \) tells us how quickly the number of infected individuals shifts, balancing between newly infected people and those recovering.
The use of differential equations in models like SIR allows for simulating past data to grasp the disease behavior or predict future outcomes. This provides a sophisticated, precise approach to understanding disease dynamics. Importantly, differential equations offer the insights needed to calculate bounds on how many individuals might become infected during an outbreak, guiding public health responses effectively.