Problem 43 - 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 \(N=100\). Assume that the spread of the disease can be described by an SIR model: $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{300} S I \\ \frac{d I}{d t}=\frac{1}{300} S I-\frac{1}{9} I \\ \frac{d R}{d t}=\frac{1}{9} I \end{array} $$ (a) Assuming that \(R(0)=0\) initially and \(I(0)=5\), calculate a bound on the maximum number of individuals who will catch the disease. (b) Assume that a vaccination program means that half of the population start out immune to the disease, i.e., \(R(0)=50\). Assume also that there are initially 5 infected individuals (i.e., \(I(0)=5\) ). Recalculate the maximum bound on the number of individuals who will eventually catch the disease.

Step-by-Step Solution

Answer

Without vaccination, 68 individuals will catch the disease; with vaccination, 36 will catch it.

Step 1: Understand the SIR Model

The SIR model divides the population into three groups: susceptible (S), infected (I), and recovered (R). The equations describe how these populations change over time due to infection and recovery.

Step 2: General Setup for Part (a)

We need to find how many individuals will eventually get infected. Initially, we have all susceptibles except for the 5 that are infected: \(S(0) = 95\), \(I(0) = 5\), \(R(0) = 0\). The goal is to find the maximum \(I(t)\) which will help determine the number who will catch the disease.

Step 3: Analyze Maximum Infection

The disease will stop spreading when \(I(t) = 0\). At the maximum infection point, \(\frac{dI}{dt} = 0\). Using \(\frac{dI}{dt} = \frac{1}{300}SI - \frac{1}{9}I = 0\), solve to find that \(S = 27\) when \(I = 0\). So, initially \(95 - 27 = 68\) will have caught the disease.

Step 4: Introduce Vaccination Impact for Part (b)

With vaccination, half the population starts immune, \(R(0) = 50\). Now \(S(0) = 45\), \(I(0) = 5\). We apply the same logic: \(\frac{1}{300}SI = \frac{1}{9}I\) leads to \(S = 9\) at maximum infection.

Step 5: Calculate for Part (b)

After vaccination, 36 individuals will have caught the disease since \(S(0) - S = 45 - 9 = 36\).

Key Concepts

Infectious Disease Modeling

Infectious disease modeling is a crucial tool used by scientists to predict and understand the spread of diseases within a population. This modeling helps in estimating how diseases like influenza or COVID-19 can spread over time and what measures can control them.

The SIR model is one of the basic models used in infectious disease modeling. It divides a population into three compartments: susceptible (S), infected (I), and recovered (R). Each of these compartments represents a different state in relation to the disease. Susceptible individuals have not yet contracted the disease but are at risk, infected individuals currently have the disease and can spread it, and recovered individuals have either recovered from the disease or possibly died, developing immunity as a result.

The transition between these compartments is described using differential equations, which allow epidemiologists to estimate how the number of susceptible, infected, and recovered individuals changes over time. Understanding these transitions helps in implementing healthcare interventions like vaccinations or quarantines to control the spread of the disease and protect the population.

Differential Equations

Differential equations are a mathematical tool used to describe the rate at which changes occur. They are crucial in models like SIR because they allow scientists to quantify how various populations shift over time.

In the context of the SIR model, differential equations establish relationships between the number of susceptible, infected, and recovered individuals in a given population. The rate of change of each compartment is expressed as an equation. For example, the equation \(\frac{dS}{dt} = -\frac{1}{300}SI\) describes how the number of susceptible individuals decreases over time due to infection. Similarly, \(\frac{dI}{dt} = \frac{1}{300}SI - \frac{1}{9}I\) describes how the infected population changes by gaining new infections and losing individuals who recover.

This mathematical framework helps predict the peak of an infection, how many people will become infected over time, and when the epidemic will die out. By understanding these equations, scientists and policymakers can make informed decisions to combat the spread of infectious diseases.

Population Dynamics

Population dynamics refers to the study of how populations change over time. This is an integral aspect of modeling the spread of infectious diseases since it helps to understand the interactions within the population and how a disease can alter those dynamics.

In the SIR model, population dynamics are crucial for determining how quickly a disease can spread. For a given population, key factors include the initial number of susceptible and infected individuals as well as how the disease is transmitted and how individuals recover or gain immunity.

• Susceptible individuals can transition to the infected state after coming into contact with an infected person.
• Infected individuals can either recover, moving into the recovered state, or continue to spread the disease.
• Recovered individuals are typically assumed immune, affecting future disease spread.

By observing the population dynamics, we can predict the maximum number of people who might be infected and identify the peak of an epidemic. Policy measures such as vaccination also play a role in these dynamics by reducing the number of susceptibles and therefore the overall number who might ever become infected. This insight assists not only in planning healthcare responses but also in preventing healthcare systems from being overwhelmed by too many simultaneous cases.