/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Consider an infectious disease w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Consider an infectious disease with immunity, but instead of assuming that immunity lasts for the individual's entire life, assume that it gradually decreases (wanes) and individuals eventually become susceptible again after an average time \(w^{-1}\). (a) Ignore births and deaths and assume individuals are infectious for an average time \(\gamma^{-1}\) and give differential equations for the numbers of susceptibles, infectives and recoverers. (b) Include births and deaths with the same per-capita rate \(a\).

Short Answer

Expert verified
(a) Without births and deaths: \(\frac{dS}{dt} = wR - \beta SI\), \(\frac{dI}{dt} = \beta SI - \gamma I\), \(\frac{dR}{dt} = \gamma I - wR\). (b) With births and deaths: \(\frac{dS}{dt} = wR - \beta SI + a(N-S)\), etc.

Step by step solution

01

Define Parameters and Variables

Let \( S(t) \), \( I(t) \), and \( R(t) \) be the number of susceptibles, infectives, and recoverers at time \( t \), respectively. We have the parameters:\[\begin{align*}\beta &: \text{transmission rate of the disease} \\gamma &: \text{recovery rate, where the average infectious period is } \gamma^{-1} \w &: \text{rate at which immunity wanes, where the average immunity period is } w^{-1}\end{align*}\]We will set up differential equations for \( S(t) \), \( I(t) \), and \( R(t) \).
02

Formulate Differential Equations Ignoring Births and Deaths

The differential equations ignoring births and deaths are:\[\begin{align*}\frac{dS}{dt} &= wR - \beta SI \\frac{dI}{dt} &= \beta SI - \gamma I \\frac{dR}{dt} &= \gamma I - wR\end{align*}\]Here, the three equations represent the rate of change of susceptibles, infectives, and recoverers. The first term in each equation accounts for gain or loss from the disease interaction or immunity waning.
03

Adjust for Births and Deaths with Per-Capita Rate

Now, include births and deaths with a per-capita rate \( a \):\[\begin{align*}\frac{dS}{dt} &= wR - \beta SI + a(N - S) \\frac{dI}{dt} &= \beta SI - \gamma I - aI \\frac{dR}{dt} &= \gamma I - wR - aR\end{align*}\]Where \( N(t) = S(t) + I(t) + R(t) \) represents the total population. The additional terms \( + a(N-S) \), \( -aI \), and \( -aR \) account for the changes due to births and deaths.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical tools used to describe how quantities change over time. They are crucial for modeling dynamics in biological systems, such as the spread of infectious diseases. In the context of disease modeling, each differential equation represents a rate of change for a specific compartment (e.g., susceptible, infective, or recovered individuals).

For example, a differential equation for the number of susceptibles might look like this: \( \frac{dS}{dt} = wR - \beta SI \). Here:
  • \( \frac{dS}{dt} \) describes the rate at which people move into or out of the susceptible class.
  • \( wR \) is the rate at which recovered individuals with waning immunity re-enter the susceptible category.
  • \( -\beta SI \) represents the loss of susceptibles who become infected due to encounters with infectives.
These simple yet powerful equations help predict how an epidemic progresses by calculating how each group evolves over time.
Infectious Disease Modeling
Infectious disease modeling is a way researchers use math to understand and predict how diseases spread. It involves creating compartments that represent different states in a population, such as susceptible, infective, and recovered individuals.

The SIR model is a common example, with its compartments:
  • Susceptibles \((S)\): Individuals who can catch the disease.
  • Infectives \((I)\): Individuals who have the disease and can spread it.
  • Recoverers \((R)\): Individuals who have recovered and are initially immune.
Disease modeling can guide public health interventions by estimating potential outbreak scenarios, the impact of control measures like vaccination, and the critical threshold needed to achieve herd immunity.
Waning Immunity
Waning immunity occurs when the protection against a disease decreases over time after recovery or vaccination. This means that recovered individuals can eventually become susceptible again.

In our model, this concept is represented by the waning rate \( w \), which affects the differential equations. Specifically, the term \( wR \) in the equation \( \frac{dS}{dt} = wR - \beta SI \) indicates how recovered individuals move back into the susceptible category.

Understanding waning immunity is critical for designing long-term disease control strategies. It influences vaccine booster schedules and helps assess the duration vaccines need to remain effective to control an outbreak.
Births and Deaths in Modeling
Including births and deaths in infectious disease modeling mirrors reality more closely. Such factors constantly alter the population structure. The birth rate adds new susceptibles to the population, while the death rate decreases all compartments proportionally.

In the adjusted equations, for example, \( \frac{dS}{dt} = wR - \beta SI + a(N-S) \), the term \( + a(N-S) \) reflects births adding to susceptibles, while the terms \( - aI \) and \( - aR \) consider deaths in infective and recovered classes.

These adjustments are essential for understanding how natural population changes affect the spread and potential control of infectious diseases. They help predict the long-term stability and fluctuation of disease patterns within diverse communities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A model for the movement of petty criminals in and out of prisons assumes that all new criminals arise from contacts of existing criminals with law abiding citizens. Assume there is a mean time \(\sigma^{-1}\) before a criminal is caught and sent to prison. The mean time for a prison sentence is \(\mu^{-1}\). Upon release from prison, a small fraction \(f\) of ex-prisoners become law abiding citizens with the remainder returning to being criminals. Formulate a model to describe the movement of criminals into and out of prison. Your model should consist of three differential equations: one for \(L(t)\) the number of law abiding citizens, a second for \(C(t)\) the number of criminals, and a third for \(P(t)\) the number of prisoners.

A new religion is spreading through a community in a remote country by missionaries recruited from the local population. The community is made up of unbelievers (with numbers denoted by \(U(t))\), converts (numbers \(C(t))\) and missionaries (numbers \(M(t))\) ). Assume that only contacts between missionaries and unbelievers result in an unbeliever becoming a convert. A constant proportion of converts each year decide to become missionaries. Formulate a system of differential equations for these populations. Your model should have the property that the total population remains constant over time. Births and deaths may be ignored and relapses to unconverted of either converts or missionaries may be neglected.

With the disease malaria in humans, the disease is carried by mosquitoes that cannot infect each other. Infectious mosquitoes can only infect susceptible humans and infected humans can only infect susceptible mosquitoes when they are bitten by a susceptible mosquito. Assume the rate of transmission is proportional to both numbers of mosquitoes and number of humans for transmission in both directions, and assume (for this model) once infected, both humans and mosquitoes never recover. Ignoring any births and deaths, develop a mathematical model for susceptible and infected humans \(S_{h}(t), I_{h}(t)\), and susceptible and infected mosquitoes \(S_{m}(t), I_{m}(t)\).

The basic reproduction number, \(R_{0}\), is the number of new infections produced by a single infective, over the the duration of time they are infectious. For the SIR model, $$ \frac{d S}{d t}=-\beta S I, \quad \frac{d I}{d t}=\beta S I-\gamma I, $$ we have \(R_{0}=\beta S_{0} \gamma^{-1}\). Consider the following model, which includes births and deaths, $$ \frac{d S}{d t}=-\beta S \frac{I}{N}+a N-a S, \quad \frac{d I}{d t}=\beta S \frac{I}{N}-\gamma I-a I, \quad \frac{d R}{d t}=\gamma I-a R $$ where \(N=S+I+R\).

Formulate a model for an infectious disease where there is immunity for only some of those who recover; others 'recover' to become permanent carriers, who can still cause infections. Thus susceptibles, \(S(t)\), may be infected by either infectious individuals, \(I(t)\), or carriers, \(C(t) .\) A carrier can infect others at a reduced rate compared to infectious individuals but shows no symptoms. (a) Assume there is a fixed proportion \(q\) of those recovering from the infection become carriers. Assume transmission rates \(\beta_{1}\) for normal infectives and \(\beta_{2}\) for carriers and assume that individuals remain infective for a mean time \(\gamma^{-1}\). Give equations for the number of susceptibles \(S(t)\), the number of infectives \(I(t)\), the number of carriers \(C(t)\), and the number of recovered who are immune \(R(t)\). (c) Give at least one example of an infectious disease that could be modelled by the equation you have developed.
See all solutions

Recommended explanations on Biology Textbooks

Responding to Change

Read Explanation

Cell Communication

Read Explanation

Biological Organisms

Read Explanation

Bioenergetics

Read Explanation

Substance Exchange

Read Explanation

Cells

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.