Purchase Solution

Differential equations to describe infection rates

Not what you're looking for?

Ask Custom Question

Use models to describe the population dynamics of disease agents.

Total population is a Constant (T).
A small group of infected individuals are introduced into a large population. Describe spread of infection within population as a function of time. This disease which, after recovery, confers immunity. The population can be divided into three distinct classes.

a) A susceptible class S, who can catch the disease
b) the infected class I - who have the disease and can transmit.
c) the removed class R, who have had the disease or are recovered, immune or isolated until recovered, or indeed, dead.

The gain in the infected class is at a rate proportional to the number of infected and susceptible persons that is rSI, where r > 0 (is a constant) r >0 is called the infection rate. The susceptibles are lost at the same rate.

The rate of removal of infected to the removed class is proportional to the number of infected people, that is aI, where a > 0 is a constant. a >0 is called the removal rate.

The incubation period is short enough to be negligible.

#1 Formulate the three differential equations describing the rate of change in the number of S, I, and R populations.

A key questions is that given r, a, and S and the initial number of infected individuals (Io) whether the infection will spread or not, and if it does how it develops with time and of course when it will start to decline. The term epidemic means that I(t) >Io for some t > 0

#2 Find the minimum number of suseptibles required to start an epidemic in terms of a and r.

Assume that a=0.5, r=0.005, So = 150 Io = 50, N = 200

#3 Is this epidemic probable or not?

#4 The global maximum number of infected population in this disease process

#5 When does the maximum in the infected class happen?

Purchase this Solution
Solution Summary

DIfferential equations to describe infection rates are examined. The models to describe the population dynamics of disease agents are provided. The solution answers the question(s) below.

Solution Preview

1.

The equations are

dS/dt = -rSI ..........1
dI/dt = rSI -aI.........2
dR/dt = aI............3

Note that the sum of the rates is 0 because total population is a constant

2.An epidemic will happen if dI/dt >0...That is infected individuals increase in number.then I(t)>Io

means (rS-a)*I>0

means rS>a since I is always positive....

means S>a/r

3.YES SINCE ...

Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Probability Quiz

Some questions on probability

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

View More Free Quizzes
Related BrainMass Solutions

  • Differential equations to describe infection

    8924 Differential equations to describe infection Use models to describe the population dynamics of disease agents. Total population is a Constant (T). A small group of infected individuals are introduced into a large population.

  • System Modelling (Output: Density)

    The first step is to build state differential equations to describe the time variant property of the states. The second step is to build output equations in terms of the state variables.

  • CBC with Differential

    Low numbers of WBCs (leukopenia) may indicate bone marrow failure (for example, due to infection, tumor, fibrosis) , presence of cytotoxic substance , collagen-vascular diseases, disease of the liver or spleen, or exposure to radiation Hemoglobin

  • Parasites in patients

    Giardia infection rates have been known to go up in the summer and backpackers or campers who drink untreated water from lakes or rivers are especially susceptible to Giardia infection.

  • Systems of Ordinary Differential Equations (4 Problems)

    37683 Systems of Ordinary Differential Equations (4 Problems) Problem 8.1 (Prob. 11, p.251) Solve the following system of equations.

  • Systems of Differential Equations : General Solution and Phase Portrait

    Find the general solution to the attached system of differential equations. x' = y y' = 2x b. Draw the phase portrait for this system and describe in words what happens to the solution for all initial conditions: x(0) = x0 and y(0) = y0.

  • Anatomy of the circulatory system and white blood cells

    527806 Anatomy of the circulatory system and white blood cell differential count 1) How does a differential leukocyte count aid in medical diagnosis? 2) In counting 100 leukocytes you are accurately able to distinguish 15 basophils.

  • System of non-homogenous first order differential equations

    Convert the problem's text to differential equations. 2. Solve a system of two non-homogeneous first order differential equations using matrices.

  • Sturm-Liouville form

    where q(x)=b(x)p(x) and r(x)=c(x)p(x) This shows how to write differential equations is in Sturm-Liouville form, and identify the weight function that would be used in defining an appropriate inner product.

View More Related Solutions