1550078481-Ordinary_Differential_Equations__Roberts_

470 Ordinary Differential Equations

For a. H = 0, b. H = 4, and c. H = 8

(i) Find the critical points of system (6) in the first quadrant and determine
the stability characteristics of each.

(ii) Use SOLVESYS or your computer software to solve system (6) on the
interval [O, 5] for the initial conditions:

1. x(O) = 1, y(O) = 1 2. x(O) = 1, y(O) = 10 3. x(O) = 10, y(O) = 1


2. x(O) = 10, y(O) = 10

For each initial condition estimate limt, 00 x(t) and limt, 00 y(t) and determine
which, if any, species becomes extinct.

10 .6 Epidemics

In chapter 3, we briefly discussed the history of epidemics and epidemiolo gy-

the scientific study of epidemics. We also introduced and studied some of the

simpler models for epidemics there. In this section, we will formulate and

analyze a few somewhat more complicated models for epidemics.

One underlying assumption which we shall make throughout this section

is that the population which can contract the disease has a constant size N.

That is, we will assume there are no births or immigrations to increase the

population size and no emigrations to decrease the population size. Since the

time span of many epidemics, such as the flu, is short in comparison to the life

span of an individual, the assumption of a constant population size is fairly

reasonable. Next, we assume the population is divided into three mutually

exclusive sets:

1. The susceptibles is the set of individuals who do not have the disease
   at time t but who may become infected at some later time.


2. The infectives is the set of individuals at time t who are infected with
   and capable of transmitting the disease.


3. The removeds is the set of individuals who have been removed at time
   t either by death, by recovering and obtaining immunity, or by being
   isolated for treatment.

We will denote the number of susceptibles, infectives, and removeds at time t
by S(t), I(t), and R(t), respectively. Under the assumptions we have made

(1) S(t) + I(t) + R(t) = N for all time t.

An assumption which was first made in 1906 by William Hamer, and which
has been included in every deterministic epidemic model ever since, is that the
rate of change of the number of susceptibles is proportional to the product of