478 Ordinary Differential Equations
Display a graph with S, I, and R. Does an epidemic occur in case (ii)? In
each case, what is limt_, 00 S(t)?
- If the disease is deadly, then the fear of the members of the susceptible
group might increase the inoculation rate to the point that it is proportional
to the product of the number of susceptibles and the square of the number of
infectives. In this instance, a model for an epidemic with inoculation is
dS 2dt = -/3SI - a.SI
dIdt = /3SI - rI
dR=rI+aSI^2
dtwhere a, /3, and r are positive constants.
Solve the following epidemic with inoculation model~~ = -.OlSI - .05SI^2
dI
dt = .OlSI - 3IdR = 3I + .05SI^2
dton the interval [O, 5] where t is in days for the initial conditions So = 500,
Io = 10 , Ro= 0. Display a graph with S , I, and R. Does an epidemic occur?
What is limt_, 00 S(t)?
- The virus AIDS, or acquired immune deficiency syndrome, had no name
when it first appeared in the United States in the late 1970s. The new disease
was identified in 1981 but was not isolated and named until 1984. The AIDS
virus is spread almost exclusively through sexual contact or blood contact.
Today with the careful screening of all blood donations, there is little risk of
contracting AIDS through blood transfusion. At present, there is no cure for
AIDS. The disease seems to have begun in the United States in the homo-
sexual male population and has since spread to the bisexual male population,
the heterosexual male population and the heterosexual female population.
Seventy-three percent of the AIDS victims in the United States are sexually
active homosexual and bisexual males and 13 of the victims are sexually
active heterosexuals. According to the Centers for Disease Control and Pre-
vention, by the end of 2002 there were an estimated 877,275 cases of AIDS in
the adult and adolescent population in the United States and an additional
9300 cases of AIDS in children under age 13. Of the adult cases, 718,002 were