1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 479

males and 159,271 were females. It is estimated that by the end of 2002, that

a total of 501,669 Americans had died of AIDS, which includes 496,354 adults

and adolescents and 5316 children under the age of 15. Stated in terms of

percentages, more that 56.53 percent of the adult and adolescent American

population who contacted AIDS prior to the end of 2002 had died by the end

of 2002.

The following model for the spread of AIDS through sexual contact assumes

the rate of increase in infection of a particular group is equal to a sum of terms

each of which is proportional to the product of the number of members in the

interacting groups which can lead to infection minus the removals. At time t

let

81 be the number of homosexual males

S 2 be the number of bisexual males

S 3 be the number of heterosexual males

S 4 be the number of heterosexual females

y 1 be the number of homosexual males infected with AIDS

y 2 be the number of bisexual males infected with AIDS

y3 be the number of heterosexual males infected with AIDS

y 4 be the number of heterosexual females infected with AIDS

A model for the spread of AIDS within these groups through sexual contact

is

(8)

where a 1 , a 2 , b 1 , b 2 , b3, c, d 1 , d2, r1, r2, r3, and r4 are positive constants. The
constants r i are the removal rates for populations Yi. The term a1Y1 ( S1 - Y1)
represents the rate of increase in infection in the homosexual male population
due to sexual contact between infected homosexual males Y1 and uninfected
homosexual males (S 1 - Y1). The term a2y2(S1 - Y1) represents the rate of
increase in infection in the homosexual male population due to sexual contact
between infected bisexual males y2 and uninfected homosexual males (S1 -y1),
and so forth.

a. Show if all removal rates are zero (that is, if r1 = r2 = r3 = r4 = 0),

the only critical points are (y1, y2,y3,y4) = (0,0,0, 0) and (y1,Y2,y3,y4) =

(S 1 , S 2 , S 3 , S 4 ). That is, if there are no removals due to death, isolation, or

recovery, then either no one has AIDS or eventually everyone contracts AIDS.