N is the whole population. N = X + Y + Z. X(0)=X0, Y(0)=Y0, Z(0)=0

The dynamics of a directly},r transmitted microparasite has been modeled by the system H =bNsXYbx I ij = 3XY+ (b+r}Y g d: =rYbZ where h, B, r are positive constants and X, Y, Z are the number of susceptibles, infectives and immune populations respectively. Here, the population is kept constant by births and deaths {with contribution from each class) balancing. Show that there is a threshold population NC such that if N <: n:="%(b+r)" the parasite cannot maintain itself in population and both infectives immune class eventually die out. quantity n r is called basic reproductive rate of infection>