HIV in CUBA Kelvin Chan & Sasha Jilkine

Developing a Model S = Susceptible I = Infected Z = AIDS Patients N = S+I = Active Population

Standard epidemic model transmission of disease as  SI Unrealistic for AIDS Disease spread through sexual contact Large difference between S and I (10,000,000 versus 99) Propose to model transmission as  = probability of getting infected from an infected partner c = # of sexual partners during lifetime I/N = probability that person is HIV Positive  effectiveness of treatment

(1)  = birth rate   = natural death rate  = transmission probability C = # of lifetime sexual partners  = rate of progression to AIDS   = AIDS death rate  = effectiveness of treatment

From (1) get Let’s look at proportions of susceptibles and infected rather than absolute numbers

 Let  =  t To find the equilibria we need to solve

If(i.e. the population is disease free) get Since s+i=1, only the second solution is true for all values of a and b Consequently, we have two equilibria: Note that the endemic equilibrium exists only if

Stability of Equilibria We were able to show that R 0 for this model was If R 0 <1, only the disease-free equilibrium exists and is asymptotically stable. At R 0 =1, a bifurcation occurs If R 0 >1, disease-free equilibrium becomes unstable, and the endemic equilibrium is asymptotically stable

R 0 = R 0 = 1.0 R 0 =

Fitting the Data The population of Cuba increased from 10 million to 11 million from 1986 to 1997 Assumed constant birth and death rate (  =0.016,     = H(1994)  =0.25     = 0.9(H(1991)-H(1993))(1-e -25(t- 1993) ) c=8

If treatment remained constant, would have seen

Conclusions Right now Cuba has very low incidence of HIV + people in absolute terms and as a proportion of the general population But the numbers are growing. There were cases of HIV in 2001 Epidemic can be stopped by decreasing Better model could be obtained by treating various socio- economic groups separately