1. HIV in CUBA Kelvin Chan & Sasha Jilkine

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

5. 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

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

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

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

9. 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

10. 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

11. R 0 =.2710843373R 0 = 1.0 R 0 = 1.807228916

12. 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,     =0.06+0.005H(1994)  =0.25     = 0.9(H(1991)-H(1993))(1-e -25(t- 1993) ) c=8

14. If treatment remained constant, would have seen

15. 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 3 485 cases of HIV in 2001 Epidemic can be stopped by decreasing Better model could be obtained by treating various socio- economic groups separately