1. Modeling the HIV/AIDS Epidemic in Cuba Presented by Raluca Amariei and Audrey Pereira 2005 PIMS Mathematical Biology Summer School

2. Outline Introduction to HIV HIV in Cuba Models Analysis of Models Output of Model III Fitting Model III to Data Extensions to the Model Conclusions

3. HIV the Virus In 2000: 36.1 million people living with HIV 390 000 in the Caribbean region Only 3230 cases in Cuba (Cuba's 0.03% infection rate is one of the lowest in the world) HIV - human immunodeficiency virus that causes Acquired Immuno- Deficiency Syndrome AIDS - weakness in the body's system that fights diseases (CD4+ cell percentage is less than 14% )

4. HIV in Cuba National Programme on HIV/AIDS established by the Cuban government in 1983: Testing blood donations Hospital surveillance - screening of patients with other STD’s, pregnant women, other hospital patients HIV screening for travellers to other countries

5. HIV in Cuba HIV seropositives placed in sanatoriums Partner Notification Programme – contact tracing and screening of sexual partners Increase in the HIV cases due to growth of tourism after 1996 Negative impact of US Embargo on Cuban Health Services

6. Model I - Parameters b = birth rate; d = death rate τ 1 = probability of acquiring HIV when in contact with an HIV+ τ 2 = probability of acquiring HIV when in contact with an AIDS sufferer k = conversion rate of HIV to AIDS (incubation period = 9 yrs) d’ = death rate for AIDS sufferers S H A b D d’ d d d τ2τ2 τ1τ1 k

7. Model I (1) S’ = b(S+H+A) - τ 1 SH - τ 2 SA - dS (2) H’ = τ 1 SH + τ 2 SA - kH - dH (3) A’ = kH – d’A – dA (4) D’ = d’A N = S + H + A = constant (b=d) → N’ = 0 → S’+H’+A’ = 0 → d’=0 Equations: No deaths from AIDS

8. Analysis of Model I Equilibrium Points: From (3): A = kH/d In (2), case (i) H = 0 → A = 0 → S = N Disease Free Equilibrium is: DFE = (N, 0, 0)

9. Analysis of Model I In (2), case (ii) H ≠ 0 divide (2) by H: τ 1 S + τ 2 kSH/d - k - d = 0 → S* = (k+d)/( τ 1 + τ 2 k/d)

10. Analysis of Model I Plug (2) and (3) in (1): H* = (d-b)S/(b+bk/d-k-d) But b=d → H* = 0. Contradiction. → there is no endemic equilibrium

11. Stability Jacobian Matrix: -τ 1 H-τ 2 Ab-τ 1 Sb-τ 2 S τ 1 H+τ 2 Aτ 1 S-k-dτ2Sτ2S 0k-d J =

12. Stability of the DFE: Jacobian Matrix at the DFE=(N, 0, 0): 0b-τ 1 Sb-τ 2 N 0τ 1 N-k-dτ2Nτ2N 0k-d J(DFE) =

13. Stability of the DFE: Eigenvalues One eigenvalue is 0: λ 1 = 0 The other two are found using the 2x2 matrix: τ 1 N-k-dτ2Nτ2N k-d Characteristic polynomial is λ 2 – tr(J’) λ + det(J’) = 0 J’ =

14. Stability of the DFE: Eigenvalues: λ 2 + (k+2d- τ 1 N)λ + d(k+d- τ 1 N)-k τ 2 N = 0 By Routh-Hurwicz Criterion (n=2), roots have negative real part when : k+2d- τ 1 N > 0 d(k+d- τ 1 N)-k τ 2 N > 0 R0R0 R 0 – the number of new infections determined by one infective introduced in a susceptible population ↔ DFE is locally asymptotically stable ↔ k τ 2 N / [d(k+d- τ 1 N)] < 1

15. Model II Assumptions: N not constant → b ≠ d. Let λ = b – d N’ = (b - d)N = λN → N(t) = N(0)e λt Let n(t) = e -λt N(t)= e -λt ( S(t) + H(t) + A(t) ) Apply the transformations for all classes: s(t) = e -λt S(t) h(t) = e -λt H(t) a(t) = e -λt A(t) d(t) = e -λt D(t) And then n(t) = s(t) + h(t) + a(t)

16. Model II Transformation of H(t): h(t) = e -λt H(t) h’(t) = -λe -λt H(t) + e -λt H’(t) = -λh(t) + e -λt ( τ 1 SH/N + τ 2 SA/N - kH - dH) = -bh - kh+ τ 1 sh/n + τ 2 sa/n Using standard incidence H/N and S/N

17. Model II s’(t) = bh + ba - τ 1 sh/n - τ 2 sa/n h’(t) = - bh - kh + τ 1 sh/n + τ 2 sa/n a’(t) = - ba + kh - d’a Equations: n = s + h + a n’ = s’ + h’ + a’ n’ = - d’a n(t) → 0

18. Model II Equilibrium Points: Still no endemic equilibrium (obtain contradiction) Disease Free Equilibrium: DFE = (n*, 0, 0)

19. Model II Stability of the DFE=(n*, 0, 0): One eigenvalue λ 1 = 0 and λ 2 + (2b+k+d’– τ 1 )λ + (b+d’)(b+k- τ 1 )-k τ 2 = 0 Routh-Hurwicz Criterion: (2b+k+d’– τ 2 ) > 0 (b+d’)(b+k- τ 1 )-k τ 2 > 0

20. Model II Therefore the DFE is stable when: (b+d’)(b+k- τ 1 )-k τ 2 > 0 ↕ k τ 2 /[(b+d’)(b+k- τ 1 )] < 1 R0R0 Comparison: ( I ) R 0 = k τ 2 N / [b(k+b- τ 1 N)] ( II ) R 0 = k τ 2 / [(b+d’)(b+k- τ 1 )]

21. Model III Equations: (1) S’ = b(S+H+A) - τ 1 SH - τ 2 SA - dS (2) H’ = τ 1 SH + τ 2 SA - kH - dH (3) A’ = kH – d’A – dA (3) D’ = d’A Assumptions: N not constant → b ≠ d

22. Model III – Equilibria Endemic Equilibrium: H ≠ 0: From (3): A = kH/(d+d’) From (2): S = (k+d)/[ τ 1 + τ 2 k/(d+d’)] Substitute in (1): H = (b-d)(k+d)/[(k+d-b-bk/(d+d’))( τ 1 + τ 2 k/(d+d’))] N not constant → no DFE

23. Model III Endemic Equilibrium: S* = (k+d)/[ τ 1 + τ 2 k/(d+d’)] H* = (b-d)(k+d)/[(k+d-b-bk/(d+d’))( τ 1 + τ 2 k/(d+d’))] A* = kH*/(d+d’)

24. Model III: Stability of the Endemic Equilibrium Jacobian matrix written in Maple:

25. Model III: Stability of the Endemic Equilibrium Characteristic Polynomial given by Maple: By Routh-Hurwicz Criterion (n=3), the endemic equilibrium is locally asymptotically stable when: a > 0, c > 0, ab > c x 3 + ax 2 + bx + c = 0

26. Data YearHIV+AIDSDeaths due to AIDS 19869952 198775114 198893146 1989121135 19901402823 19911833717 19921757132 19931028259 199412210262 199512411680 19962349992 199736312999 199836215098 1999493176122 2000545251142 Total32311284874 Given data: 1986-2000 New HIV Cases New Aids Cases Deaths each year from AIDS

27. Data

28. Fitting Model III to Data HIV Cases AIDS Cases Deaths from AIDS Legend: Given data Solution Curves of Model III

29. Fitting Model III to Data Parameters: b = 0.114 d = 0.073 τ 1 = 0.15x10 -5 τ 2 = 0.12x10 -6 k = 0.165 d’ = 0.195

30. Conclusions I: Problems Time Limitations In simulations In model development Discrete Model? Stochastic Model?

31. Extensions to the Model Suggestions for improvement: Females / Males Heterosexual / Homosexual (it started as a heterosexual disease, now 90% of seropositives are males) Exposed class - not infectious right away Include the people infected but unaware (an estimate of 20-30% of the HIV asymptomatic carriers have not been detected) Different number of sexual partners (differentiation between probabilities of transmission)

32. Extensions to the Model Suggestions for improvement: F S E D Hu A Ha Mh M Approximately 18 equations...

33. Conclusions: 3,200 HIV cases in Cuba Comparison with Canada in 2003: In Ontario - approximately same population as Cuba (12 million), but 23,863 HIV cases 12,156 HIV cases in Quebec (7 million) 11,346 HIV cases in British Columbia (3 million) 5 times more cases in QC 8 times more cases in ON 13 times more cases in BC

34. References 1. H de Arazoza and R. Lounes 2002. A non-linear model for a sexually transmitted disease with contact tracing. IMA J Math Appl Med Biol. Sep;19(3):221-34. 2. R. Lounes and H. de Arazoza 1999. A two-type model for the Cuban national programme on HIV/AIDS. IMA J Math Appl Med Biol. Jun;16(2):143-54. 3. Y.H Hsieh, de Arazoza H., Lee S.M., Chen C.W. Estimating the number of Cubans infected sexually by human immunodeficiency virus using contact tracing data. Int J Epidemiol. Jun;31(3):679-83. 4. BBC: Cuba leads the way in HIV fight. 2003 M. Bentley. http://news.bbc.co.uk/1/hi/in_depth/sci_tech/2003/denver_2003/2770631.stm

35. Thank you