Modeling the HIV/AIDS Epidemic in Cuba
Presented by Raluca Amariei and Audrey Pereira
2005 PIMS Mathematical Biology Summer School
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
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% )
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
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
Model I - Parameters b = birth rate; d = death rate τ1 = probability of acquiring HIV when in contact with an HIV+
τ2 = 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
τ2τ1
k
Model I
(1) S’ = b(S+H+A) - τ1SH - τ2SA - dS
(2) H’ = τ1SH + τ2SA - 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
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)
Analysis of Model I
In (2), case (ii) H ≠ 0
divide (2) by H:
τ1S + τ2kSH/d - k - d = 0
→ S* = (k+d)/(τ1+τ2k/d)
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
Stability
Jacobian Matrix:
-τ1H-τ2A b-τ1S b-τ2S
τ1H+τ2A τ1S-k-d τ2S
0 k -d
J =
Stability of the DFE:
Jacobian Matrix at the DFE=(N, 0, 0):
0 b-τ1S b-τ2N
0 τ1N-k-d τ2N
0 k -d
J(DFE) =
Stability of the DFE:
Eigenvalues
One eigenvalue is 0: λ1 = 0 The other two are found using the 2x2 matrix:
τ1N-k-d τ2N
k -d
Characteristic polynomial is λ2 – tr(J’) λ + det(J’) = 0
J’ =
Stability of the DFE:
Eigenvalues:
λ2 + (k+2d-τ1N)λ + d(k+d-τ1N)-kτ2N = 0
By Routh-Hurwicz Criterion (n=2), roots have negative real part when:
k+2d-τ1N > 0
d(k+d-τ1N)-kτ2N > 0
R0
R0 – the number of new infections determined by one infective introduced in a susceptible population
↔ DFE is locally asymptotically stable
↔ kτ2N / [d(k+d-τ1N)] < 1
Model II
Assumptions:N not constant → b ≠ d. Let λ = b – d
N’ = (b - d)N = λN → N(t) = N(0)eλt Let n(t) = e-λtN(t)= e-λt ( S(t) + H(t) + A(t) )
Apply the transformations for all classes:
s(t) = e-λtS(t)h(t) = e-λtH(t)a(t) = e-λtA(t)d(t) = e-λtD(t)
And then n(t) = s(t) + h(t) + a(t)
Model II
Transformation of H(t):
h(t) = e-λtH(t)
h’(t) = -λe-λtH(t) + e-λtH’(t) = -λh(t) + e-λt(τ1SH/N + τ2SA/N - kH - dH) = -bh - kh+ τ1sh/n + τ2sa/n
• Using standard incidence H/N and S/N
Model II
s’(t) = bh + ba - τ1sh/n - τ2sa/n
h’(t) = - bh - kh + τ1sh/n + τ2sa/n
a’(t) = - ba + kh - d’a
Equations:
n = s + h + an’ = s’ + h’ + a’n’ = - d’a
n(t) → 0
Model II
Equilibrium Points:
Still no endemic equilibrium (obtain contradiction)
Disease Free Equilibrium:
DFE = (n*, 0, 0)
Model II
Stability of the DFE=(n*, 0, 0):
One eigenvalue λ1 = 0 and
λ2 + (2b+k+d’–τ1)λ + (b+d’)(b+k-τ1)-kτ2 = 0
Routh-Hurwicz Criterion:
(2b+k+d’–τ2 ) > 0
(b+d’)(b+k-τ1)-kτ2 > 0
Model II
Therefore the DFE is stable when:
(b+d’)(b+k-τ1)-kτ2 > 0
↕
kτ2 /[(b+d’)(b+k-τ1)] < 1
R0 Comparison:
(I) R0 = kτ2N / [b(k+b-τ1N)]
(II) R0 = kτ2 / [(b+d’)(b+k-τ1)]
Model III
Equations:
(1) S’ = b(S+H+A) - τ1SH - τ2SA - dS
(2) H’ = τ1SH + τ2SA - kH - dH
(3) A’ = kH – d’A – dA
(3) D’ = d’A
Assumptions:N not constant → b ≠ d
Model III – Equilibria
Endemic Equilibrium:
H ≠ 0:
From (3): A = kH/(d+d’)
From (2): S = (k+d)/[τ1+τ2k/(d+d’)]
Substitute in (1):
H = (b-d)(k+d)/[(k+d-b-bk/(d+d’))(τ1+τ2k/(d+d’))]
N not constant → no DFE
Model III
Endemic Equilibrium:
S* = (k+d)/[τ1+τ2k/(d+d’)]
H* = (b-d)(k+d)/[(k+d-b-bk/(d+d’))(τ1+τ2k/(d+d’))]
A* = kH*/(d+d’)
Model III: Stability of the Endemic Equilibrium
Jacobian matrix written in Maple:
:= G
b d t1 H t2 A b t1 S b t2 St1 H t2 A t1 S d k t2 S0 k d1
Model III: Stability of the Endemic Equilibrium
Characteristic Polynomial given by Maple:
x3 ( ) d1 t1 S 2 d k b t1 H t2 A x2 k t2 S d1 t1 S 2 d1 d d1 k d1 b (
d1 t1 H d1 t2 A t1 H b t2 A b t1 S b t1 S d d b d2 d t1 H d t2 A k b d k k t1 H k t2 A ) x k t1 H b k t2 A b k d t2 S k t2 S b d1 t1 H b d1 t2 A b d1 t1 S b d1 t1 S d d1 d t1 H d1 d t2 A d1 k t1 H d1 k t2 A d d1 k d1 d2 d1 d b d1 k b
By Routh-Hurwicz Criterion (n=3), the endemic equilibrium is locally asymptotically stable when:
a > 0, c > 0, ab > c
x3 + ax2 + bx + c = 0
Data
Year HIV+ AIDS Deaths due to AIDS
1986 99 5 2
1987 75 11 4
1988 93 14 6
1989 121 13 5
1990 140 28 23
1991 183 37 17
1992 175 71 32
1993 102 82 59
1994 122 102 62
1995 124 116 80
1996 234 99 92
1997 363 129 99
1998 362 150 98
1999 493 176 122
2000 545 251 142
Total 3231 1284 874
Given data:
1986-2000
• New HIV Cases
• New Aids Cases
• Deaths each year from AIDS
Data
Fitting Model III to Data
HIV Cases
AIDS Cases
Deaths from AIDS
Legend:
Given data
Solution Curves of Model III
Fitting Model III to Data
Parameters:
b = 0.114
d = 0.073
τ1 = 0.15x10-5
τ2 = 0.12x10-6
k = 0.165
d’ = 0.195
Conclusions I: Problems
Time Limitations In simulations In model development
Discrete Model? Stochastic Model?
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)
Extensions to the Model
Suggestions for improvement:
F
S
E
D
Hu
A
Ha
Mh
M Approximately 18 equations...
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
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
Thank you