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