TB Cluster Models, Time Scales
and Relations to HIV
Carlos Castillo-Chavez
Department of Biological Statistics and
Computational Biology
Department of Theoretical and Applied
Mechanics
Cornell University, Ithaca, New York, 14853
Jun, 2002
MTBI Cornell University
Outline
• A non-autonomous model that incorporates the
impact of HIV on TB dynamics.
• Model to test CDC’s TB control goals.
• Casual versus close contacts and their impact on TB.
• Time scales and singular perturbation approaches in
the study of the dynamics of TB.
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TB in the US
(1953-1999)
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Reemergence of TB
• New York City and San Francisco had
recent outbreaks.
• Cost of control the outbreak in NYC alone
was estimated to be about 1 billion.
• Observed national TB case rate increase.
• TB reemergence became an international
issue.
• CDC sets control goal in 1989.
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Basic Model Framework
B' ( N , T , I )
F (N )
S
S
•
•
•
•
B( N , S , I )
E
E
N=S+E+I+T, Total population
F(N): Birth and immigration rate
B(N,S,I): Transmission rate (incidence)
B`(N,S,I): Transmission rate (incidence)
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kE
I
r1 I
(   d) I
r 2E
T
T
Model Equations
dS  F (N )   CS I  I ,
N
dt
dE   CS I  (  k  r2)E   'CT I ,
N
N
dt
dI  kE  (  d  r1)E,
dt
dT  r2 E  r1I   ' CT I  T ,
N
dt
N  S  E  I T
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TB control in the U.S.
CDC Short-Term Goal:
3.5 cases per 100,000 by 2000.
Has CDC met this goal?
CDC Long-term Goal:
One case per million by 2010.
Is it feasible?
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Model Construction
dN  F (N )  dI
dt
Since d has been approximately equal to zero over the past 50
years in the US, we only consider
dN  F (N ).
dt
Hence, N can be computed independently of TB.
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Non-autonomous model
(permanent latent class of TB introduced)
dL1
  ( N (t)  L1  L2  I ) I  ( (t)  A(t)  p  k  r1)L1,
dt
N (t)
dL2
 pL1  ( (t)  r2  v  A(t))L2 ,
dt
dI  (k  A(t))L  (v  A(t))L  ( (t)  d (t)  r )I .
1
2
3
dt
N (t),  (t), A(t), d (t) are known.
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Effect of HIV





3

1


(t 1983) Exp  2 (t 1983) , if t 1983;
A(t) 




0,
Otherwise.








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Upper Bound and Lower Bound For
Epidemic Threshold
R 
R 



















k
k  p  r1   d   r3  







k
k  p  r1    d   r3   












If R<1, L1(t), L2(t) and I(t) approach zero;
If R>1, L1(t), L2(t) and I(t) all have lower positive
boundary;
If (t) and d(t) are time-independent, R and R are
Equal to R0 .
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Parameter estimation and
simulation setup
Parameter

c
k
r1
r2
r3
p
Jun, 2002
Estimation
0.22
10
0.001
0.05
0.05
0.65
0.1
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Initial
Values
I(0)
87423
L1(0)
106
L2(0)
106
Parameter estimation and
simulation setup
N(t) is from census data
and population projection
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RESULTS
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CONCLUSIONS
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CONCLUSIONS
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CDC’s Goal Delayed
•
•
•
•
Impact of HIV.
Lower curve does not include HIV impact;
Upper curve represents the case rate when HIV is included;
Both are the same before 1983. Dots represent real data.
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Regression approach
Regression Equation :
LogY  11.3970  0.0597 X  0.0006 X 2
A Markov chain model supports the same result
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Cluster Models
• Incorporates contact type (close vs. casual) and
focus on the impact of close and prolonged
contacts.
• Generalized households become the basic
epidemiological unit rather than individuals.
• Use natural epidemiological time-scales in model
development and analysis.
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Close and Casual contacts
Close and prolonged contacts are likely to be responsible for
the generation of most new cases of secondary TB
infections. “A high school teacher who also worked at
library infected the students in her classroom but not those
who came to the library.”
Casual contacts also lead to new cases of active TB. WHO
gave a warning in 1999 regarding air travel. It announced
that flights of more than 8 hours pose a risk for TB
transmission.
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Transmission Diagram
knE2 NE22
knE2
knE2

S1
E2
S2
S1
S 2
Jun, 2002
kE2
I
E1
E2
S1
E1
In
I
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S2
N2
Key Features
• Basic epidemiological unit: cluster (generalized
household)
• Movement of kE2 to I class brings nkE2 to N1
population, where by assumptions nkE2(S2 /N2) go
to S1 and nkE2(E2/N2) go to E1
• Conversely, recovery of I infectious bring nI
back to N2 population, where nI (S1 /N1)=  S1 go
to S2 and nI (E1 /N1)=  E1 go to E2
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Basic Cluster Model
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Basic Reproductive Number





n
k
Q f
R 



    k    0
c
0
where
Q0  n
 
f k
k
Jun, 2002





is the expected number of infections produced by
one infectious individual within his/her cluster.
denotes the fraction who survives the latency
period and become active cases.
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Diagram of Extended Cluster Model
knE2

S2
(1  p) NIS2 2n
Jun, 2002
E2
N2
E2
knE2
pS1
kE2
E2
E1
I
E1
S1
S 2
knE2
(1  p)
In
I
MTBI Cornell University
IS1
N2 n
S1
S2
N2
 (n)
Both close casual contacts
are included in the extended
model. The risk of infection
per susceptible,  , is
assumed to be a nonlinear
function of the average
cluster size n. The constant p
measures the average
proportion of the time that
an “individual spends in a
cluster.
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R0 Depends on n in a non-linear
fashion
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Role of Cluster Size
E(n) denotes the ratio of within cluster to between
cluster transmission. E(n) increases and reaches its
maximum value at
*
pL
K
n 



K
1 1
pL
The cluster size n* is defined as optimal as it maximizes
the relative impact of within to between cluster
transmission.
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Hoppensteadt’s Theorem
(1973)
Full system
Reduced system
where x  Rm, y  Rn and  is a positive real parameter
near zero (small parameter). Five conditions must be
satisfied (not listed here) to apply the theorem. In
addition, it is shown that if the reduced system has a
globally asymptotically stable equilibrium then the full
system has a g.a.s. equilibrium whenever 0<  <<1.
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Two time Scales
• Latent period is long and roughly has
the same order of magnitude as that
associated with the life span of the
host.
• Average infectious period is about
six months (wherever there is
treatment, is even shorter).
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Rescaling
Time is measured in average infectious periods (fast time
scale), that is, = k t. The state variables are rescaled as
follow:
Where   / is the asymptotic carrying capacity.
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Rescaled Model
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Rescaled Model
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Dynamics on Slow Manifold
Solving for the quasi-steady states y1, y2 and y3 in
terms of x1 and x2 gives
Substituting these expressions into the equations for
x1 and x2 lead to the equations of motion on the slow
manifold.
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Slow Manifold Dynamics
Where
is the number of secondary
infections produced by one infectious individual in a
population where everyone is susceptible
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Theorem
If Rc0  1,the disease-free equilibrium (1,0) is globally
asymptotically stable. While if Rc0 > 1, (1,0) is unstable and
the endemic equilibrium
is globally asymptotically stable.
This theorem characterizes the dynamics on the slow
manifold
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Dynamics for Full System
Theorem: For the full system, disease-free equilibrium is
globally asymptotically stable whenever R0c <1; while R0c
>1 there exists a unique endemic equilibrium which is
globally asymptotically stable.
Proof approach: Construct Lyapunov function for the case
R0c <1; for the case R0c >1, we use Hoppensteadt’s
Theorem.
A similar result can be found in Z. Feng’s 1994, Ph.D.
dissertation.
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Bifurcation Diagram
I*
Global Transcriti cal
Bifurcatio n
1
R0
Global bifurcation diagram when 0<<<1 where  denotes
the ratio between rate of progression to active TB and the
average life-span of the host (approximately).
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Numerical Simulations
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Conclusions from cluster models
• TB has slow dynamics but the change of
epidemiological units makes it possible to identify
non-traditional fast and slow dynamics.
• Quasi steady assumptions (adiabatic elimination of
parameter) are valid (Hoppensteadt’s theorem).
• The impact of close and casual contacts can be study
using this approach as long as progression rates
from the latently to the actively-infected stages are
significantly different.
Jun, 2002
MTBI Cornell University
Conclusions from cluster
models
• Singular perturbation theory can be used to study
the global asymptotic dynamics.
• Optimal cluster size highlights the relative impact
of close versus casual contacts and suggests
alternative mechanisms of control.
• The analysis of the system for the case where the
small parameter  is not small has not been carried
out. Simulations suggest a wider range.
Jun, 2002
MTBI Cornell University