Dynamics Models for Tuberculosis
Transmission and Control
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
Ancient disease
•
TB has a history as long as the human race.
•
TB appears in the history of nearly every culture.
•
TB was probably transferred from animals to humans.
•
TB thrives in dense populations.
•
It was the most important cause of death up to the
middle of the 19th century.
Jun,2002
MTBI Cornell University
Transmission Process
• Causative agent:
Tuberculosis Bacilli (Koch, 1882).
• Preferred habitat
Lung.
• Main Mode of transmission
Host-air-host.
• Immune Response
Immune system tends to respond
quickly to initial invasion.
Jun,2002
MTBI Cornell University
Immune System Response Caricature
• Bacteria invades lung tissue.
• White cells surround the invaders and
try to destroy them.
• Body builds a wall of cells and fibers
around the bacteria to confine them,
forming a small hard lump.
Jun,2002
MTBI Cornell University
Immune System Response Caricature
• Bacteria cannot cause additional damage as
long as confining walls remain unbroken.
• Most infected individuals never develop active
TB (that is, become infectious).
• Most remain latently-infected for life.
• Infection progresses and develops into active
TB in less than 10% of the cases.
Jun,2002
MTBI Cornell University
TB was the main
cause of mortality
• Leading cause of death in the past.
• Accounted for one third of all deaths in
the 19th century.
• One billion people died of TB during the
19th and early 20th centuries.
• TB’s nicknames: White Death, Captain
of Death, Time bomb
Jun,2002
MTBI Cornell University
Per Capita Death Rate of TB
Jun,2002
MTBI Cornell University
Current Situation
• Two to three million people around the world die
of TB each year.
• Someone is infected with TB every second.
• One third of the world population is infected
with TB ( the prevalence in the US is 10-15% ).
• Twenty three countries in South East Asia and
Sub Saharan Africa account for 80% total cases
around the world.
• 70% untreated actively infected individuals die.
Jun,2002
MTBI Cornell University
TB in the US
Jun,2002
MTBI Cornell University
Reasons for TB Persistence
• Co-infection with HIV/AIDS (10% who
are HIV positive are also TB infected).
• Multi-drug resistance is mostly due to
incomplete treatment.
• Immigration accounts for 40% or more
of all new recent cases.
• Lack of public knowledge about modes
of TB transmission and prevention.
Jun,2002
MTBI Cornell University
Earliest Models
•
•
•
•
Jun,2002
H.T. Waaler, 1962
C.S. ReVelle, 1967
S. Brogger, 1967
S.H. Ferebee, 1967
MTBI Cornell University
Epidemiological Classes
Jun,2002
MTBI Cornell University
Parameters
Jun,2002
MTBI Cornell University
Basic Model Framework
B' ( N , T , I )
F (N )
S
B( N , S , I )
S
•
•
•
•
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)
Jun,2002
MTBI Cornell University
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
Jun,2002
MTBI Cornell University
Epidemiology
(Basic Reproductive Number, R0)
The expected number of secondary infections
produced by a “typical” infectious individual
during his/her entire infectious period
when introduced in a population of mostly
susceptibles at a demographic steady state.
•
•
Sir Ronald Ross (1911)
Kermack and McKendrick (1927)
Jun,2002
MTBI Cornell University
Epidemiology
(Basic Reproductive Number, R0)
Frost (1937) wrote “…it is not necessary
that transmission be immediately and
completely prevented. It is necessary
only that the rate of transmission be
held permanently below the level at
which a given number of infection
spreading (i.e. open) cases succeed in
establishing an equivalent number to
carry on the succession”
Jun,2002
MTBI Cornell University
R0
R0 






C






k
  r1  d   r2  k






k
  r2  k
• Probability of surviving the latent stage:
• Average effective contact rate
C
• Average effective infectious period
1
  r1  d
Jun,2002
MTBI Cornell University
Demography
dN  F ( N )  N  dI ,
dt
dE   C ( N  E  I ) I  (  k  r )E,
2
N
dt
dI  kE  (  d  r )I .
1
dt
F(N)=, Linear Growth
F (N )  rN , Exponentia l Growth



N
F (N )  rN 1 , Logistic Growth
K

Jun,2002
MTBI Cornell University
Exponential Growth
(Three Thresholds)
The Basic Reproductive Number is
R0 
Jun,2002





C





k
  r1  d   r2  k
MTBI Cornell University





Demography and Epidemiology
R1 
Jun,2002






C






k
r  r1  d r  r2  k
MTBI Cornell University






Demography
r


R2  *
du
Where
2  4d (C  d )(kC  mr nr )c  (d (mr  nr )  C )(mr  k )
d
(
m

n


c
(
m

k
)
r
r
r
u* 
2d (C  d )(kC  mr nr )
Jun,2002
MTBI Cornell University
Bifurcation Diagram
(exponential
growth )
r
R1
N 0
I
N
u
*
 N  0 ( R2  1)

 I   ( R2  1)
1
I
N
I 
0
I
N
I 0
0
R0
1
Jun,2002
MTBI Cornell University
Logistic Growth




βC
k


R0 


μ r1  d  μ r2  k 






R* 
2
Jun,2002
r
R0 1
k
μ d
μ  d  k  r1 R0
MTBI Cornell University
Logistic Growth (cont’d)
If R2* >1
• When R0  1, the disease dies out at an exponential
•
•
rate. The decay rate is of the order of R0 – 1.
Model is equivalent to a monotone system. A general
version of the Poincaré-Bendixson Theorem is used
to show that the endemic state (positive equilibrium)
is globally stable whenever R0 >1.
When R0  1, there is no qualitative difference
between logistic and exponential growth.
Jun,2002
MTBI Cornell University
Bifurcation Diagram
I*
Global Transcriti cal
Bifurcatio n
1
Jun,2002
MTBI Cornell University
R0
Particular Dynamics
(R0 >1 and R2* <1)
All trajectories approach
the origin. Global
attraction is verified
numerically by randomly
choosing 5000 sets of
initial conditions.
Jun,2002
MTBI Cornell University
Fast and Slow TB
(S. Blower, et al., 1995)
pSI
 S
(1  p)SI
S
Jun,2002
E
kE
E
MTBI Cornell University
I
(  d)I
Fast and Slow TB
dS     SI  S,
dt
dE  (1 p) SI  kE  E,
dt
dI  p SI  kE  dI  I.
dt
Jun,2002
MTBI Cornell University
Variable Latency Period (Z. Feng, et al,2001)
p(s): proportion of infected (noninfectious) individuals who
became infective s unit of time ago and who are still infected
(non infectious).
Number of exposed from 0 to t who are alive and still in the E class
Number of those who progress to infectious from 0 to t and
who are still alive in I class at time t
Jun,2002
MTBI Cornell University
Variable Latency Period
(differentio-integral model)
•
•
E0(t): # of individuals in E class at t=0 and still in E class at time t
I0: # of individuals in I class at t=0 and still in I class at time t
Jun,2002
MTBI Cornell University
Exogenous Reinfection
p cS I
N
cS I
 S
N
S
kE
E
E
I
rI
(   d) I
cT I
N
Jun,2002
MTBI Cornell University
T
T
Exogenous Reinfection
dS    c S I  S ,
dt
N
dE  c S I  pc E I  (k   )E  c T I ,
dt
N
N
N
dI  pc E I  kE  (d  r   )I ,
dt
N
dT  rI-cT I  T ,
dt
N
N  S  E  I  T.
Jun,2002
MTBI Cornell University
Backward Bifurcation
Jun,2002
MTBI Cornell University
Age Structure Model

dt
 

 dt
 

 dt
 

 dt
 

 dt



   s(t , a)    (a)c(a) B(t ) s(t , a)  ( (a)   (a))s(t , a),
da 
  v(t , a)   (a)s(t , a)   (a)v(t , a)   (a) B(t )v(t , a),
da 
  l (t , a)   (a)c(a) B(t ) s(t , a)  σj(t,a)  δv(t,a)  (k  μ(a))l(t,a),
da 
  i(t , a)  kl (t , a)  (r   (a))i(t , a),
da 
   j (t , a)  ri(t , a)   (a)c(a) B(t ) j (t , a) -  (a) j (t , a).
da 
Jun,2002
MTBI Cornell University
Parameters
•
•
•
•
•
•
•
•
: recruitment rate.
(a): age-specific probability of becoming infected.
c(a): age-specific per-capita contact rate.
(a); age-specific per-capita mortality rate.
k: progression rate from infected to infectious.
r: treatment rate.
: reduction proportion due to prior exposure to TB.
: reduction proportion due to vaccination.
Jun,2002
MTBI Cornell University
Proportionate Mixing
• p(t,a,a`): probability that an individual of age a has
contact with an individual of age a` given that it has
a contact with a member of the population .
• Proportionate mixing: p(t,a,a`)= p(t,a`)
Jun,2002
MTBI Cornell University
Incidence and Mixing

B(t )   i(t, a') p(t, a, a')da'
n(t, a')
0
p(t, a')   c(a')n(t, a') ,
 c(u)n(t,u)du
random mixing
0
Initial values :
s(t,0)  , v(t,0)  l (t,0)  i(t,0)  j(t,0)  0.
Boundary values:
s(0, a)  s0 (a), v(0, a)  v0 (a), l (0, a)  l0 (a),
i(0, a)  i0 (a), j(0, a)  j0 (a).
n(t, a)  s(t, a)  v(t, a)  l (t, a)  i(t, a)  j(t, a)
Jun,2002
MTBI Cornell University
Basic reproductive Number
(by next generation operator)

R0 ( )    k p (  ) ( )c( ) e  (  k )  e  (  r)  ( )dd


rk
00
 (a)  F (a)   (1 F (a))
a
F (a)  exp(   (b)db) denotes the probabilit y that a
0
susceptibl e individual has not been vacci nated at age a.
Jun,2002
MTBI Cornell University
Stability
There exists an endemic steady state whenever
R0()>1.
The infection-free steady state is globally
asymptotically stable when R0= R0(0)<1.
Jun,2002
MTBI Cornell University
Optimal Vaccination Strategies
Two optimization problems:
If the goal is to bring R0() to pre-assigned value
then find the vaccination strategy (a) that minimizes the
total cost associated with this goal (reduced prevalence to a
target level).
If the budget is fixed (cost) find a vaccination strategy (a)
that minimizes R0(), that is, that minimizes the prevalence.
Jun,2002
MTBI Cornell University
Optimal Strategies
•
One–age strategy: vaccinate the susceptible
population
at exactly age A.
• Two–age strategy: vaccinate part of the susceptible
population at exactly age A1 and the remaining
susceptibles at a later age A2.
Optimal strategy depends on data.
Jun,2002
MTBI Cornell University
Challenging Questions associated with
TB Transmission and Control
•
•
•
•
•
•
Impact of immigration.
Antibiotic Resistance.
Role of public transportation.
Globalization—small world dynamics.
Time-dependent models.
Estimation of parameters and distributions.
Jun,2002
MTBI Cornell University