Tutorials 1: Epidemiological Mathematical
Modeling Applications in Homeland Security.
Mathematical Modeling of Infectious Diseases: Dynamics and Control (15
Aug - 9 Oct 2005)
Jointly organized by Institute for Mathematical Sciences, National University of
Singapore and Regional Emerging Diseases Intervention (REDI) Centre,
Singapore
http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm
Singapore, 08-23-2005
Carlos Castillo-Chavez
Joaquin Bustoz Jr. Professor
Arizona State University
ASU/SUMS/MTBI/SFI
Bioterrorism
The possibility of bioterrorist acts
stresses the need for the development
of theoretical and practical
mathematical frameworks to
systemically test our efforts to
anticipate, prevent and respond to acts
of destabilization in a global
community
ASU/SUMS/MTBI/SFI
From defense threat reduction agency
Buildings Ports &
Airports
Response
Attribution
Urban
Treatment and
Consequence Management
Detection
Interdiction
Warning
ASU/SUMS/MTBI/SFI
From defense
reduction
agency
From
defensethreat
threat
reduction
agency
Food Safety
Medical Surveillance
Animal/Plant Health
Other Public Health
Choke Points
Urban Monitoring
Characterization
Metros
Data Mining,
Fusion, and
Management
Emergency
Management
Tools
Toxic Industrials
ASU/SUMS/MTBI/SFI
State and
Local
Governments
Ricardo Oliva:
Research Areas
•Biosurveillance;
•Agroterrorism;
•Bioterror response logistics;
•Deliberate release of biological agents;
•Impact assessment at all levels;
•Causes: spread of fanatic behaviors.
ASU/SUMS/MTBI/SFI
Modeling Challenges &Mathematical Approaches
From a “classical” perspective to a global scale
Deterministic
 Stochastic
 Computational
 Agent Based Models

ASU/SUMS/MTBI/SFI
Some theoretical/modeling challenges
•Individual and Agent Based Models--what can they
do?
•Mean Field or Deterministic Approaches--how do we
average?
•Space? Physical or sociological?
•Classical approaches (PDEs, meta-population models)
or network/graph theoretic approaches
•Large scale simulations--how much detail?
ASU/SUMS/MTBI/SFI
Ecological/Epidemiological view point
 Invasion
 Persistence
 Co-existence
 Evolution
 Co-evolution
 Control
ASU/SUMS/MTBI/SFI
Epidemiological/Control Units
 Cell
 Individuals
 Houses/Farms
 Generalized
households
 Communities
 Cities/countries
ASU/SUMS/MTBI/SFI
Temporal Scales
 Single
outbreaks
 Long-term dynamics
 Evolutionary behavior
ASU/SUMS/MTBI/SFI
Social Complexity
 Spatial
distribution
 Population structure
 Social Dynamics
 Population Mobility
 Demography--Immigration
 Social hierarchies
 Economic systems/structures
ASU/SUMS/MTBI/SFI
Links/Topology/Networks
 Local
transportation network
 Global transportation network
 Migration
 Topology (social and physical)
 Geography--borders.
ASU/SUMS/MTBI/SFI
Control/Economics/Logistics
 Vaccination/Education
 Alternative
public health approaches
 Cost, cost & cost
 Public health infrastructure
 Response time
ASU/SUMS/MTBI/SFI
Critical Response Time in
FMD epidemics
A. L. Rivas,
S. Tennenbaum,
C. Castillo-Chávez et al.
{American Journal of Veterinary Research}
(Canadian Journal of Veterinary Research)
It is critical to determine the time
needed and available to implement a
successful intervention.
The context--Foot and Mouth Disease
A
R
G
E
N
T
.I
N
A
BRAZIL
: 1-5 cases
(1- 7 days
post-onset)
1-5 cases
1
2
3
(8-14 days
post-onset)
ATLANTIC OCEAN
(May 1, 01)
Day 30 (May 22, 01)
Day 29 (May 21, 01)
Day 28 (May 20, 01)
Day 27 (May 19, 01)
Day 26 (May 18, 01)
Day 25 (May 17, 01)
Day 24 (May 16, 01)
Day 23 (May 15, 01)
Day 22 (May 14, 01)
Day 21 (May 13, 01)
Day 20 (May 12, 01)
Day 19 (May 11, 01)
Day 18 (May 10, 01)
Day 17 (May 9, 01)
Day 16 (May 8, 01)
Day 15 (May 7, 01)
Day 14 (May 6, 01)
Region 1
Day 13 (May 5, 01)
40
Day 12 (May 4, 01)
Day 11 (May 3, 01)
Day 10 (May 2, 01)
Day 9
Day 8 (April 30, 01)
Day 7 (April 29, 01)
Day 6 (April 28, 01)
35
Day 5 (April 27, 01)
Day 4 (April 26, 01)
Day 3 (April 25, 01)
Day 2 (April 24, 01)
Day 1 (April 23, 01)
Number of daily cases
Daily cases in the first month of the epidemic
“exponential”growth
Region 2
30
Region 3
25
20
15
10
5
0
The Basic Reproductive Number R0
R0 is the average number of secondary cases generated by an
infectious unit when it is introduced into a susceptible population
(at demographic steady state) of the same units.
If R0 >1 then an epidemic is expected to occur--number of
infected units increases
If R0 < 1 then the number of secondary infections is not enough
to sustain an apidemic.
The goal of public health interventions is to reduce R0 to a
number below 1.
However, timing is an issue! How fast do we need to respond?
ASU/SUMS/MTBI/SFI
Estimated
CRTs for
implementing
intervention(s)
resulting in
R_o <= 1
(successful
intervention)
1.4
days
2.6
days
3.0
days
Epidemic Models
ASU/SUMS/MTBI/SFI
Basic Epidemiological Models: SIR
Susceptible - Infected - Recovered
ASU/SUMS/MTBI/SFI
contacts probability of transmission 
 

 time  

contact
I
B(S,I)  S
N
  
S(t):
susceptible at time t

I(t): infected assumed infectious at time t
R(t): recovered, permanently immune
N: Total population size (S+I+R)
N

S


I
I
S

I
R


R

ASU/SUMS/MTBI/SFI
SIR - Equations
dS
I
 N  S  S
dt
N
dI
I
 S     I
dt
N
dR
 I  R
dt
Parameters
(1)
(2)
(3)


Per-capita death (or birth) rate

Transmission coefficient
Per-capita recovery rate


N SIR
contacts  probability of transmission 

  

(4)
 
dN d
 S  I  R  0
dt dt
(5)

unit time  
ASU/SUMS/MTBI/SFI
contact

SIR - Model (Invasion)
dS
I
 N  S  S
dt
N
dI
I
 S     I
dt
N
SN
dI
 I     I       I
dt
or
I(t)
I(t)  I(0)e
   t

R0 


ASU/SUMS/MTBI/SFI
1
Ro
“Number of secondary infections
generated by a “typical” infectious individual in a
population of mostly susceptibles
at a demographic steady state
Ro<1
No epidemic
Ro>1
Epidemic
ASU/SUMS/MTBI/SFI
Establishment of a Critical Mass of Infectives!
Ro >1 implies growth while Ro<1 extinction.
ASU/SUMS/MTBI/SFI
Phase Portraits
ASU/SUMS/MTBI/SFI

SIR
Transcritical Bifurcation
I*
I * (R0 )

R0
unstable

ASU/SUMS/MTBI/SFI
Deliberate Release of
Biological Agents
ASU/SUMS/MTBI/SFI
Effects of Behavioral Changes in a Smallpox Attack
Model
Impact of behavioral changes on response logistics and
public policy (appeared in Mathematical Biosciences, 05)
Sara Del Valle1,2
Herbert Hethcote2, Carlos Castillo-Chavez1,3, Mac Hyman1
1Los Alamos National Laboratory
2University of Iowa
3Cornell University
ASU/SUMS/MTBI/SFI
MODEL
•All individuals are susceptible
•The population is divided into two groups: normally active and less active
•No vital dynamics included (single outbreak)
•Disease progression: Exposed (latent) and Infectious
•News of a smallpox outbreak leads to the implementation of
the following interventions:
–Quarantine
–Isolation
–Vaccination
(ring and mass vaccination)
–Behavioral changes (3 levels: high, medium & low)
ASU/SUMS/MTBI/SFI
The Model
Sn
V
En
In
Q
S
Sl
R
W
E
El
I
Il
D
The subscript refers to normally active (n) or less active (l):
Susceptibles (S), Exposed (E), Infectious (I), Vaccinated (V),
Quarantined (Q), Isolated (W), ASU/SUMS/MTBI/SFI Recovered (R), Dead (D)
The Model

The behavioral change rates are modeled by a non-negative, bounded,
monotone increasing function i (for i = S, E, I) given by
i 

ai (In  I ) 1
1 bi (In  I ) day
with

ASU/SUMS/MTBI/SFI
S  E  I
Numerical Simulations
ASU/SUMS/MTBI/SFI
Numerical Simulations
ASU/SUMS/MTBI/SFI
Conclusions
•Behavioral changes play a key role.
• Integrated control policies are most effective:
behavioral changes and vaccination have a huge
impact.
•Delays are bad.
ASU/SUMS/MTBI/SFI
Mass Transportation and Epidemics
"An Epidemic Model with
Virtual Mass Transportation"
ASU/SUMS/MTBI/SFI
Mass Transportation
Systems/HUBS
Baojun Song
Juan Zhang
Carlos Castillo-Chavez
ASU/SUMS/MTBI/SFI
Subway Transportation Model
NSU
NSU
SU
SU
Subway
SU
SU
NSU
NSU
ASU/SUMS/MTBI/SFI
Vaccination Strategies
• Vaccinate civilian health-care and public
health workers
• Ring vaccination (Trace vaccination)
• Mass vaccination
• Mass vaccination if ring vaccination
fails
•Integrated approaches likely to be most
effective
Assumptions
1.The population is divided into N
neighborhoods;
2.Epidemiologically each individual is in
one of four status: susceptible, exposed,
infectious, and recovered;
3.A person is either a subway user or not
4.A ``vaccinated” class is included-everybody who is successfully vaccinated
is sent to the recovered class
Proportionate mixing
K subpopulations with densities N1(t), N2(t), …, Nk(t) at time t.
cl : the average number of contacts per individual, per unit time
among members of the lth subgroup.
Pij : the probability that an i-group individual has a contact with a
j-group individual given that it had a contact with somebody.
Proportionate mixing
(Mixing Axioms)
k>0
Pij
P 1
ij
j1
(1)
(2)
(3)

ci Ni Pij = cj Ncj PN
ji
Pij Pj 
Then
j
j
K
cl N l
l 1

is the only separable solution satisfying (1) , (2), and (3).
Definitions
Pai ai
the mixing probability between non-subway users from neighborhood i
given that they mixed.
Pa b
the mixing probability of non-subway and subway users from
neighborhood i, given that they mixed.
Pb a
the mixing probability of subway and non-subway users from
neighborhood i, given that they mixed.
Pb b
the mixing probability between subway users from neighborhood i, given
that they mixed.
Pb b
the mixing probability between subway users from neighborhoods i and j,
given that they mixed.
Paia j
the mixing probability between non-subway users from neighborhoods i
and j, given that they mixed.
Pa b
the mixing probability between non-subway user from neighborhood i and
subway users from neighborhood j, given that they mixed.
i i
i i
i i
i j
i j
Formulae of Mixing Probabilities
(depends on activity level and allocated time)
Identities of Mixing Probabilities
State Variables









i index of neighborhood
Wi number of individuals of susceptibles of SU in neighborhood i
Xi number of individuals of exposed of SU in neighborhood i
Yi number of individuals of infectious of SU in neighborhood i
Zi number of individuals of recovered of SU in neighborhood i
Si number of individuals of susceptibles of NSU in neighborhood i
Ei number of individuals of exposed of NSU in neighborhood i
Ii number of individuals of infectious of NSU in neighborhood i
Ri number of individuals of recovered of NSU in neighborhood i
Smallpox Model for NSU in neighborhood i
Ai
Si
Bi (t )
Ei
S i
Ei
Ei
Ii
(  d ) I i
ql 2 Ei
I i
ql1 S i
Ri
Ri
Model Equations for neighborhood i
Nonsubway users
dS i
 Ai  Bi (t )  ( S i  ql1 S i )
dt
dEi
 Bi (t )  ( Ei  Ei  ql 2 Ei )
dt
dI i
 Ei  (     d ) I i
dt
dRi
 I i  Ri  ql1 S  ql 2 E
dt
Qi (t )  S i (t )  Ei (t )  I i (t )  Ri (t )
Subway users
dWi
  i  Vi (t )  ( Wi  ql1Wi )
dt
dX i
 Vi (t )  ( X i  X i  ql 2 X i )
dt
dYi
 X i  (     d )Yi
dt
dZ i
 Yi  Z i  ql1Wi  ql 2 X i
dt
Ti (t )  Wi (t )  X i (t )  Yi (t )  Z i (t )
Infection Rates
Rate of infection for NSU


Ii
Bi (t)   i ai Si P˜a i
 P˜bi
  i 

 Qi
 Ti 
i   i 

  i  
Yi 
 
i   i  
  i  
Ti 
 Qi 
i   i  
Rate of infection for SU




Vi (t)   ibiW i Pa i



  
  i 
j

Y


Yi 
j 


N
Ii
i   i 
 j   j 
 Pbi
  Pb j
  i 
  i 
  
j1
j

Ti 
Ti 
T j 
 Qi
 Qi



i   i 
i   i 
 j   j 

R0 for Two Neighborhoods
(a special case)
q  0, i  0,  i  1
R0  max{ R 0,1 , R0, 2 }
   1
 a i (Ai /  )  Ai /  
R0, i   i a 




i
       d (a i Ai  bi  i ) /  (Ai   i ) /  
   1
 bi ( i /  )   i /  
  i b 




i
       d (a i Ai  bi  i ) /  (Ai   i ) /  
Two neighborhood simulations
(NYC type city)
1.
There are 8 million long-term and 0.2 million shortterm (tourists) residents in NYC.
2.
Time span of simulation is 30 days +.
3.
Control parameters in the model are: q1 and q2
(vaccination rates)
4.
We use two ``neighborhoods”, one for NYC residents
and the second for tourists.
Curve R0 (q1, q2)
=1
Plot R0 (q1, q2) vs q1 and
q2
Cumulative deaths: One day delay (q1 = q2=0.5)
Cases: One day delay (q1 = q2=0.5)
Cumulative deaths: One day delay (q1 = q2=0.8)
Cases: One day delay (q1 = q2=0.5)
Conclusions
•Integrated control policies are most effective:
behavioral changes and vaccination have a huge
impact.
•Delays are bad.
ASU/SUMS/MTBI/SFI