Download 1.what is the magnitude of the effect a delay has on tranmission 2

Influenza Vaccination Strategies when
supply is limited
Romarie “Romie” Morales Rosado
Questions
• What is the impact of having access only to a
limited number of dosages?
• What is the impact of delays in accessing the
available vaccine supply?
• What is the role of a large percentage of H1N1
asymptomatic infectious individuals?
Type A and its subtypes
Subdivided (H1-H16) , (N1-N9)
Influenza virus and transmission
http://www.youtube.com/watch?v=FvEOj
wUOzJc
Impact Pandemic Influenza
WHO declared a pandemic in June 2009, a total of 74 countries and territories had reported laboratory
confirmed infections. To date, most countries in the world have confirmed infections from the new virus.
Source:http://commons.wikimedia.org/wiki/File:H1N1_map_by_confirmed_cases.svg
Confirmed Deaths and Infections
Source: http://commons.wikimedia.org/wiki/File:H1N1_map.svg
Pandemic 2009 vs. Pandemic 1918-19
Some similarities…
•
•
•
•
Virus began to appear in the spring.
Came out of nowhere…!
Primarily attack young adults
Elderly were partially immune to 2009 disease
Source:http://news.sciencemag.org/sciencenow/2010/03/swine-flu-pandemic-reincarnates-.html?rss=1, http://ent.about.com/od/entdisordersgi/a/H1N1pandemic.htm
Prevention Methods
Methods of Prevention
Importance of Vaccination?
• Problem: Not enough vaccines for everyone
• Time constraint from identifying virus to
creating and approving vaccine
• Rich-Poor country Division
Transmission Model
P



S


V
(1   ) 
(I  J)
N
(I  J)
N
(I  J)
N
F
E


Source: G. Chowell et al Addative vaccination strategies


I
J
1

2
R


D
System of nonlinear differential
equations
I(t)  J(t)
SÝ(t)  u(t)S(t)  
S(t)
N(t)
I(t)  J(t)
VÝ(t)  u(t)S(t)  V (t)  
V (t)
N(t)
I(t)  J(t)
FÝ(t)  (1   )u(t)S(t)  
F(t)
N(t)
PÝ(t)  V (t)
I(t)  J(t)
EÝ(t)  
(S(t)  V (t)  F(t))  E(t)
N(t)
IÝ(t)  E(t)  (   )I(t)
1
JÝ(t)  I(t)  (   )J(t)
2
RÝ(t)   I(t)   J(t)
1
DÝ(t)  J(t)
2
Basic Reproductive Number
R
0
1
 (



1

( 
 )(
1
2
 )
)
Parameters
Source: G. Chowell et al Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland
R. Gani, H. Hughes, D. Fleming, T. Grifin, J. Medlock, S. Leach. Potential impact of antiviral use during influenza pandemic. Emerg Infect Dis; 11( 2005); 1355–362.
I.M.LonginiJr.,M.E.Halloran,A.NizamandY.Yang.Containingpandemicinfluenzawithantiviralagents. American Journal of Epidemiology; 159(2004); 623–633.
C.E. Mills, J.M. Robins and M. Lipsitch. Transmissibility of 1918 pandemic influenza. Nature 432 (2004):904–906.
Cases of Optimal Vaccine Strategies
• Limited Vaccine Access
• Almost unlimited Vaccine Access
• Time Delay
Optimal Control Problem
T
W 2
F (u(t))   [I(t) 
u (t)]dt
2
0
F (u* (t))  min  F (u(t))

L1 Set of functions –integral of function has finite solution we want u to belong to this set
Unconstraint Vaccine supply case
Sensitivity Analysis
• Weight Constant
• Control Upper Bound (maximum vaccination
rate)
• Efficacy of Vaccine
Weight constant
Final epidemic Size when varying
Weight Constant
Control upper bound on Final
Epidemic Size
Vaccine Efficacy
Isoperimetric Constraint
Recap on Results
• Best to apply vaccines at the beginning of the
transmission
• When transmission is weak a low vaccination
policy is effective.
Initial conditions with Delay when
R0 = 1.3
R0=1.3
Delay =10 days
Delay = 20 days
Delay = 30 days
S
172810
168750
160820
V
0
0
0
F
0
0
0
P
0
0
0
E
447
949
1786
I
255
545
1044
J
109
235
456
R
1387
4506
10851
D
7
24
57
1758
5311
12408
0
0
0
IC
C
Delay R0 = 1.3
Initial conditions with Delay when R0=2
R0=2
Run= 10 days
Run= 20 days
Run=30 days
S
167800
119310
52110
V
0
0
0
F
0
0
0
P
0
0
0
E
2329
13386
8916
I
1143
7379
6517
J
434
3039
3313
R
3291
31739
103600
D
16
161
551
4885
42318
113980
0
0
0
IC
C
Delay R0 = 2
Results
• Maximum Vaccination rate should be applied
in a timely manner!!!
• No significant difference between
unconstrained and constrained cases when R0
is low (1.3)
• Increase in vaccine efficacy and upper bound
of control results in a decrease in the amount
of vaccines that must be administered
• Delay has impact on vaccine efficacy.
Future Work
• Include asymptomatic class and understand
the impact of these individuals on disease
transmission .