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 subtypesSubdivided (H1-H16) , (N1-N9)

Influenza virus and transmission

http://www.youtube.com/watch?v=FvEOjwUOzJc

Impact Pandemic Influenza

Source:http://commons.wikimedia.org/wiki/File:H1N1_map_by_confirmed_cases.svg

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.

Confirmed Deaths and Infections

Source: http://commons.wikimedia.org/wiki/File:H1N1_map.svg

Pandemic 2009 vs. Pandemic 1918-19Some 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

VV

PP

SS

FF

JJ

RR

EE II DD

€

εμ

€

β(I + J)

N€

β(I + J)

N€

η

€

(1 −ε )μ

€

β(I + J)

N

€

κ

€

α

€

γ1

€

γ2

€

δ

Source: G. Chowell et al Addative vaccination strategies

System of nonlinear differential equations

€

˙ S (t) = −u(t)S(t) − βI(t) + J(t)

N(t)S(t)

˙ V (t) = εu(t)S(t) −ηV (t) − βI(t) + J(t)

N(t)V (t)

˙ F (t) = (1−ε )u(t)S(t) − βI(t) + J(t)

N(t)F(t)

˙ P (t) = ηV (t)

˙ E (t) = βI(t) + J(t)

N(t)(S(t) +V (t) + F(t)) − κE(t)

˙ I (t) = κE(t) − (α +1

γ )I(t)

˙ J (t) = αI(t) − (2

γ +δ )J(t)

˙ R (t) =1

γ I(t) +2

γ J(t)

˙ D (t) = δJ(t)

Basic Reproductive Number

€

0R = β(1

α +1

γ+

α

(α +1

γ )(2

γ +δ ))

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

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F(u(t)) = [I(t) +W

20

T

∫ u2(t)]dt

F(u*(t)) = minΩ 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

IC 1758 5311 12408

C 0 0 0

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

IC 4885 42318 113980

C 0 0 0

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 .