Optimal Control for a Discrete Time Inﬂuenza Modelutminers.utep.edu/pagonzalezparra/presentations/PresentationSACNAS.pdf2012 SACNAS National Conference Seattle, WA October 12, 2012

Optimal Control for a Discrete TimeInfluenza Model

Paula A. Gonzalez Parra1, Sunmi Lee2,Leticia Velazquez1,3, Carlos Castillo-Chavez4

1Program in Computational Science, University of Texas at El Paso2Department of Mathematical Sciences, Case Western Reserve University

3Department of Mathematical Sciences, University of Texas at El Paso4Mathematical, Computational and Modeling Sciences Center, Arizona State University

2012 SACNAS National ConferenceSeattle, WA

October 12, 2012

Paula A. Gonzalez Parra (UTEP) Discrete Influenza Model October 12, 2012 1 / 20

Outline

1 Introduction

2 Optimal Control Problem

3 MethodologyPontryagin’s Maximum PrincipleInterior-Point Methods

4 Numerical Solution

5 Multi-Group Discrete Influenza Model

6 Numerical simulations

7 Conclusions


Introduction

Continuous time models have been used to study influenza outbreaksand the impact of different control policies.

The identification of optimal control strategies that involve antiviraltreatment and isolation have been investigated in the continuous case.

We formulate a discrete time model and we include social distancingand antiviral treatment as control policies.


Introduction





Introduction





Epidemiological Model

The model is given by the system of difference equations:

St+1 = St(1− Gt)It+1 = StGt + (1− τt) (1− σ1) (1− δ)ItTt+1 = (1− σ2)Tt + τt (1− σ1) (1− δ)ItRt+1 = Rt + σ1 (1− δ) It + σ2Tt

(1)

Where

Gt = ρ(1− xt)It + εTt

Nt, (2)



The model is given by the system of difference equations:

St+1 = St(1− Gt)It+1 = StGt + (1− τt) (1− σ1) (1− δ)ItTt+1 = (1− σ2)Tt + τt (1− σ1) (1− δ)ItRt+1 = Rt + σ1 (1− δ) It + σ2Tt

(1)

Where

Gt = ρ(1− xt)It + εTt

Nt, (2)


Problem Formualtion

Our goal is to minimize the number of infected individuals over a finiteinterval [0, n] by using a minimal effort on treatment and socialdistancing.

The problem can be written as

min12

n−1∑t=0

(B1I2

t + B2x2t + B3τ

2t

), (3)

subject to Model 1.


Problem Formualtion

Our goal is to minimize the number of infected individuals over a finiteinterval [0, n] by using a minimal effort on treatment and socialdistancing.

The problem can be written as

min12

n−1∑t=0

(B1I2

t + B2x2t + B3τ

2t

), (3)

subject to Model 1.


Pontryagin’s Maximum Principle

The Hamiltonian at time t is defined as

Ht =12

(B1I2

t + B2x2t + B3τ

2t

)+ λi

t+1y it+1.

The necessary conditions are given by:The adjoint equation

λit =

∂Ht

∂y it,

The transversality conditionλi

n = 0,

The optimality condition

∂Ht

∂xt= 0 and

∂Ht

∂τt= 0.




Ht =12

(B1I2

t + B2x2t + B3τ

2t

)+ λi

t+1y it+1.


λit =

∂Ht

∂y it,


n = 0,


∂Ht

∂xt= 0 and

∂Ht

∂τt= 0.




Ht =12

(B1I2

t + B2x2t + B3τ

2t

)+ λi

t+1y it+1.


λit =

∂Ht

∂y it,


n = 0,


∂Ht

∂xt= 0 and

∂Ht

∂τt= 0.




Ht =12

(B1I2

t + B2x2t + B3τ

2t

)+ λi

t+1y it+1.


λit =

∂Ht

∂y it,


n = 0,


∂Ht

∂xt= 0 and

∂Ht

∂τt= 0.




Ht =12

(B1I2

t + B2x2t + B3τ

2t

)+ λi

t+1y it+1.


λit =

∂Ht

∂y it,


n = 0,


∂Ht

∂xt= 0 and

∂Ht

∂τt= 0.


Interior-Point Methods (IPM)

IPM were introduced by Karmarkar in 1984 for solving linearprogramming problems.

Contrary to the simplex method, IPM find an optimal solution bycrossing the interior of the feasible region.

Problem (3) is posed as a nonlinear programming problem,

min f (y)s.t E(y) = 0,

y ≥ 0(4)

where y = (S1, I1,T1, x0, τ0, . . . ,Sn, In,Tn, xn−1, τn−1), and

f (y) =12

(B1 |̃I|2 + B2|x |2 + B3|τ |2

)(5)







y ≥ 0(4)


f (y) =12

(B1 |̃I|2 + B2|x |2 + B3|τ |2

)(5)







y ≥ 0(4)


f (y) =12

(B1 |̃I|2 + B2|x |2 + B3|τ |2

)(5)


Description of IPM

The Lagrangian associated to (4) is

L(y ,w , z) = f (y) + E(y)T w − yT z.

The perturbed KKT conditions associated with (4) can be written as

F =

∇y L(y ,w , z)E(y)YZ − µe

= 0 (6)

A linesearch Newton’s method is applied to the perturbed KKTconditions (6).


Description of IPM


L(y ,w , z) = f (y) + E(y)T w − yT z.


F =


= 0 (6)



Description of IPM


L(y ,w , z) = f (y) + E(y)T w − yT z.


F =


= 0 (6)



Comparison between FB and IPM

Table : Comparison between Forward-Backward and Interior-Point methods.

FB IPM

Strategy # of iterations F # of iterations F1 52 0.67997 11 0.680692 54 0.33014 10 0.330183 87 0.30423 23 0.30092

For all strategies IPM reach the solution with fewer number of iterations.


Final epidemic size vs. R0

We compare the final epidemic size for different values of R0 undereach strategy.Strategy 3 yields the highest reduction of the final epidemic size.For single policies Strategy 1 has more impact in the reduction of thefinal epidemic size than Strategy 2








Limited Resources

We want to consider the case of limited resources.

An isoperimetric constraint is included

n−1∑t=0

(τi(t)Ii(t)) = k , (7)

where k represents the available number of treatment doses.

It can be written asτ T I = k


Limited Resources



n−1∑t=0

(τi(t)Ii(t)) = k , (7)




Limited Resources



n−1∑t=0

(τi(t)Ii(t)) = k , (7)




Numerical Results, R0 = 1.5

0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time(days)

Con

trol

Effo

rt

k = 53%k = 37%k = 25%k = 10%

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(days)C

um. p

rop.

of i

nfec

. ind

ivid

uals

no controlk = 53%k = 37%k = 25%k = 10%

For small values of k , the optimal control solution requires theimplementation of highest values of treatment at the beginning of theepidemic until the resources are expended.


Multi-Group Model

We explore the role of heterogeneity via a discrete time epidemiologicalmodel involving interacting groups.

The total population is divided into m subgroups according to to thecontact activity or susceptibility levels.

The fraction of susceptible individuals on group i at time t that getinfected at time t + 1 is given by

Gi = ρi

m∑j=1

(qj(1− xj(t))

(Ij(t) + εjTj(t)

Nj

)), (8)

where qi (proportionate mixing) is given by

qij = qj =CjNj

m∑k=1

Ck Nk

. (9)


Multi-Group Model




Gi = ρi

m∑j=1

(qj(1− xj(t))

(Ij(t) + εjTj(t)

Nj

)), (8)


qij = qj =CjNj

m∑k=1

Ck Nk

. (9)


Multi-Group Model




Gi = ρi

m∑j=1

(qj(1− xj(t))

(Ij(t) + εjTj(t)

Nj

)), (8)


qij = qj =CjNj

m∑k=1

Ck Nk

. (9)



The model is given by the following system of difference equations:

Si(t + 1) = Si(t)(1− Gi(t))Ii(t + 1) = Si(t)Gi(t) + (1− τi(t)) (1− σi) Ii(t)Ti(t + 1) = (1− σ)Ti(t) + τi(t) (1− σi) Ii(t)Ri(t + 1) = Ri(t) + σi Ii(t) + σTi(t).

(10)

If n denotes the final time, the optimal control problem is formulated as

min12

n−1∑t=0

m∑i=1

(BIi Ii(t)

2 + Bxi xi(t)2 ++Bτi τi(t)2) (11)

s.t. Model (10).



The model is given by the following system of difference equations:

Si(t + 1) = Si(t)(1− Gi(t))Ii(t + 1) = Si(t)Gi(t) + (1− τi(t)) (1− σi) Ii(t)Ti(t + 1) = (1− σ)Ti(t) + τi(t) (1− σi) Ii(t)Ri(t + 1) = Ri(t) + σi Ii(t) + σTi(t).

(10)

If n denotes the final time, the optimal control problem is formulated as

min12

n−1∑t=0

m∑i=1

(BIi Ii(t)

2 + Bxi xi(t)2 ++Bτi τi(t)2) (11)

s.t. Model (10).


Seasonal Influenza

According to the USA Census (27%) of the population is in Group 1,(80%) in Group 2, and (13%) in Group 3.

The biggest effort has to be applied both in the highest activity levelGroup 1 and the larger population size Group 2.

The implementation of policies reduce the final epidemic size by 13%,14%, and 11% in Groups 1, 2 and 3 respectively.


Seasonal Influenza

According to the USA Census (27%) of the population is in Group 1,(80%) in Group 2, and (13%) in Group 3.The biggest effort has to be applied both in the highest activity levelGroup 1 and the larger population size Group 2.The implementation of policies reduce the final epidemic size by 13%,14%, and 11% in Groups 1, 2 and 3 respectively.

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

Treatment in Group 1

0 50 100 150 200 250

1

2

3

4x 10

−3

Soc. Distancing in Group 1

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

without controlboth policies

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

Con

trol

Effo

rt


0 50 100 150 200 250

1

2

3

4

5x 10

−3

Con

trol

Effo

rt


0 50 100 150 200 2500

0.1

0.2

0.3

Cum

. pro

p. o

f inf

ec. i

ndiv

idua

ls


0 50 100 150 200 2501

2

3

4

5x 10

−3

Time(days)


0 50 100 150 200 250

1.1

1.2

1.3

1.4

1.5x 10

−3

Time(days)


0 50 100 150 200 2500

0.2

0.4

0.6

Time(days)



Influenza H1N1

The biggest effort both in treatment and social distancing has to beimplemented in the highest risk Group 2.

The final epidemic size is reduced by 7%, 6% and 8% in Groups 1, 2and 3 respectively.


Influenza H1N1




Influenza H1N1



0 50 100 150 2000

0.002

0.004

0.006

0.008

0.01


0 50 100 150 2000

0.5

1

1.5

2x 10

−3


0 50 100 150 2000

0.1

0.2

0.3

0.4


0 50 100 150 2000

0.005

0.01

0.015

0.02

Con

trol

Effo

rt


0 50 100 150 2000

1

2

3

4

5x 10

−3

Con

trol

Effo

rt


0 50 100 150 2000

0.2

0.4

0.6

Cum

. pro

p. o

f inf

ec. i

ndiv

idua

ls


0 50 100 150 2000

1

2

3

4

5x 10

−3

Time(days)


0 50 100 150 200

2.4

2.5

2.6

2.7

2.8x 10

−4

Time(days)


0 50 100 150 2000

0.05

0.1

0.15

0.2

Time(days)



Seasonal and Pandemic Influenza

The figure shows how treatment doses should be distributed in the caseof small and moderate value of R0.







Group 1 Group 2 Group 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Doses of Treatment on Each Group

Tre

atm

ent D

oses

H1N1 small RoH1N1 moderate Roseasonal small Roseasonal moderate Ro


Conclusions

An optimal control problem is solved by using two techniques:Forward-Backward algorithm and interior-point methods.

Interior-point methods reach the solution with fewer number of iterations.

IPM allows to incorporate inequality constraints and the isoperimetricconstraint in a more efficient way.

The use of single and dual strategies results in reductions in thecumulative number of infected individuals.

In the case of limited resources, the maximum effort in control have tobe implemented at the beginning of the epidemic.


Conclusions







Conclusions







Conclusions







Conclusions







Related Publications

P. Gonzalez-Parra, S. Lee, L. Velazquez, and C. Castillo-Chavez (2011)

A note on the use of optimal control on a discrete time model of influenza dynamics.

Math. Biosc. & Eng. 8, 183–197.

E. Fenichel, C. Castillo-Chavez, G. Ceddia, G. Chowell, P. Gonzalez-Parra, L. Velazquezet al. (2011)

Adaptive human behavior in epidemiological models.

Proceedings of the National Academy of Sciences, 108(15).

P. Gonzalez-Parra, L. Velazquez, M. Villalobos, and C. Castillo-Chavez (2010)

Optimal control applied to a discrete influenza model.

Conference Proceedings Book of the XXXVI International Operation Research Appliedto Health Services, edited by Franco Angeli Edition





Math. Biosc. & Eng. 8, 183–197.











Math. Biosc. & Eng. 8, 183–197.








Acknowledgments

Thanks to:

Special thanks to CAHSI.

CPS and College of Sicence at the University of Texas at El Paso.

American Asociation for University Women.

Carlos Castillo-Chavez and MTBI summer program.

Universidad Autonoma de Occidente, Cali–Colombia.

Thank you!


Documents

Optimal Control for a Discrete Time Inﬂuenza Modelutminers.utep.edu/pagonzalezparra/presentations/PresentationSACNAS.pdf2012 SACNAS National Conference Seattle, WA October 12, 2012