Paula A. González-Parra Compuaional Science Program, Universiy of Texas a El Paso
Ouline Inroducion Discree Epidemiological model Conrol problem Sraegies Numerical resuls Conclusions hp://healh.uah.gov/epi/diseases/flu/graphics/influenza_germ.jpg Opimal Conrol on a Influenza Model 2
Inroducion In April of 2009 he World Healh Organizaion (WHO) announced he emergence of a novel srain of A-H1N1 influenza. In June of 2009 WHO declared he oubreak o be a pandemic. Differen coninuous ime approaches have been used o sudy single influenza oubreaks. hp://i.elegraph.co.uk/elegraph/mulimedia/archi 395/swine-flu-reamen_1395505c.jpg The idenificaion of opimal conrol sraegies ha involve aniviral reamen and isolaion have also been sudied in he coninuous case. We inroduce a opimal conrol problem in order o minimize he number of infeced and dead individuals via he use of he mos ``cos-effecive" policies involving social disancing and aniviral reamen using a discree ime epidemic framework. hp://clovewo.com/archives/2009 /8/23 /healhnfiness/sf_05disance.jpg Opimal Conrol on a Influenza Model 3
Discree SAIR model The model is given by he sysem of difference equaions: S 1 1 1 1 A qs (1 G ) 1 A I (1 q) S 1 G 1 1 I 1 1 1 1 1 S G R R I D D I (1) where I G e m A N 9/28/2010 Opimal Conrol on a Influenza Model Paula A. Gonzalez-Parra 4
Discree SAITR model The model is given by he sysem of difference equaions: S 1 S G (1 ) 1 (1 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 A qs G A I q S G I T T I R R I T D D I where (2) S qs (1 G ) (1 q) S (1 G ) A I 1 2 1 R G e I m A T (1 x ) N D T Opimal Conrol on a Influenza Model 5
Final Epidemic Size In he absence of conrols we ge he final size relaion S0 S ln R0 1 S N (3) wih he basic reproducive number given by R 0 1 q q m 1 111 1. Opimal Conrol on a Influenza Model 6
The addiion of conrols replace (3) by he inequaliy l n S 0 R0 S 1 S N (4) S wc Resul : If is a soluion of (2) and S saisfies he inequaliy (3) hen wc S S 0 ( ) ln S x f x R0 1 x N 9/28/2010 Opimal Conrol on a Influenza Model Paula A. Gonzalez-Parra 7
Opimal Conrol Problem The goal is o minimize he number of infeced and dead individuals wih he judicious (cos effecive) use of social disancing and aniviral reamen measures over a finie inerval [ 0, T f ]. We compare hree differen Sraegies: Sraegy 1 : Only Social Disancing, minimize 1 2 T f 1 0 B I B D B x 2 2 2 0 1 2 subjec o model (1) and 0 x x max S 1 1 1 1 A qs (1 G ) 1 A I (1 q) S 1 G 1 1 I 1 1 1 1 1 S G R R I D D I Opimal Conrol on a Influenza Model 8
Sraegy 2 : Only Aniviral Treamen, minimize 1 2 T f 1 0 B I B D B 2 2 2 0 1 3 subjec o model (2), and 0 max S 1 S G (1 ) 1 (1 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 A qs G A I q S G I T T I R R I T D D I Sraegy 3 : Dual of Social Disancing and Treamen. T f 1 1 2 2 2 2 minimize B 2 0I B1 D B2 x B3 (5) 0 subjec o model (2), 0 and 0 x x max max 9/28/2010 Opimal Conrol on a Influenza Model Paula A. Gonzalez-Parra 9
Numerical Soluion The discree Hamilonian associaed wih problem (5) is given by: 2 2 2 2 T 0 1 2 3 λ 1y 1 H B I B D B x B Where y (,,,,, ) T S A I T R D and λ 6 1 are he sae and adjoin variables. The adjoin equaions are defined as i H, i 1,2,...,6, i y and he opimaliy condiions are: H x H 0, 0. (6) (7) Opimal Conrol on a Influenza Model 10
Adjoin Equaions For Sraegy 3: The adjoin equaions associaed wih problem (5) are: 1 1 1 1 2 3 G 1 q 1 q 1 G 2 m 2 5 SG 1 x L 1 11 1 1 1 N 1 1 1 1 N 4 4 5 SG 1 x L 1 1 2 1 2 1 N 3 3 4 5 6 B0 I SG x L 1 1 1 1 1 1 1 5 5 1 6 6 B1 D 1 1 2 3 1 1 1 1 where q +1 q L Opimal Conrol on a Influenza Model 11
Opimaliy Condiions The opimaliy condiions are: H H x 4 5 1 I 1 B 0 1 1 1 3 1 2 3 2x SG a 1 q 1 1 q 1 B 0 where I ( 1 x ) m A T G e N, a I m A T N. Opimal Conrol on a Influenza Model 12
Algorihm: (Forward-Backward mehod) Sep 1. The iniial guess x, τ and condiion y 0 is seleced. Sep 2. Solve he sae equaions (1 or 2) forward in ime. Sep 3. Solve he adjoin equaions (6) backward in ime wih he ransversaliy condiions, T f 0 Sep 4. Solve he opimaliy condiion (7). i u u old Sep 5. Check convergence. If sop; else go o Sep 2. u 0.001, Opimal Conrol on a Influenza Model 13
Numerical Resuls We presen he resuls of seleced simulaions generaed by he numerical implemenaion of Sraegy 1, 2 and 3. We compare he number of infeced individuals generaed by low R 0 (1.3-1.8) or high R 0 (2.4-3.2) wih no conrols or in he presence of single or dual opimal conrols. A sensiiviy analysis is also carried ou by sudying he robusness of our simulaions in relaionship o he weigh consans and he upper bounds of conrols. Opimal Conrol on a Influenza Model 14
Implicaion of social disancing and aniviral reamen (moderae R 0 ) Figure 1: For Ro = 1.3, The opimal conrol soluion does no required he applicaion of he permied maximum values. However, here is a srong impac in he reducion of he final epidemic size by he applicaion of each sraegy. Sraegy 3 has he mos significan reducion, almos 32% Opimal Conrol on a Influenza Model 15
Implicaion of social disancing and aniviral reamen (high R 0 ) Figure 2: For Ro = 2.4, he opimal soluion requires he implemenaion of he highes permied values for each conrol. Sraegy 3 produces a reducion of less han 22%. Even when here is a maximum conrol implemenaion, he effor is no enough and he reducion in he final epidemic size is less significan Opimal Conrol on a Influenza Model 16
Comparison of final epidemic size vs. R 0 Figure 3: By fixing he weighs B2 and B3 of each conrol funcion, he resuls show ha Sraegy 3 yields he highes reducion of he final epidemic size (more han 31% for small values of R 0 ). Opimal Conrol on a Influenza Model 17
Effec of weigh consans Figure 4: In Sraegy 1, he value of he weigh consan B2 is varied. For a small value of B2 he opimal soluion permis he implemenaion of high values of social disancing and we obain a high reducion in he final epidemic size (20%). For a large value of B2 hereducion in he final epidemic size is no significan (7.5%). Opimal Conrol on a Influenza Model 18
For Sraegy 2: Figure 5: by increasing he cos on aniviral reamen, B3, he opimal soluion permis he applicaion of smaller value for reamen. We obain an increase in he cumulaive proporion of infeced cases. In conras, when he cos is moderae he opimal soluion permis he implemenaion of high values of reamen and we ge a srong impac in he reducion of he final epidemic size 30%. Opimal Conrol on a Influenza Model 19
Final Epidemic size for Sraegies 1 and 2: Figure 6: The final epidemic size vs. R 0 for Sraegies 1 and 2 by changing he weigh consans for social disancing and aniviral reamen. When we have a moderae cos of social disancing, here is a reducion in he final epidemic size for every value of R 0 for (A); however, by changing B 3 here is no a significan difference in he final epidemic size. R 0 > 2.5 (B). Opimal Conrol on a Influenza Model 20
The effec of upper bound on he opimal conrol Figure 7: When he resources are limied and he upper bound is smaller x max = 0.07, he reducion of he final epidemic size is small (5%).However, if he upper bound is high here is a sronger impac in hereducion of hefinal epidemic size, (20%). Opimal Conrol on a Influenza Model 21
Upper bound for Sraegy 2 Figure 8: The impac of he conrol in Sraegy 2 is reduced when he resources are limied. For a small upper bound, τ max = 0.007, we ge a small reducion of he final epidemic size 5%. When he upper bound is τ max = 0.02, he final epidemic size is reduced by 13%. Opimal Conrol on a Influenza Model 22
Conclusions o The use of single and dual sraegies (social disancing and aniviral reamen) resuls in he reducion on he cumulaive number of infeced individuals. o We compare he impac of relaive coss on he effor carried ou in he implemenaion of each single sraegy (weigh consans on conrols) and also he use of limied resources (conrol upper bounds). o Dual sraegies have sronger impac in erms of he reducion in he final epidemic size, bu i is more cos-expensive. o Fuure work: we wan o include a ime delay in he applicaion of policies Opimal Conrol on a Influenza Model 23
Acknowledgmens This projec has been suppored by grans from he Naional Science Foundaion (NSF - Gran DMPS-0838705), he Naional Securiy Agency (NSA - Gran H98230-09-1-0104), he Alfred P. Sloan Foundaion; and he Presiden and Provos Offices a Arizona Sae Universiy. The Mahemaical and Theoreical Biology Insiue now hosed a he Mahemaical, Compuaional and Modeling Science Cener a ASU would like o give hanks o everybody involve wih he program for he pas 15 years. Special hanks o Sunmi Lee, Aprillya Lanz and Carlos Casillo Chavez for heir suppor and conribuions. Opimal Conrol on a Influenza Model 24
Thank you! 9/28/2010 Opimal Conrol on a Influenza Model Paula A. Gonzalez-Parra 25