Least Squares Parameter Es.ma.on

Transcription

1 Least Squares Parameter Es.ma.on Alun L. Lloyd Department of Mathema.cs Biomathema.cs Graduate Program North Carolina State University

2 Aims of this Lecture 1. Model fifng using least squares 2. Quan.fica.on of uncertainty in parameter es.mates: asympto.c sta.s.cal theory 3. Sensi.vity analysis (local linear analysis)

3 Introduc.on SIR model: Mathema.cal model for the spread of an infec.on through a popula.on by person- to- person contact This mathema.cal model describes the.me evolu.on of the state variables S, I and R, using a set of differen.al equa.ons, whose terms describe the rates at which various biological processes (infec.on and recovery) happen ds/ = βsi/n S infection I recovery R di/ = βsi/n γ I dr/ = γ I. Parameters: constants that appear in the func.ons describing these rates

4 The forward problem S infection I recovery R ds/ = βsi/n di/ = βsi/n γ I dr/ = γ I. If we know the model equa.ons, the values of the parameters and the ini.al values of the states, we can (numerically) solve the model forwards in.me to find out how the state variables will evolve Forward simula.on of S and I, given β, γ, N, S() and I() : Forward problem: Parameters dynamical behavior

5 The inverse problem (parameter es.ma.on) In many cases, we don t know the values of the parameters, but we have observa.ons of the system (e.g..me series data) Confined to bed Convalescent.-----o 25 U, w 2 a, 15 n E Z January 1. '26 ' 28' - l '3 - Februo3ry Can we figure out parameter values that would make the model exhibit behavior that is consistent with observed data? Inverse problem: dynamical behavior parameters

6 Least- Squares FiFng One approach to find unknown parameter values: fit a model to.me series data trajectory matching : find parameter values for which the model provides the best fit to a data set What do we mean by best fit? Least squares criterion: (ordinary least squares) Minimize sum of squared differences between model predic9ons and observed data - sum of squared errors hopefully familiar from simple linear regression n ( ( ) ) 2 F(θ) = y model predicted ( t i ;θ) y observed t i i=1 Here, θ is the vector of parameters

7 Least- Squares FiFng y Least squares criterion: minimize sum of squared differences between model predic.ons and observed data (sum of squared errors) y Model Predicted (t 1 ; θ) e 1 e 2 (t 1,y 1 observed) (t 2,y 2 observed) e 3 (t 3,y 3 observed) n F(θ) = y model predicted ( t i ;θ) y observed t i i=1 ( ( ) ) 2 Dots: data points (observa.ons) Red crosses: model predic.ons (for some value of the parameter vector) t e 4 (t 4,y 4 observed) Green lines indicate (model- data): called errors or residuals [Why do we use sum of squares? 1. Cannot simply add errors (cancela.on of + and - errors) 2. Sum of squared errors is beeer behaved than summed absolute errors.]

8 Minimiza.on problem Least- Squares FiFng ˆ θ = # % argmin { θ feasible} % & i=1 n ' ( y model predicted ( t i ;θ ) y observed ( t i ) ) 2 % ( % ) 1. Cannot solve this analy.cally in general (linear regression is an excep.on), so will probably need to do op.miza.on numerically Process can be non- trivial, e.g. if the func.on F(θ) has mul.ple local minima 2. If our model is an ODE, we probably don t have a formula for the model s output, so will have to numerically solve the ODE to make predic.ons If the ini.al condi.on of our ODE is unknown/uncertain, we can include it in the list of quan..es to be es.mated it 3. Es.mates depend on the data: would get a different es.mate if data was different Mathema.cally: the best- fifng parameter vector is a random variable, and its es.ma.on is a sta.s.cal process What are the uncertain.es in the es.mates we obtain? standard errors for parameter es.mates?

9 More Generally... State Space Nota.on Model: dx = f(x, t; ), where x and f are m dimensional vectors, and θ is a p dimensional vector of parameters Observa.on func.on (maps model to observable quan.ty): Ordinary least squares parameter es.mate is ˆ θ = y(t) =h(x, t; ) # % argmin { θ feasible} % & i=1 n ' ( y model predicted ( t i ;θ) y observed ( t i ) ) 2 % ( % )

10 Uncertainty Es.mates: Intui.on Nonlinear regression theory can be used to provide uncertainty es.mates Intui.on: consider two graphs of error sum of squares, based on two different models.8.7 Error Sum of Squares Parameter value Which gives us more certainty in the parameter es.mate? For 2 there is a larger range of parameter values that gives a reasonable fit In some sense, there is less informa.on about the true value of the parameter Intui.on: the curvature (2 nd deriva.ve) of the SS func.on at its minimum is an inverse measure of the informa.on Small second deriva.ve: less informa.on/more uncertainty in value Idea is formalized in the no.on of Fisher informa.on matrix (coming soon)

11 Uncertainty Es.mates: Intui.on.8.7 A couple of comments: Error Sum of Squares Parameter value By the chain rule, deriva.ves of the sum of squares func.on F ( ) = X i y predicted (t i ; ) y observed (t i ) 2 with respect to the parameters will involve deriva.ves of model predic.ons with respect to the parameters (we call these sensi.vi.es) It will be difficult to es9mate insensi9ve parameters 2. Model 2 appears to be a beeer fit (smaller value for error sum of squares) Is there a meaningful no.on of how improved the fit is?

12 Uncertainty Es.mates We need a sta.s.cal model for our observa.ons describes the origin of the errors in the data Simplest sta.s.cal model: observa.ons are the values predicted by the model under the true parameter values plus observa.on error y observed ( t i ) = y model predicted ( t i ;θ true ) + e i e i are the observa.on errors In the simplest sefng, we assume errors are independent, iden.cally distributed and have constant variance, σ 2 note: following theory is exact if errors are normal, but we don t need to assume errors are normal if we appeal to large sample size theory (asympto.c theory) can also view the following as providing a bound on uncertainty

13 We ask the ques.on: Uncertainty Es.mates Suppose we could watch the system evolve over.me on mul.ple occasions, giving us a collec.on of data sets [because we have different realiza.ons of the noise process], and es.mated the parameter values for each of those data sets... How much varia.on would we see in the parameter es.mates? Could characterize this varia.on by the variance in the values of the es.mates Could also examine the correla.on between es.mates of different parameters, captured by the covariance between their values All this informa.on is summarized in the covariance (variance/covariance) matrix, Σ Var( ˆ) cov(ˆ, ˆ) For SIR example: = cov( ˆ, ˆ) Var(ˆ)

14 Uncertainty Es.mates for Parameters Asympto.c (large sample size) theory says that the parameter es.mator (a random variable) has a mul.variate normal distribu.on, centered on the true parameter vector and with variance- covariance matrix = 2 ( ) T ( ) Here χ is the n x p matrix of sensi.vi.es, with entries (discuss calcula.on of sensi.vi.es soon!) # of data points # of parameters es.mated ( )= 1 ( ) ij (t i ; 1 ; 1 ; 2 ; 2 ; n ; n ; 1 ; 2 ; n ; p 1 C A Problem: we don t know θ or σ 2 (true values), so we make use of our es.mates

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sha1_base64="1bewaplawhwrmtko43hwjhpmdjo=">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</latexit> <latexit sha1_base64="1bewaplawhwrmtko43hwjhpmdjo=">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</latexit> Uncertainty Es.mates for Parameters Asympto.c (large sample size) theory says that the parameter variance- covariance matrix for the es.mated parameters is Here, ˆ2 (n) (ˆ ) ˆ =ˆ2 (n) (ˆ ) T 1 (n) (ˆ ) is our least squares es.mate of the parameters is (minimized value of error sum of squares) / (n p) n data points (13 for our SIR example), p es.mated parameters (2 for SIR) is the n x p matrix of sensi.vi.es, with entries (n) (ˆ ) 1 ; ˆ 2 ; ˆ n ; ˆ 1 Unbiased es.mate (n) ( ) ij (t i ; ˆ 1 ; ˆ 2 ; ˆ n ; ˆ 1 ; ˆ 2 ; ˆ n ; ˆ p 1 C A

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sha1_base64="zouzdqhsmxhww5fzr8opevm1do=">aaacd3icbzdlsgmxfibpek31vnxpjlgu3zqzkejoghvdvbbv6jryjs3uydizjbmhdhdn76kgxekuhxrzrfwecxmc15/chz855wk5w8swbvx3xdnanpmdm6+tfbexfpewa2srbd1ncrkwjqwsbomudpbi9yy3ah2msigmhdsirg+zusxnxphkfnzpiwrsrbxenovirv9nxq4u8xnuhqmgp6mv9gcojy52zv65v6m6nbcq+qvebkphh1co2au8+f2yppjfhgruuuo5ielm+c1usfhztzvlkf7jghusriiz7mbfpioybz+cwnlt2ri4x6fyfbqpzsb7zrorvtvwm7+v+ukjjzszjxkusmion4otauxmcndix2ugdviaagp4vavhf6hdcjycpmqvn8r/4x2fs1za95zvdqoj9oaemzcfuycbwfqgbnoqgso3mi9pmktc+c8om/oy7h1ypnmbmapoa+fnjodeq==</latexit> Sensi.vi.es (Local) Sensi.vity: par.al deriva.ve of some quan.ty of interest with respect to a parameter In our sefng, we will want to calculate sensi.vi.es of the state variables with respect to e.g. if y(t) = I(t), we would need to calculate (t) (t) How do we calculate these? If we had a formula for I(t), this would be easy... but we don t Instead we have to use the sensi.vity equa.ons

18 <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="q9i+jduzdzx6ehac4s/n4mixwq=">aaacgnicjvhlsgmxfm2mvwt9jbpeywfuagzwtcfqsgnywr2az1smplmg5p5knwryzaf4m+582sfac1fffc4hduusfjuv4iualbfjfmtcl6xmzxq7s9s7u3bxctlscssqanbax7hhemcej1gqognusyujocdb2rvetfvuzscxj6angceufzbdxgfmcmupbr24gcc38ppmhn2e3iri4efgl/8yudbmqps/vlkrbvjjwhwd8jv/w+jl1rbjaefv4ezb2urbfenp9mkyhi4akoltxsrpozrnfklheclpfekjhzmc6gkykzkqxtspmcuuzpg5iqu8eemoutmqkvgocelozehiq5d6e/k3xtsg46wu8sljgez1dfkqcq4wn+8a+l4ycggtaqot6rzgoiy4h9nzkogrn+curohvzdeyq81gr12vzoirogj2gu+sga1rhd6ibmoiid6nivils2cemy55nzoaxnzmcpo8/yt8i7hnq==</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="jp+t/fu8tyxv/tbdfqyyndzxru=">aaacdxicbvdltgixfl2dl8qx6tjni5rabmcmie4kcam7tosracf3soggdmbsdkzihb9w46+4caexbt278y/8bgearavputnhtve48bck6nbx9aqaxlldw19hpmy3nreye7u1ftfqgoq1jf+krhomace6xqubgsesighws7g4ve79+x5tmvndrrgfrs+x7vmcpmljqzi/yustaztgkcj++bm+igljgt5u8hsjm7ae9afokzi7mll5igsl+tlo+dsxzdbwoddoxa9ooktfushgmfwowib1inzvj6qfkuh1nthmt41jpkp6v4umzmlf/dqotr5jn66uaaz63kve/7xmahrn7yh7qwiyr6cp9ujbje+saeixkangmueqelxxwkdoejq4gctejz5lrdj7bto2exnppqrl6zpqboo4bdy4mazloekklafcvfwcm/wyj1yt9ar9tyttvmznn34a+v9g4vnm+=</latexit> sha1_base64="xgz29cpusm9ge8gpreseaq48rvu=">aaacdxicbvdlssnafj3uv62vqes3g1vonzwrgi4lbnrxwt6gcevmommhzirhzikub9w46+4cagiw/fu/butnqc2hhg4c869d+yel+jmacv6mgorq2vrg8xntb2zu6eux/qvmescw2rkiey64ginawpznmtbtjcsljtoonrzk/c+lymfwpycrdquma+yzajqv+uzjjxeikjobxzfts5+lmwqhyfqt6gqpb5atmjudxiz2tsoor7nvfjqdkmscbppwukpnw5f2k2w4xracmjfiybjgnjesgmqvlnjbjsppk2vafzdmz5a45n6uymbodreegmlad1si14m/uf1yu1fugkloljtgmwf8moodyizapcasuon6qeigtpxzezgqsiwczeozflzdj+7xmwzx7tl5u1pm4iugihamkstefaqbr1eqtrnadekiv6nv4nj6nn+n9xlow8p5d9afgxzfqo5q+</latexit> Sensi.vity Equa.ons Sensi.vi.es: par.al derivs of state variables with respect to params, e.g. )(t) For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters We can find the vector of sensi.vi.es of the states with respect to parameter θ i by differen.a.ng both sides of (1) with respect to the parameter, applying the chain rule, and switching the order of d/ and d/dθ i on the lem side: i with ini.al i (t) <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit i () = m <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="whrjnp69ctg2omuzoypfhupkic=">aaacghicbzdlssnafiynxmu9rv26gsxc3drecrorcm5cvraxaeqytcftmmfmroxhdygg1/fjqtf3hbn2zhpa2rrdwmf/zmhm+f3ysevwnaxsbk6tr6xwdoqb+/s7u2bb4fswsshanrcs7hlfm8jc1ging3vgyenicdbzxtv7vpdcpebtewyrm/yamq+5zskbbrnnu+jlq1imjbe4efsx+2iera+lyrgqd4wtsuwmquwbfqlkz4wwwc6igqk3xndqdicybc4ekoltptmlop/kkklhwdhlfyklhzmh6gkmsmnvpz4dl+fq7a+xhur8q8mz9pzgsqklj4onogmbildzy879alwh/qp/yme6ahxs+ye8ehgjnkeebl4ycmgggvhl9vxhrccfosuydsfephkz2hc126rzd/vko17euulh6arvkyuuqpdoizqiyqeat6q+/gs/fqfbif89yvo5g5qn9ktl8bs3qfq==</latexit> (2) : sensi9vity equa9ons (provided θ i is a pure parameter, not an ini.al condi.on- - - see later)

19 <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="q9i+jduzdzx6ehac4s/n4mixwq=">aaacgnicjvhlsgmxfm2mvwt9jbpeywfuagzwtcfqsgnywr2az1smplmg5p5knwryzaf4m+582sfac1fffc4hduusfjuv4iualbfjfmtcl6xmzxq7s9s7u3bxctlscssqanbax7hhemcej1gqognusyujocdb2rvetfvuzscxj6angceufzbdxgfmcmupbr24gcc38ppmhn2e3iri4efgl/8yudbmqps/vlkrbvjjwhwd8jv/w+jl1rbjaefv4ezb2urbfenp9mkyhi4akoltxsrpozrnfklheclpfekjhzmc6gkykzkqxtspmcuuzpg5iqu8eemoutmqkvgocelozehiq5d6e/k3xtsg46wu8sljgez1dfkqcq4wn+8a+l4ycggtaqot6rzgoiy4h9nzkogrn+curohvzdeyq81gr12vzoirogj2gu+sga1rhd6ibmoiid6nivils2cemy55nzoaxnzmcpo8/yt8i7hnq==</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="jp+t/fu8tyxv/tbdfqyyndzxru=">aaacdxicbvdltgixfl2dl8qx6tjni5rabmcmie4kcam7tosracf3soggdmbsdkzihb9w46+4caexbt278y/8bgearavputnhtve48bck6nbx9aqaxlldw19hpmy3nreye7u1ftfqgoq1jf+krhomace6xqubgsesighws7g4ve79+x5tmvndrrgfrs+x7vmcpmljqzi/yustaztgkcj++bm+igljgt5u8hsjm7ae9afokzi7mll5igsl+tlo+dsxzdbwoddoxa9ooktfushgmfwowib1inzvj6qfkuh1nthmt41jpkp6v4umzmlf/dqotr5jn66uaaz63kve/7xmahrn7yh7qwiyr6cp9ujbje+saeixkangmueqelxxwkdoejq4gctejz5lrdj7bto2exnppqrl6zpqboo4bdy4mazloekklafcvfwcm/wyj1yt9ar9tyttvmznn34a+v9g4vnm+=</latexit> sha1_base64="xgz29cpusm9ge8gpreseaq48rvu=">aaacdxicbvdlssnafj3uv62vqes3g1vonzwrgi4lbnrxwt6gcevmommhzirhzikub9w46+4cagiw/fu/butnqc2hhg4c869d+yel+jmacv6mgorq2vrg8xntb2zu6eux/qvmescw2rkiey64ginawpznmtbtjcsljtoonrzk/c+lymfwpycrdquma+yzajqv+uzjjxeikjobxzfts5+lmwqhyfqt6gqpb5atmjudxiz2tsoor7nvfjqdkmscbppwukpnw5f2k2w4xracmjfiybjgnjesgmqvlnjbjsppk2vafzdmz5a45n6uymbodreegmlad1si14m/uf1yu1fugkloljtgmwf8moodyizapcasuon6qeigtpxzezgqsiwczeozflzdj+7xmwzx7tl5u1pm4iugihamkstefaqbr1eqtrnadekiv6nv4nj6nn+n9xlow8p5d9afgxzfqo5q+</latexit> Sensi.vity Equa.ons Sensi.vi.es: par.al derivs of state variables with respect to params, e.g. )(t) For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters We can find the vector of sensi.vi.es of the states with respect to parameter θ i by differen.a.ng both sides of (1) with respect to the parameter, applying the chain rule, and switching the order of d/ and d/dθ i on the lem side: i with ini.al i (t) <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit i () = m <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="whrjnp69ctg2omuzoypfhupkic=">aaacghicbzdlssnafiynxmu9rv26gsxc3drecrorcm5cvraxaeqytcftmmfmroxhdygg1/fjqtf3hbn2zhpa2rrdwmf/zmhm+f3ysevwnaxsbk6tr6xwdoqb+/s7u2bb4fswsshanrcs7hlfm8jc1ging3vgyenicdbzxtv7vpdcpebtewyrm/yamq+5zskbbrnnu+jlq1imjbe4efsx+2iera+lyrgqd4wtsuwmquwbfqlkz4wwwc6igqk3xndqdicybc4ekoltptmlop/kkklhwdhlfyklhzmh6gkmsmnvpz4dl+fq7a+xhur8q8mz9pzgsqklj4onogmbildzy879alwh/qp/yme6ahxs+ye8ehgjnkeebl4ycmgggvhl9vxhrccfosuydsfephkz2hc126rzd/vko17euulh6arvkyuuqpdoizqiyqeat6q+/gs/fqfbif89yvo5g5qn9ktl8bs3qfq==</latexit> (2) : sensi9vity equa9ons Why do we need the chain rule? Because solu.on x(t) depends on (provided θ i is a pure parameter, not an ini.al condi.on- - - parameter see later) θ i

20 <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="q9i+jduzdzx6ehac4s/n4mixwq=">aaacgnicjvhlsgmxfm2mvwt9jbpeywfuagzwtcfqsgnywr2az1smplmg5p5knwryzaf4m+582sfac1fffc4hduusfjuv4iualbfjfmtcl6xmzxq7s9s7u3bxctlscssqanbax7hhemcej1gqognusyujocdb2rvetfvuzscxj6angceufzbdxgfmcmupbr24gcc38ppmhn2e3iri4efgl/8yudbmqps/vlkrbvjjwhwd8jv/w+jl1rbjaefv4ezb2urbfenp9mkyhi4akoltxsrpozrnfklheclpfekjhzmc6gkykzkqxtspmcuuzpg5iqu8eemoutmqkvgocelozehiq5d6e/k3xtsg46wu8sljgez1dfkqcq4wn+8a+l4ycggtaqot6rzgoiy4h9nzkogrn+curohvzdeyq81gr12vzoirogj2gu+sga1rhd6ibmoiid6nivils2cemy55nzoaxnzmcpo8/yt8i7hnq==</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="fqh6uyux6yh3g7+i79cws8gpoqu=">aaacgnicjvhlsgmxfm2mvwt9vv26czacknqzlehcsedgzqx7ge4pmuymdc8so6izzgp8bfc+rf+gayptlyixggczj33jdnxjqvxyflvhrmsw11bz28unre2d3ale/tnfswssganrctbllfm8ja1ging7vgyerictdzh3bjfemjs8sh8hfhmughph9znlicmesuxx5eepl6wepbnsrmtczwi/jx9ywcgdeipzzclij+bh/ipaz7ff3h8yxrfklwxjowxgtdp3nkl6r/jqebfnahycfuspjm3fe3hvlswroakiswedkmfdtqmscbun51emogyzjzsr1kfepcenz9isadukhc1miawuiu9mflbr5oaf9vnergnwei6vchpbiyij/ebps4zbthsgfdj9vsxhradd+itfxqi9ukxlhzvgjbffuhwqpvp2mgpdper+gy2egs1da9qqmgoujdkbsv48zmmsembv5mpayxmzlap8q8/gsnqmhk</latexit> <latexit sha1_base64="jp+t/fu8tyxv/tbdfqyyndzxru=">aaacdxicbvdltgixfl2dl8qx6tjni5rabmcmie4kcam7tosracf3soggdmbsdkzihb9w46+4caexbt278y/8bgearavputnhtve48bck6nbx9aqaxlldw19hpmy3nreye7u1ftfqgoq1jf+krhomace6xqubgsesighws7g4ve79+x5tmvndrrgfrs+x7vmcpmljqzi/yustaztgkcj++bm+igljgt5u8hsjm7ae9afokzi7mll5igsl+tlo+dsxzdbwoddoxa9ooktfushgmfwowib1inzvj6qfkuh1nthmt41jpkp6v4umzmlf/dqotr5jn66uaaz63kve/7xmahrn7yh7qwiyr6cp9ujbje+saeixkangmueqelxxwkdoejq4gctejz5lrdj7bto2exnppqrl6zpqboo4bdy4mazloekklafcvfwcm/wyj1yt9ar9tyttvmznn34a+v9g4vnm+=</latexit> sha1_base64="xgz29cpusm9ge8gpreseaq48rvu=">aaacdxicbvdlssnafj3uv62vqes3g1vonzwrgi4lbnrxwt6gcevmommhzirhzikub9w46+4cagiw/fu/butnqc2hhg4c869d+yel+jmacv6mgorq2vrg8xntb2zu6eux/qvmescw2rkiey64ginawpznmtbtjcsljtoonrzk/c+lymfwpycrdquma+yzajqv+uzjjxeikjobxzfts5+lmwqhyfqt6gqpb5atmjudxiz2tsoor7nvfjqdkmscbppwukpnw5f2k2w4xracmjfiybjgnjesgmqvlnjbjsppk2vafzdmz5a45n6uymbodreegmlad1si14m/uf1yu1fugkloljtgmwf8moodyizapcasuon6qeigtpxzezgqsiwczeozflzdj+7xmwzx7tl5u1pm4iugihamkstefaqbr1eqtrnadekiv6nv4nj6nn+n9xlow8p5d9afgxzfqo5q+</latexit> Sensi.vity Equa.ons Sensi.vi.es: par.al derivs of state variables with respect to params, e.g. )(t) For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters We can find the vector of sensi.vi.es of the states with respect to parameter θ i by differen.a.ng both sides of (1) with respect to the parameter, applying the chain rule, and switching the order of d/ and d/dθ i on the lem side: i with ini.al i (t) <latexit sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit 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sha1_base64="9rctmhmhmax5gl5oa45z/42rsg=">aaacehicbzdlsgmxfibp1futt6pln8ei1k2zkylullhxwcfeofnkjs2ozklyrmxdhen76kgxekuhxpzrfweuwvilb+epj4zzljzu/fumi7u8rs7s8srqwxc9tbg5t7+r39+o6shtjnrbjsdu9qrkuia+hqmmbsei8crveiplcb1xy5uwuxidw5i3a9olhs8yrwn18seuryhl3zgqffssu9epu9jnsdtivmqtskx7ji9evkezwafiy+yqnrjf7jdicubd5fjqnxlswnsp+ormesjnjtohlm2od3emhjsgot2ollori6myv+pmwjkuzc3xmpdbqebp7pdcj29xxtbp5xayxon7dtecyj8pbnh/itstai43rivyjoua4nukae+sthfwosqpnhzotgzk+8cpxtkmoxnotyovkepgfzoibdkiidz1cbk6hcdrjcwym8w4v1yd1zr9bbtdvjzwb24y+s92/+j6d</latexit> <latexit 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sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit sha1_base64="whrjnp69ctg2omuzoypfhupkic=">aaacghicbzdlssnafiynxmu9rv26gsxc3drecrorcm5cvraxaeqytcftmmfmroxhdygg1/fjqtf3hbn2zhpa2rrdwmf/zmhm+f3ysevwnaxsbk6tr6xwdoqb+/s7u2bb4fswsshanrcs7hlfm8jc1ging3vgyenicdbzxtv7vpdcpebtewyrm/yamq+5zskbbrnnu+jlq1imjbe4efsx+2iera+lyrgqd4wtsuwmquwbfqlkz4wwwc6igqk3xndqdicybc4ekoltptmlop/kkklhwdhlfyklhzmh6gkmsmnvpz4dl+fq7a+xhur8q8mz9pzgsqklj4onogmbildzy879alwh/qp/yme6ahxs+ye8ehgjnkeebl4ycmgggvhl9vxhrccfosuydsfephkz2hc126rzd/vko17euulh6arvkyuuqpdoizqiyqeat6q+/gs/fqfbif89yvo5g5qn9ktl8bs3qfq==</latexit> (2) : sensi9vity equa9ons Why are these zero? Ini.al condi.ons don t depend on parameter (provided θ i is a pure parameter, not an ini.al condi.on- - - value see later)

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sha1_base64="jp+t/fu8tyxv/tbdfqyyndzxru=">aaacdxicbvdltgixfl2dl8qx6tjni5rabmcmie4kcam7tosracf3soggdmbsdkzihb9w46+4caexbt278y/8bgearavputnhtve48bck6nbx9aqaxlldw19hpmy3nreye7u1ftfqgoq1jf+krhomace6xqubgsesighws7g4ve79+x5tmvndrrgfrs+x7vmcpmljqzi/yustaztgkcj++bm+igljgt5u8hsjm7ae9afokzi7mll5igsl+tlo+dsxzdbwoddoxa9ooktfushgmfwowib1inzvj6qfkuh1nthmt41jpkp6v4umzmlf/dqotr5jn66uaaz63kve/7xmahrn7yh7qwiyr6cp9ujbje+saeixkangmueqelxxwkdoejq4gctejz5lrdj7bto2exnppqrl6zpqboo4bdy4mazloekklafcvfwcm/wyj1yt9ar9tyttvmznn34a+v9g4vnm+=</latexit> sha1_base64="xgz29cpusm9ge8gpreseaq48rvu=">aaacdxicbvdlssnafj3uv62vqes3g1vonzwrgi4lbnrxwt6gcevmommhzirhzikub9w46+4cagiw/fu/butnqc2hhg4c869d+yel+jmacv6mgorq2vrg8xntb2zu6eux/qvmescw2rkiey64ginawpznmtbtjcsljtoonrzk/c+lymfwpycrdquma+yzajqv+uzjjxeikjobxzfts5+lmwqhyfqt6gqpb5atmjudxiz2tsoor7nvfjqdkmscbppwukpnw5f2k2w4xracmjfiybjgnjesgmqvlnjbjsppk2vafzdmz5a45n6uymbodreegmlad1si14m/uf1yu1fugkloljtgmwf8moodyizapcasuon6qeigtpxzezgqsiwczeozflzdj+7xmwzx7tl5u1pm4iugihamkstefaqbr1eqtrnadekiv6nv4nj6nn+n9xlow8p5d9afgxzfqo5q+</latexit> Sensi.vity Equa.ons Sensi.vi.es: par.al derivs of state variables with respect to params, e.g. )(t) For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters We can find the vector of i (t) <latexit 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sha1_base64="lqgurfr7xeewqju598rclf/sjp=">aaacehicbzdlssnafiyn9vbrlerszwar66ykutblwy3lcvyctqit6aqdorkwcykwkedw46u4cagiw5fufbunbrbt/whg4z/nzmz5/urwbzb1zzrwvtfwn8qbla3tnd9c/+go+juutamsyhlzyekcr6xnnaqrjdirkjfsk4/vprwu3dmkh5htzbjmbusycqdtgloyznpnuasmjkjkccjwpf5dzswyka8ntfgdhtm1apbm+flsauookitz/xbjfnqxybfuspvml4gbtq6lgecvjfusihzmh62umsmium8wyvgjdgy4iku+eecz+3sii6fsk9dxnsgbkvqstc3/avugks341gsaovo/kegfrhipehd7hkfmrea6gs679ioii6idazvnqi9ulky9a5r9tw3b5pvjunio4yoklhqizsdiga6bq1ubtr9iceat6nr6nz+pnej+3loxi5hd9kfhxdvn1nvq=</latexit> of the states with respect to parameter θ i by differen.a.ng both sides of (1) with respect to the parameter, applying the chain rule, and switching the order of d/ and d/dθ i on the lem side: The is the Jacobian matrix d (2) : sensi9vity - differen.ate equa9ons RHS of ODE w.r.t. state i with ini.al is deriva.ve of RHS w.r.t. parameter () i <latexit 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sha1_base64="whrjnp69ctg2omuzoypfhupkic=">aaacghicbzdlssnafiynxmu9rv26gsxc3drecrorcm5cvraxaeqytcftmmfmroxhdygg1/fjqtf3hbn2zhpa2rrdwmf/zmhm+f3ysevwnaxsbk6tr6xwdoqb+/s7u2bb4fswsshanrcs7hlfm8jc1ging3vgyenicdbzxtv7vpdcpebtewyrm/yamq+5zskbbrnnu+jlq1imjbe4efsx+2iera+lyrgqd4wtsuwmquwbfqlkz4wwwc6igqk3xndqdicybc4ekoltptmlop/kkklhwdhlfyklhzmh6gkmsmnvpz4dl+fq7a+xhur8q8mz9pzgsqklj4onogmbildzy879alwh/qp/yme6ahxs+ye8ehgjnkeebl4ycmgggvhl9vxhrccfosuydsfephkz2hc126rzd/vko17euulh6arvkyuuqpdoizqiyqeat6q+/gs/fqfbif89yvo5g5qn9ktl8bs3qfq==</latexit> i This is a generaliza.on of linear stability (provided θ i is a pure parameter, not an ini.al analysis condi.on- - - see later)

22 <latexit sha1_base64="o8ckqtnsexod8rscmcgexs+opxm=">aaacghicbzdjsgnbeizrxgpcoh69nazbl3fgbl2ighepecwcmtddhqsjjl3tvigoyxvpgqxjwo4ju338jhsloas/yh4eovkqrr9xmpnnr2hzu3v7c4tfxyka6urw9slra26zpofem1fstynx2qurqrr6fayzuj4jtjw/4/atrvxhhlrzxdiudhldd2o1eibhfy3mlizdqlgvuqhukksl9/su9jhst+qh9ie5j7axhblxktsveywyc84uyhefmfbvkw3dtszskefijnw65dgjtrprciz5xnrtzrpk+rtlwwyjgnldzsah5wtfobsxmq8cmny/tmrvdrqeibzpbit/+tjcz/aqug7n2jqikrr6xyaiglqrjmkqjditidoxaagvkml8s1qmmktrzfkizt+tz6f+xhhsinnzur48maqbbdifptgab7heq6hcjvg8abp8akv1qp1bl1z75pwows6swo/za2/aoz9oqa=</latexit> <latexit 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sha1_base64="jp+t/fu8tyxv/tbdfqyyndzxru=">aaacdxicbvdltgixfl2dl8qx6tjni5rabmcmie4kcam7tosracf3soggdmbsdkzihb9w46+4caexbt278y/8bgearavputnhtve48bck6nbx9aqaxlldw19hpmy3nreye7u1ftfqgoq1jf+krhomace6xqubgsesighws7g4ve79+x5tmvndrrgfrs+x7vmcpmljqzi/yustaztgkcj++bm+igljgt5u8hsjm7ae9afokzi7mll5igsl+tlo+dsxzdbwoddoxa9ooktfushgmfwowib1inzvj6qfkuh1nthmt41jpkp6v4umzmlf/dqotr5jn66uaaz63kve/7xmahrn7yh7qwiyr6cp9ujbje+saeixkangmueqelxxwkdoejq4gctejz5lrdj7bto2exnppqrl6zpqboo4bdy4mazloekklafcvfwcm/wyj1yt9ar9tyttvmznn34a+v9g4vnm+=</latexit> sha1_base64="xgz29cpusm9ge8gpreseaq48rvu=">aaacdxicbvdlssnafj3uv62vqes3g1vonzwrgi4lbnrxwt6gcevmommhzirhzikub9w46+4cagiw/fu/butnqc2hhg4c869d+yel+jmacv6mgorq2vrg8xntb2zu6eux/qvmescw2rkiey64ginawpznmtbtjcsljtoonrzk/c+lymfwpycrdquma+yzajqv+uzjjxeikjobxzfts5+lmwqhyfqt6gqpb5atmjudxiz2tsoor7nvfjqdkmscbppwukpnw5f2k2w4xracmjfiybjgnjesgmqvlnjbjsppk2vafzdmz5a45n6uymbodreegmlad1si14m/uf1yu1fugkloljtgmwf8moodyizapcasuon6qeigtpxzezgqsiwczeozflzdj+7xmwzx7tl5u1pm4iugihamkstefaqbr1eqtrnadekiv6nv4nj6nn+n9xlow8p5d9afgxzfqo5q+</latexit> Sensi.vity Equa.ons Sensi.vi.es: par.al derivs of state variables with respect to params, e.g. )(t) For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters The sensi.vity equa.ons for <latexit 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sha1_base64="lqgurfr7xeewqju598rclf/sjp=">aaacehicbzdlssnafiyn9vbrlerszwar66ykutblwy3lcvyctqit6aqdorkwcykwkedw46u4cagiw5fufbunbrbt/whg4z/nzmz5/urwbzb1zzrwvtfwn8qbla3tnd9c/+go+juutamsyhlzyekcr6xnnaqrjdirkjfsk4/vprwu3dmkh5htzbjmbusycqdtgloyznpnuasmjkjkccjwpf5dzswyka8ntfgdhtm1apbm+flsauookitz/xbjfnqxybfuspvml4gbtq6lgecvjfusihzmh62umsmium8wyvgjdgy4iku+eecz+3sii6fsk9dxnsgbkvqstc3/avugks341gsaovo/kegfrhipehd7hkfmrea6gs679ioii6idazvnqi9ulky9a5r9tw3b5pvjunio4yoklhqizsdiga6bq1ubtr9iceat6nr6nz+pnej+3loxi5hd9kfhxdvn1nvq=</latexit> i with ini.al condi.ons () = i are i 1. Sensi.vity eqns for different states with respect to a given parameter are coupled 2. Sensi.vity eqns across different parameters are not coupled so can arrange sensi.vi.es with respect to different parameters into a matrix

23 Sensi.vity Equa.ons For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters the m by p matrix of sensi.vi.es, with ini.al () = m p, sa.sfies the ODE system Equa.ons (2) are the sensi.vity equa.ons for the system (2) Sensi9vity matrix: Sensi.vi.es of different states with respect to a given parameter are arranged in a single column, different columns depict sensi.vi.es with respect to different parameters

24 <latexit sha1_base64="2wz86xapxcdloap+oek77ulhtw=">aaacbnicbzdlssnafizp6q3ww9slcinfcfuskejoghuxfewfmlamk7dhjhzikwkjubx8wncxc+gzufasfwulauft/gpj4zzkzc34v5kwqy/oskvlk6tr5fxkxubw9o65u9ewusiibzgir6lryuk5c2llmcvpnxyubx6nhw98mdc7t1rifou3ahjtn8ddkpmmykwtvnno+akt1imxuaxz5gc/fjf1zapvswqhrbbnul34gklnvvnhdcksbdruhgmpe7yvkzfnlysczhunkttgziyhtkcxxagvblqskafj7qyqhwl9qouk9/deigmpj4gnowosrnk+lpv/1xqj8s/dlivxomhipg/5cucqqnkmamaejyppngaimp4riiosc1e6uyoowz5ferhapzxbqtnx9wqjpkdynaar3acnpxba66gcsgca+p8awvxopxzlwab9pwkjgb2yc/mt6/aymlmoy=</latexit> <latexit sha1_base64="hkhfpsd37t7mbqsjksbpdanqyo=">aaacbnicbzdlssnafizp6q3ww9slcinfcfuskeiy4mzlbxubjptjdnionuzczeqsoss3voobf4q49rnc+tzo2oda+spax3/omznzbwlnsjvol1vawv1b3yhvvra2d3b37p2dtoptswilxdyw3qarypmglcp91euhwfnhac8vve79xrqvgsbvukox6eh4kfjgbtrl597iusk8xlsnqmcxrof/h+2rerts2zcs2dwavcjx79qc3iekauaejxr1xcfrfpzfsdidvrxuqstmr7snkgbi6r8blbgfjaz4dcwjojnjq5vycyhck1iqltgwe9uou13pyv1kt1eolntcsppolmhwptjnsm8kzqgelknj8yweqy81dertjkok1yfrocu7jymrtpa65tc2/q1ua9ikmmr3acz+dcbttggprqagip8aqv8go9ws/wm/u+by1zxcwh/jh18q3n+zlx</latexit> <latexit sha1_base64="2wz86xapxcdloap+oek77ulhtw=">aaacbnicbzdlssnafizp6q3ww9slcinfcfuskejoghuxfewfmlamk7dhjhzikwkjubx8wncxc+gzufasfwulauft/gpj4zzkzc34v5kwqy/oskvlk6tr5fxkxubw9o65u9ewusiibzgir6lryuk5c2llmcvpnxyubx6nhw98mdc7t1rifou3ahjtn8ddkpmmykwtvnno+akt1imxuaxz5gc/fjf1zapvswqhrbbnul34gklnvvnhdcksbdruhgmpe7yvkzfnlysczhunkttgziyhtkcxxagvblqskafj7qyqhwl9qouk9/deigmpj4gnowosrnk+lpv/1xqj8s/dlivxomhipg/5cucqqnkmamaejyppngaimp4riiosc1e6uyoowz5ferhapzxbqtnx9wqjpkdynaar3acnpxba66gcsgca+p8awvxopxzlwab9pwkjgb2yc/mt6/aymlmoy=</latexit> <latexit sha1_base64="2wz86xapxcdloap+oek77ulhtw=">aaacbnicbzdlssnafizp6q3ww9slcinfcfuskejoghuxfewfmlamk7dhjhzikwkjubx8wncxc+gzufasfwulauft/gpj4zzkzc34v5kwqy/oskvlk6tr5fxkxubw9o65u9ewusiibzgir6lryuk5c2llmcvpnxyubx6nhw98mdc7t1rifou3ahjtn8ddkpmmykwtvnno+akt1imxuaxz5gc/fjf1zapvswqhrbbnul34gklnvvnhdcksbdruhgmpe7yvkzfnlysczhunkttgziyhtkcxxagvblqskafj7qyqhwl9qouk9/deigmpj4gnowosrnk+lpv/1xqj8s/dlivxomhipg/5cucqqnkmamaejyppngaimp4riiosc1e6uyoowz5ferhapzxbqtnx9wqjpkdynaar3acnpxba66gcsgca+p8awvxopxzlwab9pwkjgb2yc/mt6/aymlmoy=</latexit> <latexit sha1_base64="2wz86xapxcdloap+oek77ulhtw=">aaacbnicbzdlssnafizp6q3ww9slcinfcfuskejoghuxfewfmlamk7dhjhzikwkjubx8wncxc+gzufasfwulauft/gpj4zzkzc34v5kwqy/oskvlk6tr5fxkxubw9o65u9ewusiibzgir6lryuk5c2llmcvpnxyubx6nhw98mdc7t1rifou3ahjtn8ddkpmmykwtvnno+akt1imxuaxz5gc/fjf1zapvswqhrbbnul34gklnvvnhdcksbdruhgmpe7yvkzfnlysczhunkttgziyhtkcxxagvblqskafj7qyqhwl9qouk9/deigmpj4gnowosrnk+lpv/1xqj8s/dlivxomhipg/5cucqqnkmamaejyppngaimp4riiosc1e6uyoowz5ferhapzxbqtnx9wqjpkdynaar3acnpxba66gcsgca+p8awvxopxzlwab9pwkjgb2yc/mt6/aymlmoy=</latexit> Sensi.vity Equa.ons For dx = f(x, t; ) (1), where x and f are m dimensional (m state variables), and θ is a p dimensional vector of parameters the m by p matrix of sensi.vi.es, with ini.al () = m p, sa.sfies the ODE system Equa.ons (2) are the sensi.vity equa.ons for the system The Jacobian matrix depends on the state variables, so the sensi.vity equa.ons (2) [a linear system of ODEs] must be solved simultaneously with the governing equa.ons (1)

25 <latexit sha1_base64="tg2ytmfgfrmwqpiwfp2wpldfera=">aaacxhicjzhlsgmxfiyzy9verfyfn26crxbvzqsgg7hgxmufe4fokzlmpo3nxejoigwyl3txjc/gg2g6lvhbfx4ifpznlvxxy8evwnbcmhcku3v7xvk5cla9pkodn3rvlejkojqskey7rdhbq9ybdol1y8li4arwc6cpi3zvlunfo/azzjebbmqccp9taloa1ztjsjtls9yjbsxeqcjwk/zt/swiqbgb1kdxtffrbe8j8jefsrwwrd7wn9grq958oj/ao9u54eucfgivrkmbbcuwtbedqwbz2ukuiwmdkjebaaxjwnqwzc3j8kvwpoxhup8qck6ud6qkugowuloyidbrm7mf+fdukib/ox5gcfaqrpc5cccq4qxtmopszbzdqqkrm+k6ytovb/r9lbyk9+ert6f43bkthpzxrrebsdvre5+gcxseb3aawekrt1eeuzdgxutrkxodzmctmdvlqgqueu/qrzlnvmmw6qw==</latexit> <latexit 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sha1_base64="fuufxqu7qwhdpqqluc6c2tk1cla=">aaachhicbzdlsgnbeevrfbtfuzdugoogmzcjgm5ewy1lbancjgw9nrrt2voguymw3yig3/fjqtf3lgq/as/wu4i+lzqclhvrxxdmfpskou+oupdi6nj4xotlanpmdm56vzciulzlbahupxqs5abvdlbbklsejzp5hgo8ds83o/vt69qg5kmx9tnsbxz8rguncyvldd8cpnrefnxjpkil2xx+xtb4khf+wqu8z2woghecmyuaiqnbfu9sx+gvcjtd136oswql747vtkmsykfdem6bkztyregqgwrpi5wyyls36otysjj9giv5xjvuxtptfqbyvidz3v8updamg4e2m+bumb9rpfo/wjonaltvyctlcrmxwbtlilhkekmxttqoshutckgl/ssthw7tiptnxybg/t75l5ys1z237h1t1vy2b2nabczbmqycb1uwbwdwcaqcan38acpzq1z7zw5z4pwiedzzhf+yhn9apyop8=</latexit> <latexit sha1_base64="fuufxqu7qwhdpqqluc6c2tk1cla=">aaachhicbzdlsgnbeevrfbtfuzdugoogmzcjgm5ewy1lbancjgw9nrrt2voguymw3yig3/fjqtf3lgq/as/wu4i+lzqclhvrxxdmfpskou+oupdi6nj4xotlanpmdm56vzciulzlbahupxqs5abvdlbbklsejzp5hgo8ds83o/vt69qg5kmx9tnsbxz8rguncyvldd8cpnrefnxjpkil2xx+xtb4khf+wqu8z2woghecmyuaiqnbfu9sx+gvcjtd136oswql747vtkmsykfdem6bkztyregqgwrpi5wyyls36otysjj9giv5xjvuxtptfqbyvidz3v8updamg4e2m+bumb9rpfo/wjonaltvyctlcrmxwbtlilhkekmxttqoshutckgl/ssthw7tiptnxybg/t75l5ys1z237h1t1vy2b2nabczbmqycb1uwbwdwcaqcan38acpzq1z7zw5z4pwiedzzhf+yhn9apyop8=</latexit> Sensi.vi.es with Respect to Ini.al Condi.ons? If an ini.al condi.on is unknown, we could include it as one of the quan..es to be es.mated using least squares Everything goes as before, except that sensi.vity equa.ons are a liele different if you are looking at sensi.vi.es with respect to an ini.al condi.on j () = e j We don t have that second term on the right side because the equa.ons right hand sides of our model don t involve ini.al condi.ons Unit vector with 1 in j th place, elsewhere Deriva.ve of ini.al condi.on i with respect to ini.al condi.on i is one, zero with respect to other ini.al condi.ons So, we now know how to calculate sensi.vi.es with respect to either parameters or ini.al condi.ons; we can calculate both together

26 Sensi.vity Equa.ons for Logis.c Growth Model Because we have an analy.c solu.on of the logis.c growth model dy = ry y 1 ; y() = y K K y(t) = 1+(K/y 1) e rt We can calculate the sensi.vi.es of y(t) with respect to K, r and y using calculus But we could also calculate those three sensi.vi.es using the sensi.vity equa.ons method... and check that the two methods give the same answer! Next slide outlines one of those calcula.ons,

27 Sensi.vity Equa.ons for Logis.c Growth Model Let s set up sensi.vity equa.on transla.ng from general nota.on General Nota.on Nota.on of our problem dx = f(x, t; ) x is y, and θ is r, f is ry(1- y/k) dy = ry 1 y K = + d Calcula.ng f/ y and f/ r, and subs.tu.ng: = r + y 1 y K Have to integrate a two dimensional system, with states y and value of sensi.vity with ini.al

28 Deriva.on of the Sensi.vity Equa.ons for the SIR Model (Appendix of Capaldi et al. 212) We have ds = βsi N di = βsi N γi The four sensi.vi.es, arranged in a matrix: " x θ = # S β I β S γ I γ % ' ' ' & Jacobian: f x " = J(S, I) = # S S ( ds ) I ( ds ) ( di ) I ( di ) % ' " = ' & # βi N βi N βs N βs N γ % ' & Deriva.ve of right hand sides of differen.al equa.ons with respect to parameters " f θ = # β β ds ds ( ) γ ( ) ( ) ( ) di γ di % ' " ' ' = SI N # SI N I & % ' &

29 Deriva.on of the Sensi.vity Equa.ons for the SIR Model We have (Appendix of Capaldi et al. 212) The four sensi.vi.es, arranged in a matrix: Jacobian: ds = βsi N di = βsi N γi f x " = J(S, I) = # S S ( ds ) I ( ds ) ( di ) I ( di ) " x θ = # % ' " = ' & # S β I β βi N βi N S γ I γ % ' ' ' & Shorthand nota.on for sensi.vi.es βs N βs N γ % ' & Deriva.ve of right hand sides of differen.al equa.ons with respect to parameters " f θ = # β β ds ds ( ) γ ( ) ( ) ( ) di γ di % ' " ' ' = SI N # SI N I & % ' &