TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE


 Christal Hoover
 1 years ago
 Views:
Transcription
1 Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs and Lohar von Wolfersdorf Absrac We invesigae wo wellknown basic opimal conrol problems for chemoherapeuic cancer reamen modified by inroducing a imedependen resisance facor. This facor should be responsible for he effec of he drug resisance of umor cells on he dynamical growh for he umor. Boh opimal conrol problems have common poinwise bu differen inegral consrains on he conrol. We show ha in boh models he usually pracised bangbang conrol is opimal if he resisance is sufficienly srong. Furher, we discuss differen opimal sraegies in boh models for general resisance. MSC 2: 49 J 15; 49 K 15; 92 C 5. Keywords: Cancer chemoherapy, opimal conrol, drug resisance. 1 Inroducion Opimal conrol problems based on mahemaical models for cancer chemoherapy have a long hisory and obained a renewed ineres in he las years Acceped for publicaion on March 13, 211. Deparmen of Mahemaics, TU Darmsad, Schloßgarensr. 7, Darmsad, Germany, Lohar von Wolfersdorf passed away on November 3,
2 Two opimal conrol problems in cancer chemoherapy 333 (cf. [26, 126]). There are furher recen papers on mahemaical models for immunoherapy and mixed immunoherapy and chemoherapy saring wih papers by A. Kuznesov and coworkers in he nineies (cf. [19,2], for insance) bu which are no in our focus here. Insead he presen paper follows he wo pioneering papers by J.M. Murray in 199 [16,17] (see also [24]) whose basic problems are modified in he following. One difficuly in applying he considered opimal conrol problems in cancer chemoherapy is he occurance of opimal soluions which are seldom or no used in medical pracice. A desired opimal soluion by he physician is he bangbang conrol consising of a saring inerval wih maximal dose of drug followed by an inerval of zeroherapy ill he end of reamen (considering one cycle of he chemoherapeuic reamen). In paricular, he herapy should heoreically end wih an inerval of zeroherapy o have he required minimum of he umor cells populaion also somewha laer han a he pracical end of reamen. To obain opimal soluions of his ype ofen a suiable choice of he objecive funcions is proposed (cf. [14, 17, 19, and 24]). The aim of he presen paper is o circumven his difficuly aking ino accoun he resisance of he umor cells agains drug (and furher using he size of he umor cells populaion a he final ime as he naural objecive funcion). Acquired and inrinsic resisance of he umor cells agains drug is an imporan bu very complex phenomen in umor herapy (cf. [8, 12]) and relaed deerminisic models [4, 12, 14] and sochasic ones [2, 3] in dealing wih i are developed recenly. In our highly simplified model we only consider a summarizing effec of resisance by inroducing a imedependen resisance facor in fron of he loss funcion of he umor cells in he deerminisic differenial equaion for he umor growh. In paricular, we do no disinguish beween drug sensiive and resisan umor cells like in [4, 12, 14]. Furher, we deal wih wo basic problems where in each problem we have wo resricions, namely he usual poinwise inequaliy for he conrol funcion (which is in he dose of he drug adminisered per uni of ime) and an inegral inequaliy for he loss funcion of he normal cells in he firs problem and for he drug dose iself in he second problem. To keep he mahemaical analysis simple oher resricions like he poinwise limi for he size of he populaion of he normal cells like in [15, 17, and 24] are no
3 334 Werner Krabs, Lohar von Wolfersdorf aken ino consideraion. There is only one dynamic equaion for he growh and suppression of he umor cells and no one for he normal cells (bu which could be easily supplemened). Boh opimal conrol problems show he desired effec ha for (properly defined) srong resisance he abovenamed bangbang conrol is he unique opimal conrol (cp. wih he resuls in [3, 26], for insance). Wih respec o general resisance we have anoher picure. In he firs problem in case of weak resisance he opimal conrol is he non desired opposie bangbang conrol wih saring inerval of zeroherapy and final inerval of maximal drug dose. On he oher hand, in he second problem for general resisance an opimal conrol similar o he desired bangbang conrol saring and ending wih a subinerval of zeroherapy is o be expeced as an example wih Gomperzian growh show (cp. wih oher forms of opimal soluions in [14, 17], for insance). So, especially wih respec o weak resisance he second problem seems preferable o he firs problem. The plan of he paper is as follows. Afer performing he mahemaical modelling in Secion 2 we invesigae he firs opimal conrol problem in Secion 3 and he second opimal conrol problem wih he example for Gomperzian growh in Secion 4. 2 Mahemaical Models We denoe he imedependen number of cancer cells in he umor by a funcion T = T (), IR +, which we assume o be differeniable wih derivaive T (). The emporal developmen of he umor cells populaion T () in a given inerval [, f ] is governed by he differenial equaion T () = [f(t ()) ϕ()l(m())]t (), [, f ], (2.1) wih he iniial condiion T () = T >. (2.2) The funcion f = f(), T describes he dynamics of he umor populaion F (T ) = f(t )T if here is no adminisraion of drugs. We assume f C 1 (IR + ) wih f(t ) >, f (T ) < for all relevan T. This is fulfilled for many of he commonly used dynamics as Gomperz, logisic (VerhulsPearl)
4 Two opimal conrol problems in cancer chemoherapy 335 and oher growh laws in an inerval [, θ] wih maximal umor populaion θ (cf. [5, 1, 16, 22, 24]). By L = L(M), M we denoe he desrucion rae of he drug level M. We assume ha his loss funcion L C 2 (IR + ) saisfies L() = and L (M) > for all relevan M. This is fulfilled, for insance, for linear and fracional linear ( sauraed ) funcion L (cf. [1, 16, 24]). The drug level funcion M = M() obeys he linear differenial equaion and he iniial condiion Ṁ() = δm() + V (), [, f ], (2.3) M() = (2.4) wih a posiive drug decay rae δ where V () denoes he drug dose ha is adminisered per uni of ime a ime [, f ]. In he following we assume δ, hus including he mahemaical limi case δ = of no drug decay. The drug dosis V = V () per uni of ime is considered as he conrol funcion in he model. We assume i o be a bounded measurable funcion, i.e. V L (, f ), and o saisfy he poinwise condiion V () A for a.a. [, f ] (2.5) where A > is a prescribed consan, he maximum drug dosis per uni of ime. Furher, below we require addiionally an inegral condiion which we regard as responsible for he compaibiliy of he reamen. The new feaure in his model is he inroducion of he funcion ϕ = ϕ(), [, f ], which is assumed o be in C 1 [, f ] saisfying ϕ() > in [, f ) and normed by ϕ() = 1. The facor ϕ in Eq. (2.1) should  in a mos simple way  describe he oal effec of inner influences (like drug resisance) and oher ones (like accompanying herapies) on he desrucion rae of he umor cells by he drug during he reamen. Especially, he influence of he drug resisance of he umor cells will be expressed by a funcion ϕ C 1 [, f ] wih ϕ() = 1, ϕ() in [, f ] and ϕ() > in [, f ]. (We call such a funcion a resisance facor in he following). The inegral condiion in problem 1 now reads
5 336 Werner Krabs, Lohar von Wolfersdorf ϕ ()L (M())d B (2.6) wih a prescribed consan B > where L (M) is he desrucion rae of he normal cells for which we assume he same properies as for he loss funcion L(M) of he umor cells above and ϕ C 1 [, f ] wih ϕ () > in [, f ), ϕ () = 1 is a weigh funcion possessing he analogous meaning for he normal cells as ϕ for he umor cells. In problem 2 we simply require ha wih a given consan B >. V ()d B (2.7) The aim of chemoherapeuic reamen is o make he umor cells populaion T ( f ) a he end of he reamen as small as possible. In view of Eq. (2.1) his can be wrien in he usual form of he minimum condiion [f(t ()) ϕ()l(m())]t ()d min. (2.8) We furher remark ha for a given V L (, f ) he soluion of (2.3), (2.4) has he form M() = M[V ]() = e δ(s ) V (s)ds, [, f ] which implies M C[, f ]. By our assumpions on f, ϕ, L we hen have T C 1 [, f ] for he corresponding soluion T = T [V ] of Eq. (2.1). Our opimal conrol problems are now defined by he minimum condiion (2.8) for he sae equaions (2.1)  (2.4) wih he consrains (2.5), (2.6) (problem 1 ) or (2.5), (2.7) (problem 2 ). Here he inegral consrains (2.6) and (2.7) can be aken, respecively, in he form
6 Two opimal conrol problems in cancer chemoherapy 337 Q( f ) B or U( f ) B (2.9) where he addiional sae funcions Q = Q() and U = U() are given by he inegrals Q() = ϕ (s)l (M(s)ds and U() = V (s)ds, respecively, or equivalenly by he addiional sae equaions wih Q() = and wih U() =, respecively. Q() = ϕ ()L (M()), [, f ] (2.1) U() = V (), [, f ] (2.11) These opimal conrol problems always have soluions as follows by adaping he exisence proof by J.M. Murray in [16] on he basis of Theorem in [1] (aking ino accoun ha for he admissible conrol V () = in [, f ] he sae equaion (2.1) has a coninuous soluion T () in [, f ] wih finie T ( f ) and because of he finie inerval [, f ] also he parameer δ = in Eq. (2.3) is possible). Finally, we simplify he mahemaical analysis for our problems slighly by applying he usual subsiuion y = lnt for T >. Then he differenial equaion (2.1) is ransformed ino ẏ() = f(e y() ) ϕ()l(m()), [, f ] (2.12) and he iniial condiion (2.2) reads The minimum condiion (2.8) akes he form y() = y = lnt. (2.13)
7 338 Werner Krabs, Lohar von Wolfersdorf [f(e y() ) ϕ()l(m())]d Min. (2.14) The opimal conrol problems o be solved are hen given by he minimum condiion (2.14) for he sae equaions (2.12), (2.13), (2.3), (2.4), and (2.1) or (2.11), respecively, under he consrains (2.5), (2.9). 3 Soluions of he firs problem We deermine opimal soluions of problem ( ), (2.5), (2.9), (2.1) as usual wih he aid of he maximum principle [9]. The Hamilonian of he problem is given by H(, y, M, Q, V, p 1, p 2, p 3, λ ) = (f(e y ) ϕ()l(m))(p 1 λ ) (3.1) +(V δm)p 2 + ϕ ()L (M)p 3 wih he parameer λ and he adjoin sae funcions p k, k = 1, 2, 3. If (ŷ, ˆM, ˆQ, ˆV ) is an opimal quadruple here exis a number λ and hree funcions p k C 1 [, f ], k = 1, 2, 3 wih (λ, p 1, p 2, p 3 ) (,,, ) saisfying he differenial equaions ṗ 1 () = f (eŷ() )eŷ() (p 1 () λ ) (3.2) ṗ 2 () = ϕ()l ( ˆM())(p 1 () λ ) + δp 2 () ϕ ()L ( ˆM())p 3 () (3.3) ṗ 3 () = (3.4) in [, f ] and he ransversaliy condiions in f p 1 ( f ) =, p 2 ( f ) =, and p 3 ( f ), p 3 ( f )( ˆQ( f ) B) = (3.5) such ha for a.a. [, f ] he maximum condiion
8 Two opimal conrol problems in cancer chemoherapy 339 ˆV ()p 2 () = max V A [V p 2()] (3.6) is valid. From (3.4) and (3.5) i follows ha p 3 is a nonposiive consan which vanishes if ˆQ( f ) < B. We define p 1 () = p 1 () λ and g() = f (eŷ() )eŷ() >, [, f ]. (3.7) Then from (3.2) we have p 1 () = g() p 1 () which gives p 1 () = p 1 () exp( From p 1 ( f ) = we obain g(s)ds), [, f ]. (3.8) p 1 ( f ) = p 1 () exp( which shows ha p 1 () for all [, f ]. We furher pu From (3.3) we ge g()d) = λ (3.9) h() = ϕ()l ( ˆM()) p 1 () ϕ ()L ( ˆM())p 3. (3.1) which yields p 2 () = e δ H() wih ṗ 2 () = δp 2 () + h(), [, f ] (3.11) H() = p 2 () + In view of p 2 ( f ) = we have e δs h(s)ds, [, f ]. (3.12)
9 34 Werner Krabs, Lohar von Wolfersdorf which implies p 2 () = e δ h()d (3.13) p 2 () = e δ e δs h(s)ds, [, f ]. (3.14) Now we disinguish he wo cases B Q A ( f ) ϕ ()L (M A ())d (3.15) where M A () = A δ [1 e δ ] if δ > o, A if δ = is he soluion of (2.3), (2.4) for V () = A a.e. in [, f ], and B < Q A ( f ) ϕ ()L o (M A ())d. (3.16) If (3.15) is fulfilled we have he opimal soluion ˆV () = A a.e. in [, f ]. So we can assume (3.16) in he following. In his case he equaliy ˆQ( f ) ϕ ()L ( ˆM())d = B (3.17) for he opimal soluions mus hold. We prove his by conradicion. If ˆQ( f ) < B we have p 3 =. In he anormal case λ = by (3.9), (3.8) and (3.1), (3.14) his implies p 1 () = p 1 () = and p 2 () = in [, f ] which conradics he condiion (λ, p 1, p 2, p 3 ) (,,, ). In he normal case λ > by (3.8) we would have p 1 () < and hence by (3.1) also h() < in [, f ] which by (3.14) yields p 2 () > in [, f ]. The maximum
10 Two opimal conrol problems in cancer chemoherapy 341 condiion (3.6) hen would give ˆV () = A for a. a. [, f ]. This implies Q A ( f ) = ˆQ( f ) < B, a conradicion o (3.16). Therefore, (3.17) and p 3 < hold rue. We furher show because of he condiion (3.17) he abnormal case λ = canno occur. Namely, from λ = as before we obain p 1 () = p 1 () = in [, f ] implying h() = ϕ ()L ( ˆM())p 3 >, [, f ] from (3.1). By (3.14) i follows ha p 2 () < in [, f ]. Then (3.6) yields ˆV () = for a.a. [, f ] which by (2.3), (2.4) leads o ˆM() = in [, f ] and by L() = o ˆQ( f ) =, a conradicion o (3.17). Summing up, in he case (3.16) equaliy (3.17) is valid and we have λ >, p 1 () < in [, f ] and p 3 <. We inroduce he funcions q() = L ( ˆM()) L ( ˆM()), () = ϕ() p 1() ϕ ()q()p 3 (3.18) so ha by (3.1) we have h() = L ( ˆM()) () wih signh() = sign (). Now we make he assumpion ha is a sricly increasing funcion in [, f ] which is fulfilled if d d () d d [ϕ() p 1() ϕ ()q()p 3 ] > in (, f ). (3.19) We disuingish he hree cases (i) () (ii) () <, ( f ) (iii) () <, ( f ) >. In case (i) we have () > () in [, f ] implying h() > in (, f ] and p 2 () < in [, f ) by (3.14). The condiion (3.6) hen yields he soluion ˆV () = a.e. in [, f ] which is no possible. In case (ii) we have () < ( f ) in [, f ] which gives h() < in [, f ] and p 2 () > in [, f ] by (3.14) again. In view of (3.6) hen ˆV () = A a.e. in [, f ] which is also no allowed in he case of (3.16).
11 342 Werner Krabs, Lohar von Wolfersdorf I remains he case (iii). By he sric monooniciy of here exiss exacly one 1 (, f ) wih ( 1 ) =, () < in [, 1 ), and () > in ( 1, f ]. This implies he analogous inequaliies for h. Therefore he funcion H in (3.12) is sricly decreasing from H() = p 2 () o H( 1 ) < p 2 () and hen sricly increasing from H( 1 ) o H( f ) =. If now p 2 () were rue we would ge H() < and hence p 2 () < in (, f ). This would imply ˆV () = a.e. in [, f ] again. Therefore, i mus be p 2 () >. Then here exiss exacly one (, 1 ) wih H( ) =, H() > in [, ) and H() < in (, f ] which implies p 2 () > in [, ) and p 2 () < in (, f ]. The maximum condiion (3.6) yields he unique opimal soluion ˆV () = { A for a.a. [, ) for a.a. (, f ] (3.2) where (, f ) can be defined as he (unique) soluion of he equaion wih ˆM() = ϕ ()L ( ˆM())d = B (3.21) { A δ [1 ēδ ] if δ >, A if δ = for [, ] A δ [eδ 1]ē δ if δ >, A if δ = for (, f ] following from (3.17) and (2.3), (2.4) wih (3.2). We summarize he resul in THEOREM 3.1 (i) Le (3.15)be fulfilled. Then problem 1 has he unique opimal soluion ˆV () = A a.e. in [, f ]. ii) Le (3.16) be fulfilled and he funcion in (3.18) sricly increasing. Then problem 1 has he unique opimal soluion (3.2) wih (3.21). REMARKS. The monooniciy assumpion on in Theorem 3.1 is an implici condiion on ϕ (and ϕ ) since he funcions p 1 by (3.8) and q by (3.18) in general depend on he opimal soluion ˆV of he problem wih he funcion
12 Two opimal conrol problems in cancer chemoherapy 343 ϕ in (2.1) (and ϕ in (2.6)). Bu his dependence can be well derived from (3.2) wih (3.21) and Eqs. (2.1), (2.3). Moreover, in he imporan paricular case L = cl wih a consan c > and L C 1 (IR + ) (cf. [16, 17]) we have q() = c and for ϕ () = 1 in [, f ] he sufficien condiion (3.19) reduces o he simple condiion ϕ() + g()ϕ() < in (, f ) (3.22) wih he posiive funcion g = f ( ˆT ) ˆT by (3.7). Furher, in he special case of Gomperzian growh f(t ) = λln θ T (λ, θ > ) we have g = λ, a consan which is independen of he opimal soluion ˆV. Condiion (3.22) is for insance saisfied, if for some γ >. ϕ() = exp( ( g(s)ds + γ)), [, f ], In general, i remains he dependence of on he (negaive) parameer p 3 or equivalenly on he (posiive) quoien p 3 / p 1 () which are no direcly expressed by he opimal soluion ˆV. To avoid his dependence we derive a furher sufficien crierion for he opimal soluion (3.2) in he sequel. LEMMA 3.2 Under he condiions (3.16) and d d [ρ()p 2()] < in (, f ) (3.23) wih a nonnegaive funcion ρ C 1 (, f ) he opimal soluion of problem 1 is uniquely deermined and has he form (3.2). Proof. Because of (3.23) he opimal soluion canno conain singular pars in subinervals where ṗ 2 () = p 2 () = and pars of he form { a.e. in [1, τ) ˆV () = A a.e. in (τ, 2 ] wih 1 < τ < 2 f where p 2 () in ( 1, τ), p 2 (τ) =, p 2 () in (τ, 2 ) and ṗ 2 (τ). Furher, he soluions ˆV () = a.e. in [, f ] and
13 344 Werner Krabs, Lohar von Wolfersdorf ˆV () = A a.e. in [, f ] are no possible in view of (3.16) wih (3.17). This proves he lemma. In view of (3.11) he condiion (3.23) can be wrien in he form [ ρ() + δg()]p 2 () + ρ()h() < in (, f ) (3.24) wih h defined in (3.1). Taking ρ() = exp( µ(s)ds), µ C(, f ) and r() = µ() + δ C(, f ) condiion (3.24) simply wries By (3.1) and (3.14) his means where r()p 2 () + h() < in (, f ). 1 () + p 3 2 () < in (, f ) (3.25) 1 () = ϕ()l ( ˆM()) p 1 () r() e δ( s) ϕ(s)l ( ˆM(s)) p 1 (s)ds, f 2 () = r() e δ( s) ϕ (s)l ( ˆM(s))ds ϕ ()L ( ˆM()). We now choose r C(, f ) such ha 2 () = in (, f ), i.e. r() = f f e δ ϕ ()L ( ˆM()) e δs ϕ (s)l ( ˆM(s))ds Then (3.25) simplifies o he condiion 1 () < in (, f ) or defining furher he quoien q 1 () = p 1() p 1 () = exp( f ( ˆT (s)) ˆT (s)ds) > (3.26).
14 Two opimal conrol problems in cancer chemoherapy 345 by (3.7), (3.8) o he inegral inequaliy f q 1 ()ϕ() e δs ϕ (s)l ( ˆM(s))ds f > q()ϕ () e δs ϕ(s)l ( ˆM(s))q 1 (s)ds (3.27) in (, f ) where q is defined in (3.18). A sufficien condiion for (3.27) is he differenial condiion f d d [q 1()ϕ()] e δs ϕ (s)l ( ˆM(s))ds f < d d [q()ϕ ()] e δs ϕ(s)l ( ˆM(s))q 1 (s)ds in, ( f ). (3.28) Summing up we obain THEOREM 3.2 i) Le (3.16) and (3.27) wih (3.18), (3.26) be fulfilled. Then problem 1 has he unique opimal soluion (3.2) wih (3.21). ii) The inegral condiion (3.27) is saisfied if he differenial condiion (3.28) is valid. REMARKS. The condiions (3.27) and (3.28) do no conain he unknown parameers p 3 and p 1 (). In he paricular case L = cl wih ϕ () = 1 from (3.28) we ge he condiion (3.22) again. Finally, we briefly deal wih he cases where in Theorem 3.1 he funcion is sricly decreasing and he inequaliies (3.19) and (3.27) in Theorems 3.1 and 3.2, respecively, are fulfilled wih he opposie signs. In paricular, his is he case if L = cl and ϕ() = ϕ () = 1 on [, f ]. Then he above analysis shows ha he unique opimal soluion of problem 1 is ˆV () = { for a.a. [, ) A for a.a. (, f ] (3.29)
15 346 Werner Krabs, Lohar von Wolfersdorf where (, f ) is he (unique) soluion of he equaion ϕ ()L ( ˆM())d = B wih ˆM() = for [, ] and ˆM() = A δ [dδ( 1] if δ >, A( ) if δ = for [, f ] following from (3.17) and (2.3), (2.4) wih (3.29) again. In case of he condiions (3.19) or (3.27), (3.28) for a resisance facor ϕ (wih some associaed ϕ ) we say ha we have srong resisance of he umor cells agains he drug, and in case of hese condiions wih he opposie sign weak resisance. 4 Soluions of he second problem Problem ( ), (2.5), (2.9), (2.11) possesses he Hamilonian H(, y, M, U, V, p 1, p 2, p 3, λ ) = (f(e y ) ϕ()l(m))(p 1 λ ) + (V δm)p 2 + V p 3 (4.1) wih he parameer λ and he adjoin sae funcions p k, k = 1, 2, 3. If (ŷ, ˆM, Û, ˆV ) is an opimal quadruple, by he maximum principle [9], here exis a number λ and hree funcions p k C 1 [, f ], k = 1, 2, 3 wih (λ, p 1, p 2, p 3 ) (,,, ) saisfying he differenial equaions ṗ 1 () = f (eŷ() )eŷ() (p 1 () λ ) (4.2) ṗ 2 () = ϕ()l ( ˆM())(p 1 () λ ) + δp 2 () (4.3) in [, f ] and he ransversaliy condiions in f ṗ 3 () = (4.4)
16 Two opimal conrol problems in cancer chemoherapy 347 p 1 ( f ) =, p 2 ( f ) =, and p 3 ( f ), p 3 ( f )(Û( f ) B) = (4.5) such ha for a.a. [, f ] he maximum condiion ˆV ()(p 3 () + p 3 ) = max V A [V (p 2() + p 3 )] (4.6) holds. By (4.4), (4.5) p 3 is a nonposiive consan which vanishes if Û( f ) < B. We remark ha in he limi case δ = in view of (2.3), (2.4) and (2.11) he quaniies U and M coincide. Hence U, p 3 could be omied and formally p 2 () + p 3 replaced by a new p 2 (). We define he funcions p 1 and g as in problem 1 wih he relaions (3.7)  (3.9). Furher we have he relaions (3.11)  (3.14) for p 2 if we replace he funcion h in (3.1) by h () = ϕ()l ( ˆM(), p 1 (), [, f ]. (4.7) Discussing he opimal soluions of problem 2 we disinguish he wo cases B f A and B < f A. For B T f A he obvious soluion is ˆV () = A for a.a. [, f ]. For B < f A we have he equaliy Û( f ) = ˆV ()d = B (4.8) and he inequaliies p 3 <, λ >, and p 1 () < in [, f ] which can be shown as above in problem 1. By (4.7) his implies h () < in [, f ] which by (3.11) and (3.14) gives ṗ 2 () δp 2 () <, p 2 () > in [, f ). (4.9) If addiionally ϕ( f ) > hen also h ( f ) < and consequenly ṗ 2 ( f ) <. From (4.9) we obain a firs resul abou he form of he opimal soluions in he case B < f A. LEMMA 4.1
17 348 Werner Krabs, Lohar von Wolfersdorf For B < f A he opimal soluions of problem 2 do no conain a soluion par of he form ˆV () = Aa.e. for [τ, f ], τ [, f ) (4.1) and if ϕ( f ) > hey do no conain singular pars in inervals of he form [τ, f ] wih τ [, f ). Proof: The asserion (4.1) for τ = follows from (4.14). For τ > we have p 2 () + p 3 in (τ, f ] and p 2 (τ) + p 3 = implying ṗ 2 (τ), bu since p 2 (τ) = [p 2 ( f ) + p 3 ] by (4.9) i mus be ṗ 2 (τ) < δp 2 (τ) and ṗ 2 <. The proof for he singular pars is a consequence of he condiion ṗ 2 () = in [τ, f ] which leads o a conradicion o ṗ 2 ( f ) < from (4.9). Lemma 4.1 shows ha he opimal soluions of problem 2 end wih an inerval of zeroherapy if ϕ( f ) >. We now give a sufficien condiion for he opimal soluions being of he (in pracice desired) bangbang conrol ype. LEMMA 4.2 Under he condiions B < f A and p 2 () < in (, f ) (4.11) he opimal soluion of problem 2 is uniquely deermined and has he form ˆV () = { A for a.a. [, ) for a.a. (, f ] (4.12) where = B/A (, f ). Proof. Since ṗ 2 () in (, f ) he opimal soluion does no conain singular pars. Furher, i does no conain pars of he forms ˆV () = for a.a. [, τ], τ (, f ] and ˆV () = { for a.a. (1, τ) A for a.a. (τ, 2 ) ( 1 < τ < 2 f )
18 Two opimal conrol problems in cancer chemoherapy 349 The firs one is impossible for τ = f because of (4.8) and for τ < f since we could have p 2 () + p 3 in (, τ) and p 2 (τ) + p 3 implying ṗ 2 (τ). For he second one we obain p 2 () + p 3 in ( 1, τ) and p 2 () + p 3 in (τ, 2 ) yielding ṗ 2 (τ) again. This proves he form (4.12) of he opimal soluion ˆV wih unique value following from (4.8). The proof can also be given direcly by using he fac ha (4.11) implies p 2 () > for all [, f ) and discussing he wo cases p 2 () + p 2 and p 2 () + p 3 >. By equaions (3.11) and (4.7) he condiion (4.11) is equivalen o where δp 2 () < φ() in (, f ) φ() = h () = ϕ()l ( ˆM() p 1 () > in [, f ) (4.13) wih φ( f ) and by (3.14) equivalen o he inegral inequaliy ψ() φ() δ e δ( s) φ(s)ds > in (, f ). (4.14) If (4.14) holds he opimal soluion is given by (4.12). In paricular, his is fulfilled for all posiive funcions ϕ in he limi case δ = suggesing ha (4.14) is no a oo srong condiion on ϕ for sufficienly small δ >. This can be underlined in he simple case of Gomperz growh (cf. [7, 1, 17, 2325]) where and a linear loss funcion In his case we find ha f(t ) = λln θ, T >, (λ >, θ > ) T L(M) = km, M, (k > ). g() = f (eŷ() )eŷ() = λ, h () = kϕ() p 1 ()e λ, [, f ], and (4.14) urns ou o be equivalen wih
19 35 Werner Krabs, Lohar von Wolfersdorf ϕ()e (λ δ) δ ϕ(s)e (λ δ)s ds > for all (, f ). (4.15) If we pu ϕ() = e (λ δ), [, f ], and assume ha λ > δ, hen i follows ha ϕ C 1 [, f ], ϕ() = 1, ϕ() < and ϕ() > for all [, f ]. Furher (4.15) urns ou o be equivalen o (1 δ( f ) > for all (, f )) δ 1 <. This shows ha (4.15) can be saisfied for sufficienly small δ > and a suiable choice of ϕ. The inequaliy (4.14) is fulfilled if we have φ() < in (, f ), (4.15) since inegraion by pars of he inegral in (4.14) yields ψ() = e δ [e δ f φ( f ) e δs φ(s)ds] > in (, f ) due o φ( f ) and (4.15). Differeniaing (4.13) and using p 1 = g p 1 we furher have where φ() = p 1 ()L ( ˆM())[ ϕ() + {g() + m()}ϕ()] m() = 1 d L ( ˆM(u) d [L ( ˆM())] = L ( ˆM(u) ˆM() L ( ˆM()). (4.16) Therefore, in view of p 1 () < in [, f ] and L (M) >, condiion (4.15) is equivalen o he differenial inequaliy
20 Two opimal conrol problems in cancer chemoherapy 351 ϕ() + [g() + m()]ϕ() < in (, f ) (4.17) where m = m() is given by (4.16) and g = g( ˆT ) by (3.17), i.e. g() = f ( ˆT ()) ˆT () <, [, f ]. (4.18) Condiion (4.17) has he same form as condiion (3.22) and is like his in general an implici condiion on ϕ. Summing up, by Lemma 4.2 and (4.14)  (4.18) we obain THEOREM 4.3 (i) Under he condiions B < f A and (4.14) he opimal soluion of problem 2 is uniquely deermined and has he form (4.12). ii) Asumpion (4.14) is saisfied if he condiion (4.17) wih (4.18) and (4.16) holds rue. REMARKS. For a linear loss funcion L we have m() = in [, f ] and he condiion (4.17) reduces o (3.22). As for problem 1 we say in case of (4.14) for a resisance facor ϕ ha here is a srong resisance of he umor cells agains he drug. We conclude he paper working ou he simple case of Gomperz growh (cf. [7, 1, 17, 2325]) f(t ) = λln θ T (λ >, θ > ) (4.19) wih a linear loss funcion L(M) = km (k > ) as an example for wha can happen for general resisance. In his case we have for y = lnt he explici expression y() = lnt e λ + λlnθ[1 e λ ] k e λ( s) ϕ(s)m(s)ds
21 352 Werner Krabs, Lohar von Wolfersdorf and he minimum condiion for y( f ) leads o he maximum condiion which can be wrien in he form e λs ϕ(s)m(s)ds max where p() = e δ p()v ()d max (4.2) e (λ δ)s ϕ(s)ds. (4.21) The maximum problem (4.2) where V L (, f ) saisfies he resricions (2.5) and (2.7) is a linear problem of he form of he NeymanPearson lemma and can be solved in explici form. Le be B < f A. For (4.19) he condiion (4.17) is equivalen o he inequaliy d d [eλ ϕ()] < in (, f ). If his is fulfilled he problem has he soluion (4.12). We consider furher he opposie case ha d d [eλ ϕ()] > in (, f ) (4.22) incorporaing he limi case of nonresisance ha ϕ() 1 on [, f ]. For δ > he funcion (4.21) has he derivaive ṗ() = e δ [F () C], [, f ] where C = e (λ δ) f ϕ( f ), F () = e δs d ds [eλs ϕ(s)]ds.
22 Two opimal conrol problems in cancer chemoherapy 353 Under he assumpion (4.22) he funcion F is sricly decreasing in [, f ] from he value F () = e δs d ds [eλs ϕ(s)]ds > o F ( f ) =. Hence we have wo cases (i) F () C where ṗ() < in (, f ) so ha p() is sricly decreasing in [, f ] and (ii) F () > C where here exiss a unique (, f ) such ha p() is sricly increasing in [, ] and sricly decreasing in [, f ] ill p( f ) =. In case (i) he opimal soluion is given by (4.12). In case (ii) he opimal soluion has he form ˆV () = { a.e. in [, 1 ) and ( 2, f ] A a.e. in ( 1, 2 ) (4.23) where 1, 2 wih < 1 < < 2 < f are uniquely deermined by he equaions A( 2 1 ) = B, p( 1 ) = p( 2 ). In he paricular case ϕ() = 1 on [, f ] we have (e (λ δ) f e (λ δ) )e δ if λ > δ p() = ( f )e δ if λ = δ (e (λ δ) e (λ δ) f )e δ if λ < δ and C = e (λ δ) f, F () = { λ λ δ [e(λ δ) f 1] if λ δ λ f if λ = δ. Therefore, case (i) occurs if λ f 1 for λ = δ, δe (λ δ) f λ for λ > δ, and λe (δ λ) f δ for λ < δ, and case (ii) under he opposie inequaliies. We remark ha he opimal soluion (4.23) in case (ii) sars and ends wih an inerval of zeroherapy. References [1] F.H. Clarke, Opimizaion and Nonsmooh Analysis, Wiley, New York, [2] A.J. Coldman, J.H. Goldic, A model for he resisance of umor cells o cancel chemoherapeuic agens, Mah. Biosci. 65 (1983)
23 354 Werner Krabs, Lohar von Wolfersdorf [3] A. J. Coldman, J.M. Murray, Opimal conrol for a sochasic model of cancer chemoherapy, Mah. Biosci. 168 (2) [4] M.I.S. Cosa, J.M. Boldrini, and R.C. Bassanezi, Drug kineics and drug resisance in opimal chemoherapy, Mah. Biosci. 125 (1995) [5] M. Eisen, Mahemaical Models in Cell Biology and Cancer Chemoherapy, Lecure Noes in Biomahemaics, Vol. 3, SpringerVerlag, Berlin HeidelbergNew York, [6] K.R. Fiser, J.C. Panea, Opimal conrol applied o compeing chemoherapeuic cellkill sraegies, SIAM J. Appl. Mah. 63 (23) [7] C.L. Frenzen, J.D. Murray, A cell kineics jusificaion for Gomperz equaion, SIAM J. Appl. Mah. 46 (1986) [8] J.H. Goldie, Drug resisance in cancer: A perspecive, Cancer and Measasis Reviews 2 (21) [9] A.D. Joffe, V.M. Tichomirov, Theorie der Exremalaufgaben, Deuscher Verlag der Wissenschafen, Berlin, [1] W. Krabs, S.W. Pickl, Modelling, Analysis and Opimizaion of Biosysems, SpringerVerlag, Berlin Heidelberg New York, 27. [11] U. Ledzewicz, T. Brown and H. Schüler, Comparison of opimal conrols for a model in cancer chemoherapy wih L 1  and L 2 ype objecives, Opimizaion Mehods and Sofware 19 (24) [12] U. Ledzewicz, H. Schüler, Drug resisance in cancer chemoherapy as an opimal conrol problem, Discree and Coninuous Dynamical Sysems, Ser. B 6 (26) [13] R.B. Marin, M.E. Fisher, R.F. Minchin, and K.L. Teo, A mahemaical model of cancer chemoherapy wih an opimal selecion of parameers, Mah. Biosci. 99(199) [14] R.B. Marin, M.E. Fisher, R.F. Mincin, and K.L. Teo, Lowinensiy combinaion chemoherapy maximizes hos survival ime for umors conaining drugresisan cells, Mah. Biosci. 11 (1992)
24 Two opimal conrol problems in cancer chemoherapy 355 [15] A.S. Maveev, A.V. Savkin, Applicaion of opimal conrol heory o analysis of cancer chemoherapy regimes, Sysems and Conrol Leers 46 (22) [16] J.M. Murray, Opimal conrol for a cancer chemoherapy problem wih general growh and loss funcions, Mah. Biosci. 98 (199) [17] J.M. Murray, Some opimal conrol problems in cancer chemoherapy wih a oxiciy limi, Mah. Biosci. 1 (199) [18] J.M. Murray, Opimal drug regimes in cancer chemoherapy for single drugs ha block progression hrough he cell cycle, Mah. Biosci. 123 (1994) [19] L.G. de Pillis, W. Gu, K.R. Fiser, T. Head, K. Maples, A. Murugan, T. Neal, and K. Yoshida, Chemoherapy for umors: An analysis of he dynamics and a sudy of quadraic and linear opimal conrols, Mah. Biosci. 29 (27) [2] L.G. de Pillis, A. Radunskaya, A mahemaical umor model wih immune resisance and drug herapy: An opimal conrol approach, J. Theor. Medicine 3 (21), [21] L.G. de Pillis, A. Radunskaya, The dynamics of an opimally conrolled umor model: A case sudy, Mah. Compu. Modelling 37 (23) [22] G.W. Swan, Opimal conrol analysis of a cancer chemoherapy problem, IMA J. Mah. Appl. Med. Biol. 4 (1987) [23] G.W. Swan, Applicaions of Opimal Conrol Theory in Biomedicine, Marcel Dekker Inc., New York Basel [24] G.W. Swan, Role of Opimal Conrol Theory in Cancer Chemoherapy, Mah. Biosci. 11 (199) [25] G.W. Swan, T.L. Vincen, Opimal conrol analysis in he chemoherapy of IgG muliple myeloma, Bull. Mah. Biology 39 (1977) [26] A. Swierniak, U. Ledzewicz, and H. Schäler, Opimal conrol for a class of comparmenal models in cancer chemoherapy, In. J. Appl. Mah. Compu. Sci. 13 (23)
Newton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationRenewal processes and Poisson process
CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationBasic Assumption: population dynamics of a group controlled by two functions of time
opulaion Models Basic Assumpion: populaion dynamics of a group conrolled by wo funcions of ime Birh Rae β(, ) = average number of birhs per group member, per uni ime Deah Rae δ(, ) = average number of
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationOPTIMAL PRODUCTION SALES STRATEGIES FOR A COMPANY AT CHANGING MARKET PRICE
REVISA DE MAEMÁICA: EORÍA Y APLICACIONES 215 22(1) : 89 112 CIMPA UCR ISSN: 1492433 (PRIN), 22153373 (ONLINE) OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY A CHANGING MARKE PRICE ESRAEGIAS ÓPIMAS DE
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationand Decay Functions f (t) = C(1± r) t / K, for t 0, where
MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationTime variant processes in failure probability calculations
Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TUDelf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More information4. The Poisson Distribution
Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationDoctoral Students Conference th 7 th July, 2009 Thessaloniki, Greece
Docoral Sudens Conference h 7 h July, 2009 Thessaloniki, Greece 6 h Susainable Developmen and he Bulgarian Naional Sraegy for Regional Developmen 20052015: 2015: an applicaion of a hree dimensional heoreical
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationUsing RCtime to Measure Resistance
Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a builin ADC (Analog o Digial Converer)
More informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationA NOTE ON UNIT SYSTEMS
Tom Aage Jelmer NTNU eparmen of Peroleum Engineering and Applied Geophysics Inroducory remarks A NOTE ON UNIT SYSTEMS So far, all equaions have been expressed in a consisen uni sysem. The SI uni sysem
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms)
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More information13 Solving nonhomogeneous equations: Variation of the constants method
13 Solving nonhomogeneous equaions: Variaion of he consans meho We are sill solving Ly = f, (1 where L is a linear ifferenial operaor wih consan coefficiens an f is a given funcion Togeher (1 is a linear
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationOptimal market dealing under constraints
Opimal marke dealing under consrains Eienne Chevalier M hamed Gaïgi Vahana Ly Vah Mohamed Mnif June 25, 2015 Absrac We consider a marke dealer acing as a liquidiy provider by coninuously seing bid and
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationCommunication Networks II Contents
3 / 1  Communicaion Neworks II (Görg)  www.comnes.unibremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationStrategic Optimization of a Transportation Distribution Network
Sraegic Opimizaion of a Transporaion Disribuion Nework K. John Sophabmixay, Sco J. Mason, Manuel D. Rossei Deparmen of Indusrial Engineering Universiy of Arkansas 4207 Bell Engineering Cener Fayeeville,
More informationINSTRUMENTS OF MONETARY POLICY*
Aricles INSTRUMENTS OF MONETARY POLICY* Bernardino Adão** Isabel Correia** Pedro Teles**. INTRODUCTION A classic quesion in moneary economics is wheher he ineres rae or he money supply is he beer insrumen
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More information