On the age-, time- and migration dependent dynamics of diseases

On the age-, time - and m igration dep endent dynamics o f diseases Ralph Brinks Institute for Biometry and Epidemiology German Diab e tes Cen ter Duesseldorf, German y This pap er generalizes a previously published differen tial equation that de- scrib es the relation betw een the age-sp ecific incidenc e, remission, and mor- talit y of a disease with its prev alence. The underlying mo del is a simple compartmen t mo del with three states (illness-death mo del). In con trast to the former w o r k, migration- and calendar time-effects are included. As an ap- plication of the theoretical findings, a h ypo thetical example of an irrev ersible disease is treated. Keywor ds: Incidence; Remission; Mort a lit y; Prev alence; Illness-Death Mo d el; Com- partmen t mo del; Epidemiology . 1 Intro duction With a v iew to basic epidemiological parameters such as incidence, prev alence and mor- talit y o f a disease, it has b een pro v en useful to consider simple illness-death mo dels [8] as sho wn in Figure 1 . D ep ending on the con text, sometimes these are referred to as state mo dels or compartmen t mo dels. Here w e consider three states: Normal or non- diseased with nu m b er of p eople denoted as S (susceptible), the diseased s tate with num b er C (cases) and the death state. The transition in tensities b etw een the states henceforth are denoted with the sym b o ls as in F igure 1: incidence i , remission r and mortalit y rates 1 m 0 and m 1 . In general, the in tensities dep end on calendar time t , ag e a and sometimes also on the duration d of the disease. 1 The express ions r ate and density ar e syno nymously used in this article. 1 Figure 1: Simple illness-death mo del Mo dels of this kind a r e quite common, see fo r example [8], [9] or the text b o ok [7]. Murra y and Lop ez ( [1 3] and [14]) ha v e considered suc h a compart men t mo del with rates b eing indep enden t from calendar time t and duration d . In the con text of the Glob a l Bur den of D i s e ase s tudy of the World He al th Or g a nization the y used f ollo wing system of o r dinary differen tial equations (OD Es) to describ e the transitions b et w een the three states: d S d a = − ( i + m 0 ) · S + r · C d C d a = i · S − ( m 1 + r ) · C . (1) By this system the c hanges in t he nu m b ers of the non-diseased and diseased p ersons aged a are related to the in tensities as in Figure 1. Age pla ys here the role of temp ora l progression. This homogeneous linear system o f ODEs lo oks relativ ely harmless, but is limited due to it s hea vy assumptions. By a n easy calculation it can b e sho wn that Eq. (1) implies the p opulation b eing stationary . Let N ( a ) := S ( a ) + C ( a ) denote the total n um b er of p ersons aliv e in the p opulation aged a . F or a ∈ [0 , ω ] with N ( a ) > 0 define the age-sp ecific prev alence p ( a ) := C ( a ) C ( a ) + S ( a ) . (2) Then from Eq. (1) it follow s d N d a = d S d a + d C d a = − m 0 · S − m 1 · C = − N · [(1 − p ) · m 0 + p · m 1 ] . The term (1 − p ) · m 0 + p · m 1 is the ov erall mortality m in the p o pulation. Hence, it ho lds d N d a = − m · N , whic h is the defining equation of a stationa ry p opulat ion, [16]. Although the mo del of a stationar y p opulation is wide ly used in demograph y , real p opulations merely are stationar y . Moreo v er, the inclusion of the v alues S and C is disturbing. It w o uld b e b etter if Eq. (1) could be expressed in terms o f the a g e-sp ecific prev alence 2 (2), what indeed can b e ac hiev ed. In [1] it has b een sho wn, that syste m (1) can b e transformed in to the follow ing one-dimensional OD E of Riccati t ype: d p d a = (1 − p ) ·  i − p · ( m 1 − m 0 )  − r · p. (3) The imp ortance of Eq. (1) and (3) is ob vious. F or giv en incidence-, remission- and mortalit y-rates plus an initial condition, the age profile of the num b ers of patien ts a nd the prev alence is uniquely determined, resp ectiv ely . T o state it clearly , the “forces” inci- dence, remission and mortality uniquely prescrib e the prev alence - not only qualitativ ely but in these quan titativ e t erms. This is called the forwar d pr oblem : w e infer from t he causes – the f o rces – to the effect – the nu m b ers of diseased or the prev a lence, resp ec- tiv ely . If in the scalar Riccati OD E (3) the a ge-profiles of the prev alence, mortality and remission are kno wn, one can directly solve Eq. (3) fo r the incidence. This is the inve rs e pr oblem – we conclude from the effec t to the c ause. This allows, for example, cross- sectional studies b eing used f o r incidence estimates, where otherwise lengthy follow-up studies are needed. F or an example on real da t a , see [1]. Rec en t ly , it has b een pro v en that the in v erse problem is ill-p osed [2]. The article is organized as follo ws: I n the next section Eq. (3) is generalized allo wing dep endency on calendar time a nd migratio n. The cen tral result is a partial differen tial equation (PDE). Similar to the O DE, in the general case there is a forward and an in v erse problem f o r the PDE, to o. These are analyzed in a simulated register data of a h ypot hetical c hro nic disease in the section thereafter. Finally , the results are summed up. 2 General equation of disease dynamics In this section the simple illness-death mo del of Fig ure 1 is generalized. The rates i, r , m 0 and m 1 henceforth dep end on age a a nd calendar time t , but are ass umed to b e indep enden t from the duration d . F urthermore, let the num b ers of the non-diseased S ( t, a ) and diseased p ersons C ( t, a ) a ged a at time t b e non-negativ e a nd partia lly differen tiable. Define N ( t, a ) := S ( t, a ) + C ( t, a ). Additionally , let σ ( t, a ) and γ ( t, a ) denote those pro p ortions of N ( t, a ), suc h that σ ( t, a ) · N ( t, a ) and γ ( t, a ) · N ( t, a ) are the net migration r a tes of non- diseased a nd diseased p ersons aged a a t time t , resp ectiv ely:  ∂ ∂ t + ∂ ∂ a  S = σ · N − ( i + m 0 ) · S + r · C  ∂ ∂ t + ∂ ∂ a  C = γ · N + i · S − ( m 1 + r ) · C . (4) After in tro ducing the age-sp ecific prev a lence p ( t, a ) in year t , p ( t, a ) := C ( t, a ) C ( t, a ) + S ( t, a ) , 3 for ( t, a ) ∈ D := { ( t, a ) ∈ [0 , ∞ ) 2 | C ( t, a ) ≥ 0 , S ( t, a ) ≥ 0 , C ( t, a ) + S ( t, a ) > 0 } the system (4) can b e transformed in to an equation similar to (3): Theorem 2.1. L et S ( t, a ) and C ( t, a ) b e given by Eq. (4) , then p ( t, a ) is p artial ly differ entiable in D and it hol d s  ∂ ∂ a + ∂ ∂ t  p = (1 − p ) [ i − p ( m 1 − m 0 )] − r p + µ, (5) wher e µ := γ (1 − p ) − pσ describ es the imp act of migr ation. Pr o of. F ollows from applying the quotien t rule to p ( t, a ) = C ( t,a ) C ( t,a )+ S ( t,a ) and using (4). Ob viously , if the incidence- and mortality rates do not dep end on the calendar time t , then from Eq. (5 ) with µ ≡ 0 it follow s (3 ). Hence, Eq. (3) do es not dep end on the stationary p opulatio n assumption. F o r applications in epidemiology it is imp ortan t that solutions of Eq. (5) a re me an- ingful , i.e. p ( t, a ) ∈ [0 , 1] for all ( t, a ) ∈ D . Therefor w e note: Theorem 2.2. F or al l ( t, a ) ∈ D fol low i n g statements ar e e quivalen t: (1) p ( t, a ) = C ( t,a ) S ( t,a )+ C ( t,a ) is a solution of Eq. (5) . (2) S ( t, a ) = (1 − p ( t, a )) · N ( t, a ) and C ( t, a ) = p ( t, a ) · N ( t, a ) ar e solutions to Eq. (4) . Pr o of. This follo ws b y inserting the expressions into the PDEs. By Theorem 2.2 a solution p ( t, a ) o f Eq. (5) can b e written as p ( t, a ) = C ( t,a ) N ( t,a ) with N ( t, a ) = S ( t, a ) + C ( t, a ). F o r ( t, a ) ∈ D this implies p ( t, a ) ∈ [0 , 1] . The migration term µ will b e analyzed f urther now. Let ϕ := σ + γ b e the ov erall migration rate. W e split all migrat ion rates f , f ∈ { ϕ, σ , γ } in to a p ositive part f + ≥ 0 (immigration) and a negative part f − ≥ 0 (emigration): f = f + − f − , for f ∈ { ϕ, σ, γ } . Moreo v er, for ϕ − ( t, a ) > 0 define p ( m ) − ( t, a ) := γ − ( t,a ) ϕ − ( t,a ) the pre v a lence of the disease in the emigran ts and for ϕ + ( t, a ) > 0 define p ( m ) + ( t, a ) := γ + ( t,a ) ϕ + ( t,a ) the prev a lence in the immigran ts. Prop osition 2.1. With the n o tations as ab ove it holds µ ( t, a ) =                ϕ + ( t, a ) · p ( m ) + ( t, a ) − ϕ − ( t, a ) · p ( m ) − ( t, a ) − ϕ ( t, a ) · p ( t, a ) , for ϕ − ( t, a ) , ϕ + ( t, a ) > 0 ; ϕ + ( t, a ) · h p ( m ) + ( t, a ) − p ( t, a ) i , for ϕ − ( t, a ) = 0 , ϕ + ( t, a ) > 0 ; − ϕ − ( t, a ) · h p ( m ) − ( t, a ) − p ( t, a ) i , for ϕ − ( t, a ) > 0 , ϕ + ( t, a ) = 0 ; 0 , for ϕ − ( t, a ) = ϕ + ( t, a ) = 0 . 4 Pr o of. F or all ( t, a ) ∈ D it holds µ = γ − ϕ · p. By splitting this expression in to p o sitiv e and negativ e parts, the Prop osition fo llows. With the assumption t ha t the prev alence of those aged a at time t who immigrate is the same of those who emigrate, say p ( m ) ( t, a ), then it holds µ = ϕ  p ( m ) − p  . (6) Hence, if the prev alence p ( m ) of the migran ts is the same as of those who sta y , p ( m ) ≡ p , the c ha nge in prev alence ( ∂ ∂ a + ∂ ∂ t ) p do es not dep end on migra tion. This is an imp or t an t result, b ecause in illness-death mo dels the assumption of absence of migration is often made. In our framew ork this restriction is not necessary . Ev en if there is migra tion, but the prev alence in the migra nts is the same as in the residen t p opulation, then the prev alence is not affected b y migr a tion. The solution of Eq. (5) can b e obtained b y the metho ds of characteris tics [15]. Let an initial condition o f the form p ( a, 0) = p 0 ( a ) b e g iv en, then we ha v e a so called Cauc hy problem, whic h has a unique solution if the right-hand side of the PDE is sufficien tly smo oth [15]. This solution is calculated a s follo ws. Assume, the prev alence fo r those aged ˜ a in y ear ˜ t has t o b e calculated. First, rearrang e ( ∂ ∂ t + ∂ ∂ a ) p suc h that  ∂ ∂ a + ∂ ∂ t  p = α ( t, a ) + β ( t, a ) p + γ ( t, a ) p 2 . Second, solv e the initial v alue problem giv en by following Riccati OD E: d y ( τ ) d τ = α ( τ + a 0 , τ ) + β ( τ + a 0 , τ ) y + γ ( τ + a 0 , τ ) y 2 , (7) and initial v alue y (0) = p 0 ( a 0 ) where a 0 := ˜ a − ˜ t. Then, an easy calculation shows that y ( ˜ t ) = p ( ˜ t, ˜ a ) is the desired v alue. 3 Application on a simulated register In this section, the a pplicatio n of the ab ov e-form ulated Cauc hy problem on a sim ulated register of a c hronic disease is sho wn. Since the disease is assumed to b e irrev ersible, it holds r ≡ 0 . F irst, we address a direct problem: F ro m given ag e-sp ecific prev alence in some p oint in time t 0 w e wan t to deduce the age-sp ecific prev alence in t 1 , t 1 > t 0 , b y applying Eq. (5) with µ ≡ 0. Second, an in v erse problem is formulated. Assume in the ye ar t 0 the functions p 0 = p ( t 0 , · ), i ( t 0 , · ), m 0 ( t 0 , · ) and m 1 ( t 0 , · ) w ere measured. If in y ear t 1 , t 1 > t 0 , the age profile of the prev alence p ( t 1 , · ) is given, the question arises: ho w has the course of the a g e-sp ecific incidence ch anged in the meantime? This is an in v erse problem, b ecause w e infer from the effect (prev alence in t 1 ) on the causes. Here w e will form ulate a simple, straigh t forw ard solution by an optimization approac h. Both problems will b e treated based on da t a of a sim ula t ed register. The register is designed suc h that in a p erio d of 150 years a ll p ersons are track ed from birth to death. 5 F o r eac h p erson, the date of an ev en tual diag nosis of the c hronic disease is recorded. F o r the sim ulatio n, the fo llowing assumptions are placed as a basis: 1. In eac h calendar y ear 0 to 150 2,000 p eople are b orn. The births during the calendar y ear follow a uniform distribution. 2. The morta lit y of the non-diseased p ersons is of Strehler-Mildv an t yp e and is giv en b y the equation m 0 ( t, a ) = exp( − 10 . 7 + 0 . 1 a ) · (1 − 0 . 00 2 ) ( t − 20) + . The notatio n ( t − 20 ) + denotes the p ositiv e comp onen t of the express ion ( t − 20). The exp onen tial term approx imates the curren t mortality of men in German y , the second factor tak es the increasing life exp ectancy into accoun t. 3. The incidence is described by the equation i ( t, a ) = ( a − 30) + 3000 · 0 . 99 ( t − 50) + . (8) 4. The relativ e risk of death is constan t for all ages and times: R ( t, a ) = m 1 ( t, a ) m 0 ( t, a ) = 2 . After the sim ulation, eac h p erson in the register is represen ted b y fo ur pieces o f infor- mation: 1) A unique iden tification num b er (an integer), 2) Calendar y ear of birth, 3) The p erson’s age in y ears at diagno sis (0 if the p erson do es not fall ill), 4) Age of death of the p erson in ye ars. En tr ies 2) - 4) in the registe r are giv en to three decimals, which corresponds to a precision of one da y . The ide n tification n um b er of the p erson is an o ngoing c oun ter. The date of birth (in calendar y ears) is giv en b y the sim ulat ed year, the decimals a re dra wn from a uniform dis tribution [0 , 1) . T o dec ide if a th us far non-diseased p erson b orn in y ear τ b ecomes ill or dies without the disease, a comp eting risk approach in a discrete ev en t sim ulation (DES) is accomplished. Based on the cum ulative distribution function of the common risk (to t a l intensit y i ( τ + a, a ) + m 0 ( τ + a, a )), the age a 0 of ev ent is dra wn by the inv erse transform sampling (in v ersion metho d, [5]). Based on a comparison b et w een i ( τ + a 0 , a 0 ) and m 0 ( τ + a 0 , a 0 ) , it is decided whether the onset of the disease or the de ath without disease o ccurred. In the first cas e a 0 represen ts the 6 age at disease’s onset, in t he second case, a 0 is the a ge of death. If the p erson gets the disease, the age of death is sim ulated (conditional on reac hing the age a 0 ). As in the calendar y ears 0 to 150 exactly 20 00 p eople are b o rn eve ry y ear, the h y- p othetical register con tains 151 · 2000 = 3 02 , 000 p ersons. Then, the ev ents of the h ypot hetical register are tra nsformed in to a L exis diagr a m of fiv e ye ars interv als [10]. This allo ws an e asy extraction of the p erson-y ears and the n um b ers of ev en ts in the corresp onding a g e- and p erio d classes. In b oth test cases, in the direct and the inv erse pro blem, w e assume information to b e giv en only in t w o p oints in time, t 0 and t 1 . Of course, three or more p oin ts in time w o uld b e a dv a n tageous, but with re sp ect to applicabilit y in epide miological con texts, the test problems try to mimic a minimalistic setting. 3.1 Di rect p ro blem Assume w e ha v e measured the a ge profile p 0 = p ( t 0 , · ) of the prev alence in t 0 , and the age-sp ecific incidence i ( t 0 , · ) and mortality densities m 0 ( t 0 , · ) and m 1 ( t 0 , · ). F urthermore, at a later p oint in time t 1 > t 0 let the age-sp ecific r ates i ( t 1 , · ) and mortalities m 0 ( t 1 , · ) and m 1 ( t 1 , · ) be giv en. T he direct problem refers to the question: what can b e said ab out the age-sp ecific prev alence p ( t 1 , · ) in t 1 ? T o answ er t his question, age-sp ecific incidence and mortality ra t es at tw o time p oints t 0 = 120 and t 1 = 140 (y ears) a re extracted from the register. In addition, the age- sp ecific prev alence p ( t 0 , · ) is collected at t 0 . Figure 2 show s the extracted a g e-sp ecific incidence densit y (dashed lines) in t 0 (red) and t 1 (blue) in comparison with the theo- retical v alues (solid lines). No w consider the Cauch y problem that is giv en b y Eq. (5) with the initial condition p ( t 0 , · ) = p 0 . F or the solution one needs t he functions i ( t, · ), m 0 ( t, · ) and m 1 ( t, · ) for all time p oints t b et wee n t 0 and t 1 . F or this, the function v alues are interpola ted affine- linearly . The initial v alue problem Eq. (7) is solved numeric ally using the MA TLAB 2 function ode45 . If w e compare the n umerical solution o f the Cauc hy pro blem in y ear t 1 = 140 with the actually observ ed prev alence in the year 140, one gets the result as sho wn in Figure 3. Visually this giv es a f a irly go o d agreemen t b et w een the predicted curv e with the actually observ ed age-sp ecific prev alence. The ma ximum a bsolute deviation is 0.0146, whic h means that in this example the prev alence can b e pre dicted up to 1.5 percen t p oints . The larg est deviation is in the oldest age class, when w e hav e only a few cases of the disease. 3.2 I nverse p roblem In epidemiological studies, it is more lab orious to measure incidence rates than prev a - lences. Hence, in practice, the follo wing in vers e problem is m uc h more imp o rtan t than 2 The MathW ork s, Natic k, Massach usetts, USA 7 0 20 40 60 80 100 0.000 0.005 0.010 0.015 Age (years) Incidence Figure 2: Age-sp ecific incidence densit y extracted from the register (dashed lines) at t 0 = 120 (red) a nd t 1 = 140 (blue) in comparison with the theoretical v alues (solid lines). the direct problem of the previous section. Assume in the ye ar t 0 = 12 0 the functions p 0 = p ( t 0 , · ), i ( t 0 , · ), m 0 ( t 0 , · ) and m 1 ( t 0 , · ) are kno wn. Moreov er, in y ear t 1 = 140 let the age profile of the prev alence p ( t 1 , · ) b e give n. The functions m 0 ( t 1 , · ) and m 1 ( t 1 , · ) are also assumed to b e kno wn (f or example from other epidemiological studie s). The question then is, how w ell the incidence i ( t 1 , · ) in the y ear t 1 can b e deriv ed from this information. F or simplicit y , w e assume that the incidence of i ( t 1 , · ) in t 1 can b e expressed as a pro duct i ( t 1 , · ) = i ( t 0 , · ) · (1 − h ) , (9) where h ∈ [0 , 1] . The upp er limit for h stems f r o m the fact, that incidence rates ar e non- negativ e. The low er limit reflects the prior kno wledge, that incidence has not increased in t 1 compared to t 0 : i ( t 1 , a ) ≤ i ( t 0 , a ) , for all a ∈ [0 , ∞ ) . Equation (9) corresp onds to a prop ortiona l hazards approa c h, whic h is used widely in epidemiology . T o solv e this in v erse problem, w e form ulate an optimization problem. F o r given h ∈ [0 , 1] a nd i ( t 0 , · ) b y Eq. ( 9 ) the function i ( t 1 , · ) is defined. If furthermore p ( t 0 , · ), 8 20 30 40 50 60 70 80 90 100 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Age (years) Prevalence Figure 3: Numerical solution of the direct problem (da shed line) compared to the ob- serv ed prev alence in y ear t 1 = 140 (solid line). m 0 ( t 0 , · ) , m 0 ( t 1 , · ) , m 1 ( t 0 , · ) and m 1 ( t 1 , · ) are kno wn, then we are in the situation to calculate a unique function ˆ p h ( t 1 , · ) b y solving the Cauc h y problem described in the pre- vious subsection 3.1. The solution ˆ p h ( t 1 , · ) of the direct problem dep ends on h. W e can compare ˆ p h ( t 1 , · ) with the measured prev alence p ( t 1 , · ) in the register. Th us, w e seek for h ∗ ∈ [0 , 1] that minimizes the Euclidean distance b et w een ˆ p h ( t 1 , · ) and p ( t 1 , · ) : h ∗ = arg min h ∈ (0 , 1) Z A t 1 | ˆ p h ( t 1 , a ) − p ( t 1 , a ) | 2 d a, (10) where A t 1 = { a ∈ [0 , ∞ ) | ( t 1 , a ) ∈ D } . Figure 4 sho ws the Euclidean distance b etw een the prev alence p ( t 1 , · ) in the register and the solution ˆ p h ( t 1 , · ) as a function of h . F ro m the graph in F igure 4 it is ob vious that the square of the distance is minimized at ab out h ∗ = 0 . 25. Since fr om the 50th calendar year the incidence decreases b y 1 % p er y ear and a p erio d of 20 y ears w as considered, a facto r 1 − h of ab o ut 0 . 99 20 = 0 . 82 = (1 − 0 . 18) is exp ected. The rev ealed v a lue h ∗ = 0 . 25 is ab o ut a factor of 1.4 to o large. 4 Discussion In this w ork w e dev elop ed a new equation linking incidence-, remission- and mortalit y- rates with prev alence of a disease. In con trast t o former w orks, the assumptions of 9 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.005 0.01 0.015 0.02 0.025 h Squared distance Figure 4: Euclidean distance (as in Eq. ( 1 0)) as a function of h . There is a unique minim um h ∗ = 0 . 25. stationary p opula t io ns, indep endence from calendar time and zero net migratio n hav e b een released. The new equation has a wide range of applicability in epidemiological, health care and health economic contexts . Ho w ev er, it has sev eral limitations. First, Eq. (5) needs the remission rate r and mortalit y ra te m 1 of the diseased to b e independent from the duration d of the disease. In real diseases indep endence from duration is o nly an approximation. F or man y infectious diseases, immun e response is dependen t on the time since o nset of the disease. Also in c hronic diseases duration since onset pla ys a ma jor role. F or example, the age- a nd sex-adjusted morta lit y due to coro nary heart disease roughly doubles for eac h 10 -y ear increase in diab etes duration. The all- cause mortality increases b y a factor of 1.2 p er 10-y ear duration, [4]. Second, alt hough the new equation is not limited to the case µ ≡ 0, in practical appli- cations informatio n ab out the health of immigran ts and emigra nts is seldom obta inable. By Prop osition 2.1 reasonable know ledge of prev alence in all migran ts is necess ary to accurately treat the case µ 6 = 0. T o give an example, coun tries with large-scale im- migration programs suc h as Canada o bserv e a so-called he althy immigr ant e ff e ct with resp ect to c hr o nic diseases : immigran ts are healthier than residen ts, [11]. Assumed that the emigrants from Canada ha v e the same prev alence as t he residen ts, it would follow µ 6 = 0 . How ev er, surv eys ab out the health stat us of emigrants are missing. The reason is ob vious, Canada’s ta xpay er-funded health care system is in terested in measuring health of those who immigrate, but not in those who emigrate. Hence, informatio n is la c king 10 and assumptions hav e to b e made. Third, Eq. (5) only considers prev alence in migrants at the momen t of emigration or immigration. O f course, large scale immigra tion is likely to c hang e the incidence of the disease in t he p opulation, b ecause immigran ts’ health adapts to the new en vironment. There are many examples where immigrants from the dev eloping coun tries increase in- cidence o f diab etes and related complications when adopting we sternized lifest yle, [1 2]. The opp osite ma y a lso b e true, in Canada immigra n ts contin ue to hav e a lo w er relative risk of c hronic conditions compared to the native-born, ev en man y y ears af t er immigra- tion, [11]. Beside theoretical considerations, w e use the new equation in a sim ulated register of a h ypot hetical c hr o nic disease. The register has been sim ulated b y Monte Carlo tech niques and has b een a na lyzed by a numerical implemen tation of the new equation. T o chec k the practical applicabilit y of the analysis, the sim ulation a nd the analysis ha v e b een strictly se parated, i.e. ne ither w as the P DE use d in simulating t he register, nor w as information other tha n explicitly men tioned, used as input for the simulation exploited in the analysis. The PDE has only b een used in the analysis of the direct and inv erse problem. In the direct problem, the prev alence at the later p oint in time t 1 could b e predicted from the prev alence in t 0 t w en t y y ears earlier with a high accuracy . Of course, the obt a ined accuracy is a result of the structure inheren t to the sim ulation. The solution of b oth, direct and in v erse problem, uses an a ffine-linear interpolatio n for the incidence- and morta lit y rates b etw een t 0 and t 1 . In t he sim ulated register this works w ell, b ecause it refle cts the trends in the inciden ce and mortalities. Affine-linear in terp olation will imp ose problems if the incidence trend turns around b et wee n t 0 and t 1 . An example for a ch ange of trends can b e found in [3 ]: from 1995 to 2004 incidence of diab etes is fo und to b e rising with a n a v erage of 5 .3% p er y ear in all age classes, and from 2 005 to 20 07 incidence is declining with 3.1% p er y ear. In the inv erse problem, the incidence in t 1 w a s r econstructed from the observ ed prev a- lence in t 1 . Pro vided that the righ t-hand side of Eq. (5) is sufficien tly smo oth, existence of h ∗ ∈ [0 , 1] f o llo ws from the contin uous dep endency of the solution of the Cauch y problem o n h from the compact in terv al [0 , 1]. Contin uous dep endency can b e seen b y noting that the solution constructed b y the metho ds o f characteristics inherits its smo othness prop erties from the smo othness of the right-hand side of Eq. (7), [6]. The question remains why the result (in terms of h ∗ ) is ab out a factor 1.4 to o large. The approac h in solving the inv erse problem is the pro p ortional hazards a ssumption Eq. (9). Indeed, the sim ulation considers a decline o f exp onen tial t ype, see Eq. ( 8 ). Although the exp onential in this case can approximated by an affine-linear in terp olation function quite w ell, it app ears that the solution of the inv erse pro blem reacts quite sensitiv ely o n inaccuracies. This is in line with our observ ation, that the inv erse problem is ill-p osed [2]. 11 References [1] Brinks R (20 1 1). A new metho d for deriving incidence rates from prev alence data and its application to demen tia in Germany . http://arxiv .org/abs/1112.2720v1 [2] Brinks R (2 012). On characteristic s of an ordinary differen tial equation and a r elated in v erse problem in epidemiology . http://arx iv.org/abs/1 208.5476v1 [3] Carstensen B, Kristensen JK, Ott o sen P , Borch-Johnsen K (2008). The Danish Na- tionalDiab etes Register: tr ends in incidence, prev alence and mortality . D iab etolog ia 51 (12):2187- 96. [4] F ox CS, Sulliv an L, D’Agostino RB Sr, Wilson PW (2004) The significant effect of diab etes duration on coronary heart disease mortalit y: the F ramingham Heart Study . Diab etes Care 27 (3):70 4-8. [5] Gilli M, Maringer D, Sch umann E (201 1) Numerical Metho ds and O pt imizatio n in Finance. Academic Press, W altham, MA. [6] Hale JK (1 980) Ordinary differen tial equations. Rob ert Krieger Publishing, Malabar, FL. [7] Kalbfleisc h JD, Pren tice RL (2 0 02) The Statistical Analysis of F ailure Time Data, 2nd edn. John Wiley & Sons, Hob oken, NJ. [8] Keiding N, Hansen BE, Holst C (199 0 ) Nonparametric estimation of disease incidence from a cross-sectional sample of a stationa r y p opulatio n. Lect Notes Biomath 86: 36- 45. [9] Keiding N (1991 ) Age-sp ecific incidence and prev alence: a statistical p ersp ectiv e. J R Statist So c A 154:371 -412. [10] Lexis W (1903) Abhandlungen zur Theorie der Bev¨ olk erungs- und Moralstatistik. G usta v Fisc her, Jena, http://dspa ce.utlib.ee /dspace/bitstream/10062/5316/4/le xis_abhandlocr.pdf Accesse d 14 August 2011. [11] McDonald JT, Kennedy S ( 2003) Insigh ts into t he healthy immigran t effect: health status and health service use of immigrants to Canada. So c Sci & Med 59: 1613- 27. [12] Misra A, Ganda OP ( 2 007) Migrat ion a nd its impact on adip osity and t ype 2 diab etes. Nutrition 23 (9):696-708 [13] Murra y CJL, Lop ez AD (1994) Quantifying disabilit y: da t a , metho ds and results. Bull W orld Health Organ 72 (3):481- 4 94. 12 [14] Murra y CJL, Lo p ez AD (1996 ) G lo bal and regional descriptiv e epidemiology of disabilit y: incidence, prev alence, health exp ectancies a nd ye ars lived with disabilit y . In: Murray CJL, Lop ez AD (ed) The Global Burden of D isease. Harv ard Sc ho ol of Public Health, Boston, pp 201 -46. [15] P oly anin AD, Zaitsev VF, M oussiaux A (2002) Handb o ok of First Order P artial Differen tial Equations. T aylor & F rancis, London. [16] Preston SH, Coale AJ (1982) Age structure, g ro wth, attrition, and accession: A new syn thesis. P op Index 48 (2):217- 5 9. 13