From reaction kinetics to dementia: A simple dimer model of Alzheimer’s disease etiology

Oligomers of the amyloid β-protein (Aβ) have been implicated in the pathogenesis of Alzheimer’s disease (AD) through their toxicity towards neurons. Understanding the process of oligomerization may contribute to the development of therapeutic agents, but this has been difficult due to the complexity of oligomerization and the metastability of the oligomers thus formed. To understand the kinetics of oligomer formation, and how that relates to the progression of AD, we developed models of the oligomerization process. Here, we use experimental data from cell viability assays and proxies for rate constants involved in monomer-dimer-trimer kinetics to develop a simple mathematical model linking Aβ assembly to oligomer-induced neuronal degeneration. This model recapitulates the rapid growth of disease incidence with age. It does so through incorporation of age-dependent changes in rates of Aβ monomer production and elimination. The model also describes clinical progression in genetic forms of AD (e.g., Down’s syndrome), changes in hippocampal volume, AD risk after traumatic brain injury, and spatial spreading of the disease due to foci in which Aβ production is elevated. Continued incorporation of clinical and basic science data into the current model will make it an increasingly relevant model system for doing theoretical calculations that are not feasible in biological systems. In addition, terms in the model that have particularly large effects are likely to be especially useful therapeutic targets.

and Cizas et al. [3].Standard errors for the Lambert data in the control (oligomer concentration of 0) are based on a worst-case estimate. The figure markings obscured the error bars and we chose the half-width of the largest marker as the standard error as part of the calculation. The study of Lambert et al. did not specify whether the error bars displayed were standard errors or standard deviations. We assume standard errors. Such variations only change σ by a modest scaling factor. Table B: Dimensionless parameters. With 1 chosen these serve as constants for the asymptotic calculations. For the values displayed, we useS =S G . All parameters except for are O (1). The bottom parameters ensure that the slow timescales over which κ and S change are on the scale of 1/ 2 . Fig. A: Viabilities at various oligomer concentrations after 24 hours. We fit the model to viability data [2,3]. The errors bars represent two standard errors.     [21]. B: incidence curve [22]. C: HV curve.

Parameter Estimation
We present our analyses to estimate the model parameters. In general, we seek to estimate parameters that are representative of a healthy brain. We understand that no two brains, let alone no two people, are exactly alike, therefore defining the parameters for a "healthy brain" is formally impossible. Nevertheless, for the purpose of creating a model, we have made our best estimates based on what information is available in the literature. We would hope that these estimates are within an order of magnitude of "formally representative."

Viability Modeling
One important metric for brain health is cell viability, which we denote V (t) and define to be the number of viable neurons per unit volume, relative to the optimal value in a healthy brain with no neuronal death. Through aging and various insults, the value of V will decrease. We let D represent the concentration of dimers. For a given neuron, we denote T > 0 to be its age when it dies. We use survival analysis and hazard functions to model this [1].
Over a short time window δt, we assume Pr(t < T ≤ t + δt)|T > t) = σDδt for some σ > 0. In other words, given that a cell lives as long as time t, the probability it dies over the next interval of length δt is proportional to δt (and to the dimer concentration). Alternatively, we can model cell death as an instantaneous Poisson (memoryless) process with rate σD. As a differential equation, we have To estimate σ, we choose two experiments [2,3] from the literature studying oligomer toxicity on neurons. There are many papers that study this phenomena, but most study only cultured neurons in isolation or in the presence of unrealistically high Aβ concentrations. Lambert et al. measured cell death in mouse brain slice cultures treated with various levels of oligomers for 24 hours. Cizas et al. performed similar experiments but with mixed neuronal-glial cell cultures. For our fitting, we assume: The cell viability V begins at maximum V = 1.
The experiments directly measure V (in expectation, the fraction of surviving cells should indeed be V ).
General oligomer (dimers, trimers, n-mers) toxicity and dimer toxicity are the same.
The values σ and D do not vary over the duration of the experiment.
Cell death is induced only by the dimers and thus, at time t, we have V (t) = exp(−σDt).
After estimating values from their respective published graphs, converting cell death to cell survival in Lambert et al., and normalizing both datasets by the survival of their respective controls (see Table A), we fit for σ in (1) with a Maximum Likelihood estimate [4] (see Fig A). We obtainσ = 4.9 ± 0.4 M −1 s −1 (SE) as the characteristic dimer toxicity.  [3].Standard errors for the Lambert data in the control (oligomer concentration of 0) are based on a worst-case estimate. The figure markings obscured the error bars and we chose the half-width of the largest marker as the standard error as part of the calculation. The study of Lambert et al. did not specify whether the error bars displayed were standard errors or standard deviations. We assume standard errors. Such variations only change σ by a modest scaling factor. Viabilities at various oligomer concentrations after 24 hours. We fit the model to viability data [2,3]. The errors bars represent two standard errors.
To provide the survival fractions of Table A, we divided the survival rates from the experiments by their respective controls and added errors in quadrature to report the standard errors.
We note that we could allow V to be spatially dependent as well due to D potentially varying in space. Equation (1) would be unchanged, however, i.e., there would not be any explicit spatial dependence and all spatial variation would be implicit through the varying concentration of dimers.
It is important to remark that the effects of oligomer toxicity are significantly smaller in the Lambert study. Fitting individually, we find that with Lambert alone, the value isσ = 1.0 ± 0.1 M −1 s −1 (SE) and with Cizas alone, the value isσ = 8.9 ± 0.6 M −1 s −1 (SE). Changing σ up or down by factor of 2 would not alter our main results as we operate only with representative constants for the parameters and the formal asymptotic analysis done is robust, i.e., results do not appreciably change, with these sorts of perturbations. There is one notable exception: varyingσ would have a similar effect to varyingD in that the neuronal death rate would change and, in the HV study, the rate of change of volume would be different. However, the rest of the model has a dependence upon γ, which would then compensate for a changedσ.

Incidence and Prevalence
For this modeling, we assume the brain tissue of interest (such as the hippocampus) has a uniform distribution of monomers and oligomers and a uniform viability measure V (t). We begin by considering H(t), the "survivorship function," being the "healthy" fraction of the population still alive at age t that does not have AD. We have H(t) = Pr(AD has not developed by age t|alive at age t).
We again use a survival analysis. Over a short time interval, δt, we assume i.e., the probability someone develops AD in the interval (t, t + δt] if they did not have it up to time t is proportional to the percentage of remaining neurons lost over the interval (t, t + δt], where γ > 0 is a dimensionless proportionality constant that we expect to be on the order of unity. This is motivated by the fact that just a small number of neurons may be highly influential in memory formation [5] and we are describing the probability one of these influential neurons dies over the interval of width δt. In expectation, and with infinitesimal time steps, With H(0) = 1 and V (0) = 1, the solution is that To model prevalence, we note that Alzheimer's patients live, on average, T D = 7.1 years after diagnosis. Without specifically modeling the factors that lead to mortality, we assume that all Alzheimer's patients die T D years following diagnosis. This factor is relevant in computing the prevalence, P (t), which should be the fraction of the population (AD and non-AD) still alive at age t that has AD. This can be expressed as July 20, 2021 4/23 Neglecting the edge case of t < T D , the numerator is the fraction of patients with AD within the population who are still alive at time t (they cannot be diagnosed earlier than t − T D ). The denominator is all people still alive, with and without Alzheimer's disease (to still be alive, they must develop AD later than t − T D ). The incidence (fractional rate people are diagnosed per unit time), −H (t)/H(t), is A more rigorous derivation of the prevalence and incidence is provided later.

Monomer and Dimer Concentrations
We denote parameters of interest with a bar to indicate a characteristic value in a healthy brain.
Monomer Production RateS: The monomer production rate for Aβ42 is approximately 330 fM s −1 [6]. Combining this with the fact that a typical Aβ42/Aβ40 ratio is 0.1 [7], assuming this ratio also reflects their production rates, and focusing solely upon these polymers, we estimate the Aβ monomer production rate is 3.63 × 10 −12 M s −1 .
It has been found that β-secretase activity increases with age [10]. One group reported an increase of ≈ 65% over 100 years [11]. Fukumoto et al. [12] reported that in AD patients the increase was as much as 63% relative to normal controls. These authors also found that β-secretase activity increases by 58% over 50 years in Down's Syndrome patients but that α-secretase (the enzyme responsible for cleaving APP to release truncated peptides of Aβ) activity remains relatively constant throughout life for those with and without Down's Syndrome. We assume that monomer production rate is linearly proportional to β-secretase activity. This leads to values of λ G S = 4.85 × 10 9 s and λ D S = 2.72 × 10 9 s where, for example, we model monomer production in the general population asS G (1 + t/λ G S ).
Dimer ConcentrationD: From Lue et al., the range of total soluble Aβ was measured both for Aβ42 and Aβ40. The average concentrations of these peptides were 0 and 1.9 pg/g, respectively, for healthy people, and 15.5 and 66.5 pg/g, respectively, for those with AD [14]. Using molecular weights of 4514 g/mol for Aβ42 and 4329.9 g/mol for Aβ40, and assuming brain tissue has a density of approximately 1 g/cm 3 , this results in dimer concentrations of 2.19 × 10 −13 M for normal individuals and 9.16 × 10 −12 M for AD patients. We chooseD = 1 pM, which is approximately the geometric mean of the two measurements for modeling.
Extra Assumptions. To estimate additional parameters, we make a series of assumptions: 1. monomers and dimers are in rapid equilibrium so in describing the dimer concentration, we always have that µD ≈ νM 2 ; and 2. the loss of monomers due to forming higher-order structures (trimers and higher) is negligible in comparison to the loss due to clearance so that, in combination with the previous assumption,M =S/κ.

Estimatingμ andν:
Experiments measuringμ [15] have foundμ = 12700 s −1 , but from experiments observing the conversion of oligomers to monomers, the time scale is on the order of days (soμ ≈ 10 −5 s −1 ) [16]. There is thus a large range of possible values forμ. We take the geometric mean of these two values to inferμ = 0.4 s −1 . Then from the ratio , we computeν = 115 M −1 s −1 .
We note that while obtaining estimates for bothν andμ is an interesting exercise, in our model, the disease dynamics depend upon their ratio and not the values individually.

Diffusion Coefficients
Estimates for diffusivities of monomers and dimers are 1.4 × 10 −6 cm 2 /s and 1.1 × 10 −6 cm 2 /s, respectively [17]. The tortuosity of brain tissue is ≈ 1.6 [18]. The observed diffusivity of a chemical species D * is related to its unimpeded diffusivity D and the tortuosity of its environment ι by This results in the diffusivities used in our study.

Mathematical Details
Here we provide some of the mathematical steps done to arrive at the results presented in the main body of the paper.

Negligible Higher Order Oligomers
The concentration of HOOs (trimers and above) is negligible within our parameter regime, thus they are not included in the model. This is a mathematical justification that supplements the focus upon dimers based on a quantitative measurements [19]. We shall denote O j to be the concentration of oligomers of order j. Thus M = O 1 and D = O 2 as special cases. We assume that oligomer growth and loss is done through monomer addition/loss; furthermore, we assume that the oligomer growth rate (addition of a monomer) and oligomer dissociation rate (loss of a monomer) are given by those of the dimerization rate ν and dimer dissociation rate µ. Then in the absence of diffusion (as we are looking for represenative scales), we havė where the dot represents a time derivative. If we assume that O j ↓ 0 as j → ∞, i.e., the concentration of oligomers of arbitrarily large size tends to 0 and that the system July 20, 2021 6/23 can reach an equilibrium state, then at the equilibrium, by computing ∞ j=3Ȯ j , which is the rate of change of the total concentration of HOOs, we have and with this, in general, . are negligibly small. Asymptotically, νO1 µ is on the order of 2 , where is a small parameter defined in the next section. None of the results of the asymptotics would change with their inclusion.

Scalings
It is mathematically convenient to work in a dimensionless framework to have better scaled variables and fewer parameters. A dimensionless framework also allows "small" terms to be identified which are used in formal asymptotics to furnish highly accurate but approximate solutions. The equations presented in the main body of the paper have been converted back to dimensional form. For brevity, we present the analysis with S =S G ; the other case ofS =S D can be handled mutatis mutandis.
We denote x for spatial position. Within Eqs. (1), (2), and (7), of the Main Manuscript, we perform a change of variables according to where all overlined values represent dimensional scales, and τ, m, d, v, z, along with all tilde-variables are dimensionless. We remark the rate constants are allowed to be time-dependent. We choose the scalest =κ −1 ,M =St,D =νM 2 µ , andx = √ D Mt . This yields the dimensionless system of equations with values appearing in Table B. The use of 0 < 1 is suggestive of formal asymptotics.

ODEs
We first consider equations Eqs. (2)-(4) in the absence of diffusion so that the equations are ordinary differential equations and we solve them subject to m(τ = 0) = d(τ = 0) = 0, with an initial concentration of monomers and dimers of zero, and with v(τ = 0) = 1, i.e., the cell viability is initially at its maximum. We begin by finding asymptotic solutions for m and d and deal with v later.
Parameter Definition Value Since the monomer clearance rate and activity of the β−secretase varies over decades (where τ = O(1/ 2 )), assume a similar behavior in other rate constants and model this bys for O(1) functions with O(1) rates of changes s ,ν s ,μ s ,σ s , ands * s (τ ) = 1 +τ . The functionsκ s andκ * s withκ * s (τ ) = 1 −τ are O(1) with O(1) rates of change but may become o(1) as the argument ofκ * s approaches 1. We assume all the rates change over a slow, O(1/ 2 ) timescale. The subscripts in functions such as κ s signify a slowly evolving function. The constants λ 0S,G , λ 0S,D , and λ 0κ are O(1) with λ 0S,G used for the general population and λ 0S,G used for the Down Syndrome population -values appear in Table B. As we have approximate forms for how S and κ vary, we are more precise about the form their dimensionless representations take.
The system admits multiple time scales but our approach is to treat the system as a set of inner-outer matching problems. Over a very fast time scale (indicated by a subscript f ), July 20, 2021 8/23 Over the "normal" timescale we posit that m ∼ m (n) and d ∼ d (n) for O(1) functions m (n) and d (n) to obtain which can be trivially matched to the innermost τ f -region with In order to observe the system response to changing rate constants, we consider a slow timescale τ s = 2 τ . Becauseκ * (τ ) = 0 atτ = 1, for our analysis, we assume that 1 − τ s /λ 0κ = O(1). Here, we take m ∼ m (s) and d ∼ d A composite solution on τ 1/ 2 with 1 − 2 τ /λ 0κ being O(1) can be obtained as We shall refer to these leading order solutions as m 0 and d 0 , respectively. From (4), we have that This can be evaluated to We focus on an O(1) description of v so that a uniformly valid approximation for τ 1/ 2 and 1 − July 20, 2021 9/23 The solutions presented in the main body of the paper are for the biologically relevant timescale of decades, i.e., τ = O(1/ 2 ) with 1 − 2 τ /λ 0κ = O(1).
As a mathematical remark, when 1 − 2 τ /λ 0κ = O( ), a different asymptotic regime is obtained. However, this analysis is not relevant to the problem at hand and we do not analyze it.

PDEs
In analyzing the PDE models, there are a number of asymptotic balances that are possible, i.e., depending on the spatial, temporal, or concentration scales that we look at, different terms in the equations dominate the system behavior. We study one of the most biologically relevant balances.
Throughout this analysis, we will focus upon the local effects of a perturbation in the monomer production. We consider a radially symmetric perturbation so that spherical symmetry and various simplifications can be applied. The solutions presented are obtained from the Green's functions provided later.

Spatial Model Results
We Ifs =s 0 + ρ 1, r < R 0, r ≥ R where R denotes the dimensionless radius of a hypothetical sphere where monomer production is increased, then using (22) and (21) with α = 1, β =κ, φ =s, we can explicitly solve for m 0 at steady state giving and The fact that we only consider the steady-state here is motivated by our observation in the ODE model that over an O(1) time in τ , the viability loss is O( 2 ), which is negligible. The most important dynamics occur over the τ s timescale whereby m 0 and d 0 can be taken as steady-state values.

Solutions via Green's Functions
The preceding solutions were obtained using Green's functions. In general, to solve the spherically symmetric problem for h(r, t), we find that where The function Θ denotes the Heaviside step function, Also, if φ does not depend on time, the steady-state solution can be found from with G now given by

Asymptotic Validity
There are precise ranges over which the asymptotic results will be valid. Taking the system out of the well-defined scalings will result in approximations that become less valid and could ultimately fail. We summarize here the important considerations in using these formulas.

ODEs
Some essential conditions are listed below: 1. The value defined in Table B is small, i.e. much smaller than 1.
We remark that no conditions are placed upon γ.

PDEs
For the PDE analysis, we presented the solution at steady state and the rate parameters should not be changing. The solution is valid provided items 1 and 2 of the ODE conditions above hold and the spatial and temporal scales are O(1).

Uncertainty Quantification and Damage Distributions Notation
Here we adopt some notational conventions and assumptions. We shall denote to be the (possibly) time-dependent cell damage rate and AD development rate where U (0) = U 0 , ω(0) = ω 0 , and Ξ U (t) and Ξ ω (t) are scaling functions that combine the effects of aging and lifestyle factors. We will generally consider that U 0 is either fixed or that it has a distribution within the population, likewise for ω 0 . But for simplicity, we assume that within a population, Ξ U (t) and Ξ ω (t) are the same for all people. In the static model, Ξ U = Ξ ω = 1 but in the dynamic model, both Ξ U and Ξ ω increase. Using only how κ(t) and S(t) vary, we The distinction between Ξ U and Ξ ω is not necessary in the current model. But in a more general setting where possibly γ is time-dependent, the functions will not be the same.
We denote the integration operator We denote g U to be a probability density for U 0 . And we denote g ω to be a probability density for ω 0 . When U 0 and ω 0 are discrete, the densities are understood to allow for δ−function so that integration represents a sum. In reality g U and g ω are probability measures. Integrations with respect to probability measures are understood to range over the entire set of values for the corresponding random variable.
We shall denote g U (u) = Pr(U 0 = u) with the understanding that if g is a density then g(u)du = Pr(u ≤ U 0 < u + du) for a differential du. We adopt similar notation with g ω (w) = Pr(ω 0 = w). We furthermore denote g U (u|E) as the probability density for U 0 = u conditioned on the event E and likewise for g w (w|E). Through Bayes theorem, we have July 20, 2021 12/23

General Hazard Function
Let T be a random variable denoting the time of an event (such as AD development or the death of a neuron). Let Γ(t) be the Hazard function for the time of an event. Let dt denote an infinitesimal time interval. Then, by definition, The probability density for T , call it f (t) satisfies Thus, by the definition of conditional probability, .

Letf (t) =
t 0 f (s)ds be the cumulative distribution function for T . Then Thus, given a hazard function, the probability density function can be computed. And Note that if t(sup 0≤s≤t Γ(s)) 1, then Pr(T ≤ t) ≈ t 0 Γ(s)ds by a Taylor expansion.

Viability
Recall that we denote V (t) to be the fractional density of healthy neurons relative to their initial healthy value. Let N 0 be the initial number of healthy neurons in a small volume with N (t) being the number of viable neurons in that volume a time t later. For each neuron, let T be time of death for each neuron with hazard function U 0 Ξ(t). Then the probability density function (pdf) for the time of death is Then we have immediately that Pr(T > t) = exp(−U 0 I 0,t Ξ U ) from (26) with Γ = U 0 Ξ U . Thus, at time t, the number of viable neurons, N (t), is a Binomial random variable with As N 0 ↑ ∞, to describe a large number of neurons, the variance ↓ 0. Thus, the mean and all percentiles converge to exp(−U 0 t 0 Ξ U (s)ds). We therefore use this as the value of V (t) for anyone with neuronal damage rate U 0 .

Prevalence and Incidence
For someone to have AD at age t, they must have been diagnosed later than t − T D because our model assumes death occurs T D years after diagnosis. Thus, the prevalence of AD at age t is the number of people diagnosed later than t − T D who were diagnosed on the interval (t − T D , t] divided by the number diagnosed later than t − T D . In all of this there is an implicit assumption that the individuals did not die from other causes by age t. For simplicity, we assume all other causes of death act independent of AD. Within the general population, the hazard function of AD development is ω 0 Ξ ω . We denote A(a, b) to be event that an individual is diagnosed with AD on the time interval (a, b]. Let N 0 (t) here denote the number of people diagnosed later than t. Let pt(t) denote Pr(A(t −t, t)|A(t −t, ∞)). Performing the calculation, we have Let N P (t) denote the number of people diagnosed on (t − T D , t]. Then We observe that pt(t) The incidence, for an infinitesimal time step dt, is the fraction of people diagnosed on (t − dt, t] divided by dt. Let N I,∆t denote the number diagnosed on (t − ∆t, t]. Then I(t) = N I,∆t ∆tN0(t−∆t) . Thus letting ∆t ↓ 0: The Central Limit Theorem tells us that P (t) and I(t), for large enough populations, will be normally distributed according to ) and N (ω 0 Ξ(t), ω0Ξ(t) ∆tN0(t) ), respectively. If ∆t is truly taken to 0, the variance of the incidence diverges. In practice, observations are done over small, but finite time windows.
The mean values are precisely those in Eqs. (8) 2 and (9) 2 of the Main Manuscript. Furthermore, we see that for large enough populations, all the percentiles converge to the mean values. As AD data are observed over very large population sizes, we do not plot error bars in our model predictions.
In general, with a finite initial population of N 0 (0), we can estimate the 95%-confidence interval for a population under observation as approximately within 1.96 standard deviations of the mean, where the standard deviation is inversely proportional to the square root of N 0 (0). In (29) and (31), the denominator is time-dependent, however, so once people have died from AD, this approximation will begin to lose accuracy.
As later depicted in Fig C, the 95% confidence window does observe this scaling. For instance, at age 60, the dynamic model with fixed ω(t) = ω 0 Ξ ω (t) having a single value ω 0 =ω and with an age-related scaling, we find the 95% confidence windows for prevalence and incidence are 0.430 − 0.563% and 0.0352 − 0.127%/y, respectively. With N 0 = 40, 000 treated as constant and with ∆t = 0.25 y, Eqs. (29) and (31) predict widths of the 95% confidence windows of 0.133% and 0.0918%/yr, respectively. This is completely consistent with the observed intervals of 0.136% and 0.111%/yr, respectively.
Another thing to note is that at leading order, P (t) = ω 0 t t−T D Ξ ω (s)ds, which is proportional to ω 0 . Thus, if monomer production S(t) increased by a factor Ω, prevalence would increase by a factor of ≈ Ω 2 . Also, the doubling times for incidence and prevalence, when fitted to exponential growth, do not depend upon ω 0 .

Lifetime Risk
The pdf for diagnosis at age t is given by ω 0 Ξ ω (t) exp(−ω 0 t 0 Ξ ω (s)ds) and thus, by integrating from t = 0 to t = T , we can compute the cumulative risk up to age T .
We can also compute the risk of developing AD over an interval t 1 to t 2 given it has not occurred up to t 1 via

Average Initial Neuronal Damage Rate U 0 within AD Patients
We consider the possibility that U 0 has a distribution of values within the population. We suppose that U 0 ∼ g U (u). We suppose further that E[U 0 ] = U * is the mean value and Var[U 0 ] = Σ * 2 is the variance. To exploit various asymptotic approximations, we operate under the assumptions: In practice many distributions fit this requirement. Note also that I t−T D ,t Ξ U = O(T D ). We wish to understand if the average value of U 0 among those with AD is higher than U * , i.e., if those with AD may have a larger rate of neuronal loss, on average. To this end, we seek the density g U (u|AD at age t) = Pr(AD at age t|U 0 = u)g U (u) Pr( AD at age t|U 0 = u)dg U (u) .
Colloquially, it is the density for U 0 given that someone has AD. We wish to understand if is larger than U * , the mean value of U 0 . Pr(AD at age t|U 0 = u) = 1 − exp(γuI t−T D ,t Ξ U ) is the prevalence of AD with U = u, which, by Taylor series, is approximately γuI t−T D ,t Ξ U , for small enough γT D u.
Also, by compliment, Pr(no AD at age t|U 0 = u) = exp(γuI t−T D ,t Ξ U ) ≈ 1 − γuI t−T D ,t Ξ U . Note that to not have AD at age t implicitly means being alive at that age, so the same conditional requirement that diagnosis takes place after t − T D is present.
Thus, with formal asymptotics, We have used Var(·) = E(· 2 ) − E(·) 2 . We can also compute the expected value of U 0 given someone does not have AD. Using similar notation and approximations, we have g U (u|no AD at age t) = Pr(no AD at age t|U 0 = u) To leading order, this is U * . As a check, we have that We can see these approximations are accurate in Fig B. Effects of Distribution of AD Development Rate ω 0 Here we consider calculating the prevalence and incidence with a distribution of ω 0 -values in the population. Here g ω is now describing the density for ω 0 .
This allows for two important calculations. The prevalence is defined by Pr(A(t − T D , t)|A(t − T D , ∞)) and the incidence is found by limt ↓0 1 t Pr(A(t −t, t)|A(t −t, ∞)). In particular, with the parameters of the example distribution of Table 2 of the Main Manuscript, we have U − =Ū /10 with p − = 0.765 and U + = 3.93Ū with p + = 1 − p − = 0.235. Let ω − = γU − and ω + = γU + . The results are

General Inequalities
For arbitrary 0 <t < t, is, for each t, the integral of a concave function C(ω) = 1 − e −ω t t−t Ξ(s)ds over a probability distribution. Thus, it is generally true, by Jensen's Inequality [20], that A consequence of (33) is that for an arbitrary distribution g ω within a population, the prevalence and incidence (for finite time steps) of AD will not exceed the prevalence and incidence in a population where every individual has the same value of ω 0 . The same argument also shows that for an arbitrary distribution g U within the population and with γ and t fixed, the average healthy neuronal density V (t) will be at least as large as the healthy neuronal density in a population where every individual has the same value of U 0 . The inequality is reversed because the exponential decay for V is a convex function.

Stochastic Simulation Details
The pseudo-code for our stochastic simulations can be summarized below. We used M = 400, N = 40, 000, t end = 90 yr, dt = 0.25 yr. Note that for an individual, U (t) = U 0 Ξ(t), where U 0 may be fixed or chosen from a distribution and Ξ(t) may be constant (in the static model) or age-dependent (in the dynamic model).
for i = 1 to M: # number of simulations M t = 0 for j = 1 to N: # number of people N # pick U0 from sample distribution choose value U0~g(u) initialize person with U0 add to population while t < t_end: for j = 1 to N: if person j has had AD more than T_D: die continue endif # generate exponential distribution # with hazard function choose I~Exp(gamma*U(t)) if I < dt: person j gets AD endif t <-t + dt endwhile endfor
Plots of the clinical and model incidence and prevalence are provided in Fig F, in addition to the HV for this alternative model. The agreement is better, and through a line of best fit, we find doubling times of 8.94 yr and 10.1 yr for the incidence and prevalence, respectively.  [21]. B: incidence curve [22]. C: HV curve.