Modelling lifespan reduction in an exogenous damage model of generic disease

We model the effects of disease and other exogenous damage during human aging. Even when the exogenous damage is repaired at the end of acute disease, propagated secondary damage remains. We consider both short-term mortality effects due to (acute) exogenous damage and long-term mortality effects due to propagated damage within the context of a generic network model (GNM) of individual aging that simulates a U.S. population. Across a wide range of disease durations and severities we find that while excess short-term mortality is highest for the oldest individuals, the long-term years of life lost are highest for the youngest individuals. These appear to be universal effects of human disease. We support this conclusion with a phenomenological model coupling damage and mortality. Our results are consistent with previous lifetime mortality studies of atom bomb survivors and post-recovery health studies of COVID-19. We suggest that short-term health impact studies could complement lifetime mortality studies to better characterize the lifetime impacts of disease on both individuals and populations.

The emergence of novel diseases -such as COVID-19, Ebola, SARS, Zika, avian flu, or monkeypox -is a worsening trend. 1 Every new disease raises urgent questions about how they could impact infected individuals and the population at large.Yet observational studies offer answers only in retrospect.How can a priori knowledge inform us before new diseases are studied and characterized?One approach is to identify potentially universal effects of disease.This approach may also be useful for existing diseases that are not yet fully characterized.
3][4][5][6][7][8][9] For example, short-term mortality due to COVID-19 rises approximately exponentially with age -more than 30-fold from 55 to 85 years. 10,11 any infectious diseases also exhibit long-term complications, exemplified by post-acute 'sequelae' (PAS) -for example, SARS and MERS, 12 Ebola, 13 Zika, 14 'long COVID', 15 and COVID complications. 16Surprisingly, we do not know the long-term effects of most PAS, how they depend on age, or how they compare to the impact of short-term mortality.This is because there are very few long-term, large-scale studies of the impact of acute disease; most studies are limited to less than 5 years.One notable exception is the study of lifetime mortality impacts of exposure to the atomic bombs at Hiroshima and Nagasaki. 17,18 hile this does not represent the effects of disease, it does represent the long-term effects of acute exogenous damage.
Understanding age-effects of disease is particularly important.For example, assuming that short-term mortality is the only impact of acute diseases implies that immunization of older individuals will typically 19 save more years of life than immunizing younger individuals. 11,20 owever, if post-acute health impacts of disease -including PAS -lead to substantial shortened lifespans then immunizing young individuals could save more years of life.Resolving these questions of age-effects for individual diseases is not easily done, since lifetime observational studies require many decades.
A promising a priori approach is to computationally model the age-effects of disease.This first requires a model of normal aging.Encouragingly, aging populations exhibit simple and universal behavior.Average human mortality rates exhibit an exponential increase with age known as Gompertz' law, 21 which is reminiscent of the increased short-term mortality of disease with age.Individual health can be captured by the frailty index, which measures damage and dysfunction. 22Before death, individuals accumulate damage approximately exponentially with age, 23 leading to worsening individual health. 24The random but inexorable accumulation of damage during aging can be modelled at the individual level by a complex network of binary health attributes (healthy or not), 25 where damage propagates stochastically across static links (edges).][28][29] A GNM model provides a dynamical context for propagating damage due to disease.We can model the onset of disease by treating it as an exogenous event that further damages an individual.As such, we can also consider any exogenous damageand are not specifically limited to disease.While the generic nature of the health attributes in the GNM precludes a detailed study of specific diseases, its generic nature allows us to identify and characterize potentially universal effects of disease in 1 arXiv:2305.06808v2[q-bio.PE] 29 Jul 2023 aging individuals.
We will consider the effects of disease timing (onset age), severity, and duration.We will first consider excess mortality (fatality) rates due to disease.To assess the long-term impact of diseases we also need to consider years of life lost due to damage originating from disease.We can use years of life lost within different time horizons to compare short and long-term impacts of disease.We also develop and explore a simplified phenomenological model of how exogenous damage leads to earlier mortality.

Generic Network Model (GNM) of Disease and Exogenous Damage
The GNM represents individual health by an undirected scale-free network. 30Links, defining network topology, are static.Nodes are dynamic binary health attributes -either damaged or not.A summary measure of individual health is the frailty index ( f ), 22,24 which is the fraction of damaged nodes.An undirected scale-free network is generated using the Barabási-Albert preferential attachment model, 31 with an average node degree ⟨k⟩ and scale-free exponent α GNM .Nodes are initially undamaged at age t = 0, but damage at a rate Γ + = Γ 0 exp(γ + f i ), where f i is the fraction of damaged neighbours for node i. Damaged nodes repair at a rate Γ − = (Γ 0 /R) exp(γ − f i ), though repair has a negligible effect on population statistics in practice.Individual mortality occurs when the two most connected nodes are both damaged.We use previously determined GNM parameters 26,28 that approximate sex-combined USA population health and mortality statistics 32 for ages t ≳ 20: ⟨k⟩ = 4, α GNM = 2.27, Γ 0 = 0.00183, γ + = 7.5, with small repair (γ − = 6.5 and R = 3.0) and N = 10 4 nodes.For simplicity and clarity we do not use a false-negative correction 26 to reduce the range of f to [0, 1 − q] -i.e.we use q = 0 and have f ∈ [0, 1].Stochastic dynamics are exactly sampled. 33All plotted data corresponds to at least 10 6 simulated individuals.Errorbars for averages, unless indicated, are smaller than point sizes.All times are in years.
The GNM models damage from all sources that arises during the aging process, including the propagation or amplification of earlier damage.It then captures mortality effects due to that damage.Since the GNM is parameterized from population health and mortality statistics, it implicitly includes many extrinsic events such as disease or injury -the usual stressors of living.As such we expect that the GNM will allow us to model the effects of an individual disease, which we here consider as additional or perturbative to the normal aging process in order to estimate its effect.
We will not model details of the disease process, rather we will simply assume the disease starts (e.g.due to infection) at some onset age t on and lasts for a duration τ.In a similar spirit we will assume that the disease has a fixed severity or magnitude m.In terms of the GNM, our model disease damages a fraction m of nodes at the onset age t on .While formally m ∈ [0, 1], we do not damage already damaged nodes so m is kept small.We exclude individuals from analysis who have initial damage f > 1 − m.For m ≤ 0.02 no individuals are excluded, while for m = 0.05 a small fraction (10 −4 ) are excluded for t on ≥ 90.At the end of the disease (at t on + τ) a fraction r of the applied damage is removed.The fraction r of damage that is removed is a recovery or "resilience" parameter.For acute diseases we typically use r = 1, while chronic diseases could be modelled with r = 0 (equivalently, τ → ∞).Since we model disease by introducing exogenous damage m at time t on , and allow for a fraction r to be repaired after τ through resilience, we can use the same model for any exogenous damage.The effect of our model disease is illustrated in Fig. 1a with respect to the frailty index f .The control population with no disease is indicated by the grey dashed line.We see that even with r = 1 there is excess damage ∆ f left in the individual after the end of the disease.This residual damage leads to long-term mortality effects that we characterize.We compare these long-term effects with the short-term acute effects that we also characterize.
We measure long-term mortality using the average reduction in lifespan (∆t tot ) and also by the average years lost within a window of w years after the disease (∆t w ), assuming the mortality rate of the control population after that window.All disease results are with respect to a large control population with no disease (m = 0).The excess probability of death due to the disease corresponds to an excess Infection Fatality Rate (IFR) as compared to the control population.

GNM Results
Our GNM model disease has a significant impact on long-term health, as shown by the average frailty index ( f ) vs age for large simulated populations that received a disease (blue points and solid line) or did not (grey dashed line) in Fig. 1a.With maximal resilience (r = 1, our default acute disease) all of the damage introduced at t on is removed after τ.Nevertheless excess damage propagates within the GNM and remains at t on + τ, as indicated by ∆ f .For a variety of onset ages, and for selected durations τ as indicated, we show ∆ f in Fig. 1b.We see that ∆ f increases with onset age, and also that the individual variability of propagated damage (indicated by the shaded regions) is large.This reflects the stochastic nature of damage propagation within the GNM.
In Fig. 2a, we show the excess mortality during an acute disease (IFR) vs onset age t on .The IFR increases monotonically with t on for all m and τ investigated, and maintains an approximately exponential age dependence similar to the all-causes mortality curve (µ, grey squares).In Fig. 2b we show the total years lost due to disease (∆t tot ) vs the onset age.Strikingly, we  A disease is represented by exogenous damage of severity m inserted at onset time t on ; a fraction r of the original damage is then removed after duration τ.Excess damage that is left at t on + τ is indicated by ∆ f .The average damage vs age, as assessed by the frailty index ( f , the fraction of damaged nodes within the GNM), for an acute disease with r = 1, m = 0.05, t on = 50 and τ = 5 is indicated by the blue points.A control population (with m = 0) is indicated by the grey dashed line, and is well approximated by an exponential f = ae αt where a = 0.0548 ± 0.0009, α = 0.0314 ± 0.0003, and t is the age -as indicated by the solid grey curve.(b) Excess damage.Increase in the frailty index at the end of an acute disease, ∆ f at t = t on + τ, with severity m = 0.02 vs onset age t on , with duration τ as indicated by legend and r = 1.The shading indicates the standard deviation of ∆ f .All ages and times, in this and other figures, are in years.see that the average reduction in lifespan is highest for younger populations (note the log-scale).There are two mechanisms that could contribute to the reduction of lifespan of younger individuals.The first is that mortality during the disease leads to more years of life lost for younger individuals -who have more years left in their life expectancy.The second is that long-term mortality effects could be worse for younger individuals.We can separate these effects by considering different observation windows w after the disease.
In Fig. 3a we show the average years lost ∆t w within a window of duration w after the end of the disease.We account for all excess mortality between t on and t on + τ + w.Just considering deaths during the disease (w = 0, yellow open triangles), we find that older populations have the largest number of years lost -as observed with, e.g., COVID-19. 20Even though younger individuals have more lifespan left to lose, it is not enough to offset their much lower IFR.However, for younger ages years lost due to deaths during the disease account for only a small fraction of the total years lost.As we increase w, ∆t w increases, and its peak shifts towards younger ages.The largest lifetime impact (∆t ∞ ≡ ∆t tot , blue squares) is for the youngest individuals, in agreement with Fig. 2b.This effect holds for a wide range of τ and m parameter values, see Supplemental Figs.S2 and  S3.Strikingly, the peak (mode) of lifespan impact only moves away from the oldest ages with long observation windows of w ≳ 20 years.The ratio of lifespan reduction ∆t tot /∆t 0 exceeds 100 for the youngest onset ages, and does not strongly depend on duration τ or severity m (Supplemental Fig. S1).The ratio will further increase for lower resilience (r < 1) since acute mortality, IFR, and acute life lost, ∆t 0 , are unchanged but mortality after the disease is increased due to larger residual damage ∆ f .For example, in Fig. 3b with r = 0 we show that ∆t tot is more than ten-fold larger than with r = 1.

Phenomenological Model of Disease and Exogenous Damage
While the GNM allows for stochastic and high-dimensional individual health trajectories, the connection between modelling assumptions and phenomenological behavior is obscured by its complexity.A simpler model would be more interpretableallowing us to see how and when our modelling assumptions lead to the behavior we see.A simpler model would also be easier to generalize.While other mean-field versions of the GNM exist, 26,28 here we develop a simple model that is directly rooted in the observed aging phenomenology: damage accumulates non-linearly with age and this damage drives mortality.The essential simplification here is that the health-state is described only by the average damage -rather than by the many interconnected nodes of the GNM.This phenomenological model complements our network-based simulations using the GNM, and can be easily modified for different phenomenological assumptions.
We start with the observation that the average damage, or frailty index, increases approximately exponentially with age f 0 (t) = ae αt . 35From the GNM, we have α ≈ 0.031 (and a ≈ 0.055, see Fig. 1) which is consistent with observational estimates for adults with t ≳ 20 (α ≈ 0.035 ± 0.02 35 ).We assume that exogenous damage, such as from disease or injury, forms part of -and behaves similarly to -the damage exhibited during aging.As such it satisfies the differential equation d f /dt = α f and any exogenous damage m grows exponentially thereafter.By including resilience, we then have simple expressions for the average damage before, during, and after the disease: where is the propagated damage at the end of the acute disease (at t end = t on + τ, and with resilience r).This phenomenological damage model is already considerably simplified compared to the GNM: we have a single deterministic health state variable ( f ) rather than N = 10 4 distinct and stochastic health-nodes.By comparing our expression for the propagated damage ∆ f (Eqn.2) with Fig. 1b, we see that the phenomenological model has a single value of ∆ f that is independent of onset age t on while the GNM has a broad range of ∆ f with an average that increases with t on -though by much less than the individual variability.
We also need an explicit mortality model.We use the well-established but phenomenological Gompertz law of µ 0 = be βt , 36 whereby the mortality rate of adults increases exponentially with age.We estimate β ≈ 0.089 (and b ≈ 4.3 × 10 −5 , see Fig. 2a).We then assume that the increasing mortality rate results only from the increasing frailty-index f (t).To obtain the correct time-dependence for mortality from f 0 ∝ e αt we have This expression will hold for both the disease and control populations, since by assumption the mortality is expressed only through the health.With a disease, for t > t end we can express this as where f end = f 0 (t end ) = ae α(t on +τ) is the control (non-disease) frailty at the end of the disease.Note that a chronic disease corresponds to a disease with no resilience -i.e.r = 0.A similar expression for the hazard applies during the disease, with the ratio ∆ f / f end replaced by m/ f on .The lifetime mortality rates, µ(t), uniquely determine the survival statistics. 37In Fig. 4a we present the death age distributions for several disease parameter values.The disease has two lifespan-shortening effects: a short-term, acute effect that increases mortality during the disease, reducing lifespan by ∆t short ; and a long-term, chronic effect that shifts the death age distribution to younger ages, further reducing lifespan by ∆t long .In Fig. 4b we numerically calculate the ratio of acute to chronic effects.As with the GNM, we see that long-term effects dominate for younger individuals whereas short-term effects dominate for older individuals, and are essentially independent of disease severity mτ.
We can also obtain simpler expressions for mortality effects -particularly in the 'weak' limit of small m and τ.These are useful to develop an understanding of the origins of the effects exhibited by diseases in the GNM.

Long-term effects
While short-term survival mediates long-term effects, this coupling is small in the weak limit.For simplicity, here we will condition on short-term survival -i.e.assume that individuals are alive at t end = t on + τ with excess damage ∆ f .Since mortality is determined by health, then the addition of exogenous damage ∆ f at t end effectively ages an individual by ∆t long where f 0 (t end + ∆t long ) = f 0 (t end ) + ∆ f .This is independent of the form of the mortality law.We obtain ). ( This expression neglects a monotonic memory term which is small for young t on , but that significantly decreases ∆t long at old t on (Supplemental Eqn.S75).Note that ∆t long estimates the increase in biological age following disease. 35Using Eqn. 2, and assuming small severities m we obtain ∆t long ≈ ∆ f /(α f 0 (t end )).Further assuming small durations τ we obtain Since mortality only depends on f , ∆t long estimates the long-term reduction in lifespan after the survival of mild diseasesexcluding any short-term mortality during the disease.Since f (t) increases with age, ∆t long is largest in the youngest individuals -independent of disease parameters m, τ, and r.For imperfect resilience, with r < 1, chronic effects typically dominate the long-term impact of disease-survivors and ∆t long ≈ m(1 − r)/ [α f 0 (t on )]; these chronic effects are independent of τ.We observed that COVID-19 has r < 1 whereas seasonal flu does not (see below).

Short-term effects
We can use the hazard µ(t) in Eqn. 4 to solve for the survival probability S(t), using dS/dt = −µS (details are in the supplemental).Conditional on being alive S = 1 at t on we obtain where f on is the frailty at t on .The probability of mortality by the end of an acute disease is 1 − S(t end ) therefore we obtain the excess short-term mortality ∆p death due to the acute disease by the difference in the survival function between using f on = f 0 (t on ) and f on + m at t on .For small m and τ we obtain We see that ∆p death ∝ e (β −α)t on is highest for older individuals since β > α.This is consistent with the observation of increasing short-term mortality with age in many diseases.

Comparing short-and long-term effects
To compare short-and long-term effects, we need to estimate the years of life lost due to death during the disease -all within the small m and τ limit.We can approximate the remaining lifespan ∆t D from the survival curve by imposing S(t on + ∆t D ) = 1/e, this approximates the survival curve as a step function.Using Eqn. 7 we obtain ∆t D = β −1 ln(1 + β /µ 0 (t on )).The years of life lost during acute disease is then ∆t short = ∆p death ∆t D which gives In the limit of small m and τ, the ratio of short to long-term lifespan effects is then where we have also allowed for maximal recovery after the disease (r = 1).Interestingly, this ratio is independent of disease details.We note that ln(1 + x)/x ≈ 1 for x ≈ 0 and monotonically decreases towards 0 with increasing x = β /µ, i.e. with decreasing age.At large ages ∆t short /∆t long ≈ β /α > 1, so that short term mortality during disease affects lifespan more than long-term effects.Conversely, at sufficiently young ages, we expect long-term mortality effects after the disease to have greater impact on lifespan than short-term mortality during the disease.From our estimates of α and β , ∆t short /∆t long = 1 for µ ≈ 0.024.From all-causes mortality statistics from the U.S. population (Fig. 2a, grey squares) we have µ ≲ 0.024 for ages t on ≲ 70, implying that ∆t short < ∆t long for onset ages < 70.So, our phenomenological model indicates that most people would have a greater reduction of lifespan due to premature death long after the disease than from death during the disease.Similar results are observed away from the small m and τ limit (see Fig. 4) and in the GNM (see Fig. 3a).

Long-term excess relative risk (ERR) and the Life-Span Study (LSS) of Atom-bomb survivors
The Life-Span Study (LSS) of approximately 120,000 survivors of the atomic bombs dropped on Nagasaki and Hiroshima has tracked excess lifetime mortality due to radiation exposure for more than 50 years, and found that excess relative risk decreased with age of exposure and was approximately linear with dosage. 17,18 eaths due to solid-tumor cancer predominate the excess mortality.
Our phenomenological model allows for any source of exogenous damage m, not just from disease.We recast it in terms of excess long-term hazard to be able to directly compare with the LSS analysis.Using Eqn. 4 with τ = 0 we obtain If we linearize in the hazard in ∆ f we obtain where on the right we show a model of excess relative risk (ERR) from the LSS 17 -here the covariates ⃗ c such as sex, city, and birth year are indicated (ERR ≡ γ(⃗ c)de θt on ).Qualitatively both the LSS and our approach have excess absolute risk 17 declining with age of exposure t on and with linear dose-response (∆ f or d in Sv).We can identify θ = −α.Their model estimates α = 0.045 (90% CI: [0.031, 0.060]), 17 which is consistent with our estimate of 0.031.We suggest that the increased radiation sensitivity at younger exposure ages reported by the LSS 17 may be a general effect of increased damage sensitivity at younger exposure ages.
Our phenomenological model also suggests different risk models that could be used with LSS data, such as including nonlinear effects with Eqn.11.Using α = −θ and β = 0.089 (Fig. 2a), we estimate ∆ f /a = 0.98d, where d is the exposure dose in Sieverts (Sv). 17This implies that the dose and the propagated damage ∆ f are approximately equal, when expressed in natural units.Since survivable doses range up to 5 Sv, the linearized approximation may be worse for younger individuals.

Parameterizations of COVID-19, influenza and Ebola
Using published IFRs we estimated disease severity, m, for COVID-19, 38 , influenza 39 and Ebola, 6 Table 1.Studies of both COVID-19 40 and influenza 39 recorded health in terms pre-disease vs post-recovery frailty, ∆ f .This allowed us to estimate the resilience parameter for those diseases, r.Each column of Table 1 includes parameter estimates taken from the literature for populations at particular ages, including τ, IFR, and ∆ f , together with our phenomenological model estimates for m and r using Supplemental Eqns.S2 and S3, respectively (where possible).Observe that resilience was not significantly different from 1 for influenza, but resilience was significantly lower for COVID-19.This may explain why COVID-19 is observed to have large long-term chronic effects 15,16 : Eqn.6 predicts that r < 1 effects will dominate the chronic disease effects.See supplemental for details.
Disease severity, m, depends on individual robustness -and is used to set the scale for both IFR and ∆ f .Note that while m > 1, we observe physiologically reasonable ∆ f ≪ 1.We observed that as individuals age, their robustness follows a U-shaped curve: increasing from infancy to adulthood and then decreasing with advanced age (Supplemental Fig. S5).In the case of COVID-19, this decreasing robustness with adult age paralleled the expected changes to frailty, f , suggesting a loss of robustness with increasing frailty.Consistent with this, comorbidities both increase the frailty index 22 and are major risk factors for mortality due to COVID-19 2 .
The frailty index includes both physical and mental deficits 22 .A large UK study found that individuals whom suffered from severe COVID-19 showed reduced cognitive impairment ∼ 2 years post-infection comparable to effectively aging ∼ 10 years 41 .Using Eqn. 5 we can estimate a generic aging effect from our model.Our ∆ f indicates an effective aging of ∆t long = 6 years for a median-aged 57.5 year-old -comparable with the observed cognitive aging 41 .

Discussion
We have developed and explored a three-parameter model of generic acute disease, which is built upon a generic network model (GNM) of organismal aging (age of onset t on , severity m, and duration τ).We evaluated short-term mortality outcomes using the excess infection fatality rate (IFR) and long-term mortality outcomes using the average reduction in lifespan due to the disease (∆t tot ).We found that while mortality during acute diseases is highest for older populations, the total reduction in lifespan is highest for younger populations.The majority of the years of life lost for younger populations are due to premature deaths later in life.Older populations have worse short-term outcomes because they have greater frailty f (worse health), which leads to a greater likelihood of death during the disease.Younger populations lose more years of life both because there is more to lose and more time for propagated damage ∆ f to impact mortality at the end of life.
Our results are qualitatively consistent with higher short-term mortality for older populations as reported for many acute diseases, including COVID-19, 10 SARS and MERS, 2 influenza, 4,5 Ebola, 6 varicella (chickenpox), 7,9 and meningococcal disease. 8While the 1918 ("Spanish") flu pandemic had much higher than expected mortality for younger adults, this appears to be a special (non-generic) case 42 partially due to the effects of age-varying immunological history. 5,43 15][44][45][46][47][48][49] We predict that such post-acute effects should increase with acute severity m, in qualitative agreement with, e.g., studies of long-COVID. 50Similar severity dependence is seen in ICU (intensive care unit) survivors. 51Our disease model is essentially one of exogenous damage, and so should be more general than just acute disease.Long-term studies of hip-fracture survivors have shown significant excess relative risk that is approximately independent of attained age 52,53 in agreement with our simple phenomenological model (Eqn.11).Atomic bomb survivors provide a unique long-term dataset for exogenous damage due to radiation 17 -with exposure ages ranging from 0 − 60 and with more than 50 years of followup.In agreement with our findings, lifetime risks are greatest for younger exposure ages t on .
5][56] Disease frequency typically increases with age, 57 consistent with declining robustness.Robustness and resilience can be considered individual and disease-specific parameters since, e.g., vaccinations or prior exposure increase robustness to infectious disease while, e.g., medical care can improve recovery.Robustness could affect the frequency and/or severity of disease for older individuals (e.g.t on and m).Resilience could affect recovery and duration (r and τ).Our results are for a fixed severity (m) so direct comparisons between ages require caution.Nevertheless, the ratio ∆t short /∆t long is conditioned on the disease occurring, and is largely independent of disease severity (Fig. 4b).The observation that the lifespan impact of disease can be much worse than the acute impact of disease for younger individuals is therefore independent of robustness.
Our model explicitly includes resilience through r.Smaller resilience (r) should lead to larger ∆ f and thus worse long-term effects.Since resilience is expected to decrease with age, 55,56 we would expect more long-term effects in older individuals.The result would be a smaller ratio of ∆t short /∆t long for older individuals.
Our disease model has no explicit age dependent dynamics, so all effects occur via individual health.We expect that short-term mortality will be worse with either worse health or older ages.Consistent with this, the prognosis of disease generally worsens with a higher frailty index f . 24,58 ultiple concurrent diseases are expected to combine additively through f , although saturation or exclusion effects may occur for severe or overlapping multimorbidities, respectively.While our phenomenological model has no age effect for ∆ f at a given m, our GNM exhibits increasing ∆ f with age.Furthermore, we expect that declining robustness with age (or declining health) will lead to larger m and so larger long-term health impacts (∆ f ).Such effects are observed.For example, disability following hospitalization increases more with age 59 , and more following ICU admission with frailty 60 .Frailty hinders recovery from influenza 39 .Age is a risk-factor associated with post-COVID-19 conditions. 15,50  and with PAS of chikungunya virus disease. 47onsistent with this picture, we observed that our estimates for disease severity, m, increased with age.For COVID-19, m increased exponentially with age: commensurate with f and consistent with a loss of robustness with increasing frailty.Although we did not have data to estimate age-related changes to resilience, we did observe that the seasonal flu showed nearly perfect resilience whereas COVID-19 indicated incomplete recovery (r < 1).This could help explain the prevalence of COVID-19 PAS. 15,16 arameterizing additional specific diseases will facilitate future studies to investigate disease-specific effects on lifetime mortality.
Most studies of post-acute mortality effects only have a w ≲ 5 yr observation window.We found that w ≳ 20 yr is needed to observe the largest mortality impacts, which we predict occur for smaller onset ages.Larger observation windows w are needed.For shorter w ≲ 20 windows, general health measures such as the frailty index f 61 can be used to assess excess damage ∆ f due to the disease.The effective cognitive aging of approximately 10 years due to long COVID-19 41 is consistent with our generic estimates using Eqn. 5.The relative ease with which mental deficits can be measured may make them a convenient way to measure follow up health post-infection.
Our GNM disease model is stochastic and exhibits considerable individual variability in e.g., excess post-acute damage ∆ f (see Fig. 1b).For real diseases, we expect additional variability in the acute severity (m).Our models are restricted to adults (with t ≳ 20), due to similar restrictions on the GNM, frailty f , and Gompertz's law.We expect adult males to experience worse short-term mortality risk, including both acute and chronic effects, due to their higher baseline risk (Supplemental Fig. S6b).This sex-effect is seen in parasite-associated mortality 62 and most infectious diseases 62,63 .
Our simple phenomenological theory shares with the full disease model our assumptions that residual damage and mortality are determined by health via f .Subject to these assumptions, the qualitative agreement of our models indicates the potential universality of our results.From the phenomenological theory we see the key role of the exponential growth rates of mortality and frailty, β and α respectively.Empirically we have β > α, so short-term excess IFR (∆p death ) grows with age.Our phenomenological theory also indicates that post-survivor years of life lost ∆t long is universally greatest for younger adults -a consequence of α > 0.
We infer universal aspects of disease through the effects of direct (m) and secondary damage (∆ f ) in an aging population.We find large long-term effects at young onset ages.Including such age-effects in epidemic models, such as for COVID-19, 16,19 would help us better understand and mitigate the impacts of disease on societies.Researchers typically ask if it is better to vaccinate the old to reduce direct risk, or vaccinate the young to reduce overall infection prevalence. 19Similarly, cost effectiveness of e.g.rotavirus vaccine 64 or allocation of COVID-19 vaccine 20 often only consider mortality during disease.Often neglected are the potential chronic effects due to propagated damage, which we find are worse for the young.Our results could have significant implications for how we prioritize medical interventions across age.Long-term observational studies of health and mortality after acute disease or exposure are needed to better capture lifetime disease impacts.

S2 Validation for COVID-19, influenza, and ebola
Here we provide details of fitting observed health and mortality data to our phenomenological model, and provide fits for selected diseases.Both the GNM and our phenomenological model are founded on two key hypotheses: (1) acute mortality is due to damage, and (2) this damage should cause secondary propagated damage.Secondary hypotheses are that robustness, via m, and resilience, via r, may vary by disease or due to individual risk factors (including health and age).While the main text primarily explores the key hypotheses, this supplemental section validates the secondary hypotheses using easily available data from influenza, COVID-19, and Ebola.Using acute mortality data we can estimate m (and hence robustness effects).If individual health is followed post-recovery, we can also estimate r (and hence resilience effects).
Several studies have shown evidence of residual or collateral damage post-disease.Increased disability in activities of daily living ?and increased clinical frailty score ?have both been observed after COVID-19 recovery.Similar effects are seen in other coronavirus': Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS).SARS and MERS show long term deficits in fitness capacity and mental health -including increased stress -for up to a year post-recovery ? .These deficits can lead to collateral, propagated damage due to the negative health effects of stress and dysfunction during the disease together with, e.g., the lack of positive effects of exercise.The main text deals with the consequences of this propagated damage.In general, residual damage may also be due to finite resilience r < 1, i.e. acute damage that was not fully recovered from.
Frailty has been observed to increase both after COVID-19 ? and also after hospitalization due to influenza A/B ? .In Figure S4 we present data from Lees et al's study of hospitalizations due to confirmed influenza.They observed a marginally significant increase in the FI (frailty index f ) post-influenza.We use this influenza data from Lees et al, and data from COVID-19 ?, to estimate r (resilience) and m (including any robustness effects) for these diseases.We will also estimate m from Ebola mortality data ?-where without health information we are unable to estimate r.We approximated that individuals were in the hospital for 12.9 days -this is the average length of stay for COVID-19 ?, which is similar to influenza.?Error bars are standard error in the mean.

Influenza
The key statistic for estimating m is the infection fatality rate (IFR).Estimating r requires an additional estimate of ∆ f .m represents the disease severity and is the fraction of damaged health attributes.∆ f represents the residual damage after some fraction, r, of the initial m is repaired -in addition to secondary propagated damage.
The IFR is simply the difference in survival between control and disease groups during the acute period, where the survival of the control, S c , and disease S d , are defined by the corresponding terms in the preceding equation.The disease parameters are the age of onset, t on , the disease strength, m, and duration τ.While we do not know the forms of S c and S d from the GNM, we can compute analytic forms for the phenomenological model, see Section S3 (S 2 using Eqns.S22 and S23).This model assumes Gompertz' law and also that all mortality occurs due only to changes in the frailty index, f (or "FI").
For the phenomenological model we can algebraically invert Eqn.S1 to yield the m-estimator, where b and β are Gompertz fit parameters from the healthy population, α ≈ 0.031 is the FI growth exponent and f on is the control-group FI at the start of the disease.
Using published IFR data we estimated m for several diseases using Eqn.S2. m captures both the intrinsic severity of the disease and the individual's resistance to that disease i.e. robustness.Increases to m with age reflect decreases to robustness (and vice versa).In Figure S5 we present m as a function of age for COVID-19 (a) and Ebola (b).As we would expect, robustness increases from infancy to adulthood, causing m to decrease in both diseases.Also expected is that robustness then decreases with increasing age during adulthood, causing m to increase.This increase of m is much faster with COVID-19 (note log-scale) than with Ebola.For COVID-19 the increase of m (decrease of robustness) approximately parallels the increase of the FI (frailty, f ).The different behavior of COVID-19 and Ebola supports the hypothesis that robustness is disease dependent.Note that these robustness effects are in addition to the age-effects of acute mortality discussed in the main paper with constant m.The combination of age effects are qualitatively consistent with known mortality risk factors for coronavirus' such as COVID-19, including a strong age dependence and the magnifying effects of comorbidities -which increase f -such as hypertension, diabetes and chronic lung disease ? .using S(t) = 1 − t 0 p(t)dt.Finally, a useful result for later is to apply integration by parts using p(t) = −dS/dt (Eqn.S13),

S3.3 Disease
The effects of the disease on survival can be formalized by calculating the FI, f , as a function of disease parameters.Using Eqn.S6 we can then calculate the mortality risk and therefore the survival using the risk formalism.?The disease parameters are • the severity, m, equal to the increase in FI during the disease, • the duration, τ, • the age of onset, t on , and • the resilience, r, equal to the fraction of m that is repaired at the end of the disease.
For convenience, we define t end ≡ t on + τ (S19) as the end time of the disease.The control is the special case with m = 0. We can track f because we know how much damage we're adding and therefore we know, ae αt t < t on (ae αt on + m)e α(t−t on ) = ( f (t on ) + m)e α(t−t on ) = (a + me −αt on )e αt t on ≤ t < t end (( f (t on ) + m)e α(t end −t on ) − rm)e α(t−t end ) = f (t, m = 0) + ∆ f e α(t−t end ) = (a + ∆ f e −αt end )e αt t end ≤ t (S20) where ∆ f ≡ m(e ατ − r) and t end ≡ t on + τ.Observe that the disease is equivalent to introducing some initial damage, thus increasing the FI by m.Conversely, we can view adding initial damage as aging the individual, for gained age, δ , defined by Eqn.S21.See Section S3.5 for details.We can then calculate the hazard using Eqn.S6 to yield,  (S86) As discussed in Section S3.8, the µ term in the denominator is always positive and < 1, increasing from a small correction in young ages to order unity by approximately age 100.Hence dropping this term will give a lower limit that's tight (good) at younger ages but poor at older ages.

Figure 1 .
Figure 1.(a) Model disease.A disease is represented by exogenous damage of severity m inserted at onset time t on ; a fraction r of the original damage is then removed after duration τ.Excess damage that is left at t on + τ is indicated by ∆ f .The average damage vs age, as assessed by the frailty index ( f , the fraction of damaged nodes within the GNM), for an acute disease with r = 1, m = 0.05, t on = 50 and τ = 5 is indicated by the blue points.A control population (with m = 0) is indicated by the grey dashed line, and is well approximated by an exponential f = ae αt where a = 0.0548 ± 0.0009, α = 0.0314 ± 0.0003, and t is the age -as indicated by the solid grey curve.(b) Excess damage.Increase in the frailty index at the end of an acute disease, ∆ f at t = t on + τ, with severity m = 0.02 vs onset age t on , with duration τ as indicated by legend and r = 1.The shading indicates the standard deviation of ∆ f .All ages and times, in this and other figures, are in years.

Figure 2 .
Figure 2. (a) Mortality.Excess probability of death during the disease (IFR) vs onset age (t on ) for acute diseases with duration τ as indicated, and m = 0.02.Square grey markers indicates the all-causes mortality rate (per year) vs. age from the U.S. population (2010). 34Exponential fit (solid black line): (4.3 ± 0.3) × 10 −5 exp [(0.089 ± 0.001)t on ].Male (M) and female (F) sub-populations are as indicated.(b) Lifespan reduction.The average total reduction in lifespan due to disease, ∆t tot , vs. onset age t on for severity m = 0.02 and duration τ as indicated by legend, with r = 1.Chronic disease corresponds to τ = ∞ (or r = 0).

Figure 3 .
Figure 3. Lifespan reduction for different observation windows.(a) The average years lost ∆t w vs t on for different observation windows w past the end of acute disease (with r = 1).The effects of mortality during the disease (w = 0) are largest for older individuals, even though the younger individuals have more lifespan left to lose.The effects of lifetime mortality (w → ∞) are largest for younger individuals, demonstrating the impact of residual damage.All with τ = 1 and m = 0.02.(b) ∆t tot for a chronic disease (r = 0).The lifetime effects (w → ∞) are much larger than in Fig. 3a.

Figure 4 .
Figure 4. Phenomenological model.(a) Effect of varying m and r on death age.The control distribution (black, dot-dashed line) is shifted towards lower ages by the disease.With resilience (dashed lines), two phases emerge: an acute phase during the disease (ages 20-30) and a chronic phase after the disease ends, due to propagated damage.Each phase contributes to the overall loss of life due to the disease.Without resilience (solid line, r = 0) the two phases merge into a single short-lived persistent phase.(τ = 10, t on = 20) (b) Acute vs chronic effects.Ratio of expected life lost during acute phase vs chronic phase, ∆t short /∆t long .The ratio increases approximately exponentially with increasing age of onset, t on , nearly independently of disease severity (mτ).(τ = 10 −3 , 10 −4 ≤ m ≤ 10 −1 , r = 1)

Figure S4 .
Figure S4.Changes to followup health due to influenza hospitalization.Older individuals (average age 80) were measured for frailty index (FI, f ) before, at and after hospitalization for influenza A or B. The band indicates the expected change to FI over the study period for the control group -it is essentially constant.Data were extracted from survivor data in Fig.2of Lees et al.?  We approximated that individuals were in the hospital for 12.9 days -this is the average length of stay for COVID-19 ?, which is similar to influenza.?Error bars are standard error in the mean.

Table 1 .
Disease Parameter Estimates for Specific Ages (95% CI) Figure S5.Disease intensity m estimates, age-dependent.Using infection fatality rate (IFR) data we estimate m as a function of age, which captures age-related changes to robustness (which decreases as m increases).(a)COVID-19.We observed a strong age dependence for COVID-19.For adults, our estimates for m increased exponentially with increasing age (blue solid line), paralleling the expected changes to the FI (green dashed line; same scale).This suggests that robustness may be a function of frailty.Source data: COVID-19 forecasting team, Lancet, 2022 ?.(b)Ebola.We observed a small increase in susceptibility at young ages in both (note scales) COVID-19 (a) and Ebola (b).For Ebola, the youngest group (< 1 years old) had higher mortality data than the oldest group (45+).Once individuals reach maturity, however, the disease effect, m, seems to be approximately constant with age for Ebola: with at most a linear age-dependence.Finally, note the much stronger effect of Ebola compared to COVID-19, with m > 1 at all ages -reflecting the much higher IFR.f captures an individual's health state, so that for very strong diseases with m > 1, acutely ill individuals are less healthy than the oldest healthy individual.Source data: Agua-Agum et al., N Engl J Med, 2015 ?.The oldest age group (45+) was imputed as age 60 for convenience.The conditional distribution given that death occurs after some reference time, t r , is , m) ensures normalization, ∞ 0 p(t > t r , m)dt = 1.The associated survival function is For small τ and m we can use Eqn.S67 and Eqn.S84,⟨∆t⟩ short ⟨∆t⟩ long ≈ S 1 (t on )S 2 (t end , m = 0) mτ S 1 (t on )S 2 (t end , m = 0) mτ f on r + 1−r ατ 1 − µ(t end ,m=0) m) f (t,m=0) β /α e βt t < t on b ( f (t on )+m)e −αton a β /α e βt = b f (t,m) f (t,m=0) β /α e βt t on ≤ t < t end b ( f (t end ,m=0)+∆ f )e −αt end a β /α e βt = b f (t,m) f (t,m=0)β /α e βt t end ≤ t.