^{1}

^{2}

The authors have declared that no competing interests exist.

We use a simple SIR-like epidemic model integrating known age-contact patterns for the United States to model the effect of age-targeted mitigation strategies for a COVID-19-like epidemic. We find that, among strategies which end with population immunity, strict age-targeted mitigation strategies have the potential to greatly reduce mortalities and ICU utilization for natural parameter choices.

In this paper, we use a simple age-sensitive SIR (Susceptible, Infected, Removed) model integrating known age-interaction contact patterns to examine the potential effects of age-heterogeneous mitigations on an epidemic in a COVID-19-like parameter regime. Our goal is to demonstrate the qualitative point that for an epidemic with COVID-19-like parameters, age-sensitive mitigations can result in considerably less mortality and ICU usage than homogeneous mitigations, among strategies which end with population immunity.

We model mitigations which result in a 70% reduction in transmission rates among all of the population except for a relaxed group. In our models, natural transmission rates correspond to an _{0} of 2.7. (In S3 Section in

Note that while it is not at all surprising that targeting mitigations at higher risk groups can greatly reduce mortality during mitigations, our main contribution is to emphasize that this result still holds even if transmission rates for all groups are eventually relaxed. In particular, as we discuss in Section 2, the fact that this holds for COVID-19 depends on the coincidence that age-specific mortality rates for COVID are very strongly anti-correlated with age-specific contact patterns during periods of normal social interaction.

Related work in the context of MERS can be found in [

It seems intuitively obvious that preferentially targeting at-risk populations for mitigations may reduce disease mortality by a large factor. Our goal in this paper is to show a more subtle (but still simple) qualitative effect; namely, that even if transmission rates return to normal in the future and the epidemic ends only when population immunity is sufficient to survive reintroduction of infection, mitigations which were age-targeted can still achieve a large mortality reduction. For this question, the extent to which mortality can be greatly reduced by targeted mitigations depends heavily on the interaction between mortality rates and contact patterns. If the population most at risk of death from infection can sustain an epidemic without any role played by the younger population, then preferentially targeting mitigations at the at-risk population may accomplish relatively little, since when they are eventually lifted, the at-risk population will suffer high levels of infection anyways.

To examine this interplay between mortality rates and contact patterns, our model integrates the contact matrix generated for the United States in [

Age-group | Hospitalization rate | ICU rate for hospitalized cases | IFR |
---|---|---|---|

0 to 9 | 0.1% | 5.0% | 0.002% |

10 to 19 | 0.3% | 5.0% | 0.006% |

20 to 29 | 1.2% | 5.0% | 0.03% |

30 to 39 | 3.2% | 5.0% | 0.08% |

40 to 49 | 4.9% | 6.3% | 0.15% |

50 to 59 | 10.2% | 12.2% | 0.60% |

60 to 69 | 16.6% | 27.4% | 2.2% |

70 to 79 | 24.3% | 43.2% | 5.1% |

80+ | 27.3% | 70.9% | 9.3% |

The goal of this paper is to explore the implications of this interplay between contact patterns and total mortality from COVID-19, under the assumption that some approximation to normal interaction patterns eventually resumes (or at least, the

In this paper we explore the extent to which the age-distribution of the infected can be shifted by mitigation strategies which do not perfectly isolate or separate populations based on age, but simply target transmission reductions at older populations.

We use a simple SIR-like model acting on

We let _{1}, …, _{n}) gives the death rate for each group separately.

Given a vector

The basic dynamics of the model are captured by the following vector differential equations:

Here ⊙ denotes coordinate-wise multiplication, and

We emphasize that by using a simple SIR model, we are disclaiming the goal of making precise predictions regarding, e.g., the timing of infection peaks under various scenarios. Instead our goal is to understand the impact of age-targeted strategies relative to homogeneous ones.

When modeling mitigation strategies, the contact matrix _{0} value). More realistically, we can incorporate known patterns of inter-age-group interactions. To do this, we use contact matrices generated by [

One feature of using a non-constant contact matrix ^{5} infections in a much larger population, we do not expect those to be uniformly distributed by age, but instead distributed in a way which depends on

To understand the dependence on _{0} is often measured in the field). We call this the _{1}, …, _{n}) be the vector giving the fraction of the total population within each age group. If the

In particular, _{1} denote its eigenvalue, which is the (unique) largest eigenvalue of

To tie our model to reported estimates of _{0} for COVID-19, we wish to compute a scaling factor _{0} value computed early in an epidemic (before a substantial fraction of the population is infected), given an infection distributed according to the early-stable proportion vector _{1}, thus for an infected population with age distribution proportional to

In particular, we see that _{0} value is that which ensures that

See [

We use the recovery rate

Note that we do not apply a correction for asymptomatic cases, which may make our analysis conservative (pessimistic). Comparable (but not identical) estimates may be found from other sources. For example, see [

Weighting the age groups by their population sizes, these mortality rates would correspond to assuming an overall IFR (infection fatality rate) of roughly 1%. But because of the age-variation of the mortality rate, the IFR depends to a large extent on the expected age-profile of the infected population. In particular, we will see that the numbers in

Source code for our model can be found at the entry for this paper at

There has been considerable variation over time and space between the correspondence of COVID case severity and ICU utilization. For example, the fatality rates for COVID patients receiving ICU care with known resolution have ranged 26% for Lombardy, Italy [

In the present manuscript, we do not aim to predict precise levels of ICU utilization, but instead use a coarse model of ICU utilization simply as a rough proxy for utilization of scarce healthcare resources from very severe cases, for the purpose of making relative comparisons between strategies. Motivated by the parameter choices in [

Our simulations all begin from an initial infection affecting 100, 000 individuals (distributed proportional to the early-stable proportion vector) in an otherwise fully susceptible population of size roughly 3 × 10^{8}. In all of our scenarios, we assume that normal transmission levels are linearly resumed between the 9- and 15-month marks from the start of the simulation.

As a first example,

In all our scenarios, we assume that transmission rates linearly resume between the 9 month and 15 month points.

For our age-targeted mitigations, we consider relaxing mitigations just on those under 40, just on those under 50, and just on those under 60.

Since it is natural to expect targeted mitigations to be based on household-level of risks, because of cohabitation of younger and older adults, we consider, in each case, a scenario where only 2/3 of the younger population is subject to normal transmission levels.

In each of these scenarios, depicted in Figs

It might be natural to suspect that age-specific strategies are simply trading mortalities in one group for mortalities in another. However, we find in our models that age-targeted restrictions can dramatically reduce mortalities among older populations with very small impacts on mortality in younger populations; see

The bottom row of this table shows mortalities in each age group for the optimum homogeneous mitigations scenario, as a fraction of the total population of the age group. Note that this is

Some evidence has been presented that young children are less susceptible to infection from COVID-19 than adults; for example none of 234 tested children under 10 tested postive for COVID-19 in Vo, Italy, despite some living in households with infected members [

Separately, there is some evidence that even once infected, children are less infectious than adults: Countries tracking infection events by age (including, e.g., Iceland, where schools for young children have remained open) have seen very few events of young children infecting adults [

If a large fraction of the younger population played no role in transmission of COVID-19, this could actually undermine the results in the present paper, since our findings depend on the ability of the younger population to bear a greater burden of population immunity than the older, at-risk population. (This would not be possible if all observed epidemic growth was actually driven primarily by the older population).

To address the influence of this hypothetical issue, we have modeled additional scenarios where children are 50% less susceptible to infection and/or 90% less likely to transmit once infected. We find that these changes do not have a large enough impact on the transmission dynamics of the epidemic to impact our findings (S5 Fig in

Each of these scenarios corresponds to a modified contact matrix; decreased susceptibility for an age group is modeled by scaling the corresponding rows, decreased infectiousness is modeled by scaling the corresponding columns. In _{0} of 2.7 for the whole population. (Note that scaling this modified contact matrix to _{0} = 2.7 requires older age-groups to have higher transmission rates than results from rescaling the contact matrix from

The top two panels are generated using the modified contact matrix described in

_{0} values for COVID. The fact that mortalities are nevertheless decreased is caused by the fact that the number of people ever infected is smaller. This is can be understood through the fact that contact patterns are more heterogeneous for the scenario where young children play a lower role (as measured by the uniformity of the Peron-Frobenius eigenvector) which leads to the expectation of a lower attack rate. For example, in scenarios where reduced transmissibility and infectiousness apply to everyone under 20, hetereogeneity would decrease and IFR and total mortality would move in the same direction.

We have considered a model of age-heterogeneous transmission and mitigation in a COVID-19-like epidemic, which is simple but also tied to current estimates of both disease parameters and U.S.-specific contact patterns. We find that age-targeted mitigations can have a dramatic effect both on mortality and ICU utilization. However, we also find that to be successful, age-targeted mitigations may have to be strict. Our scenarios modeling moderate mitigations on the restricted group (shown in S8, S11 and S14 Figs in

Importantly, we find that while relatively good strategies exist in a range of scenarios, so long as mitigations on restricted groups can be strict or very strict, the precise choices which minimize ICU utilization and deaths are sensitive—for example, to the fraction of younger people which will actually be released from mitigations.

We also find that if only moderate mitigations are possible on the population subject to mitigations, then the discrete set of age-targeted mitigations we considered fared poorly.

We view our modeling as demonstrating a qualitative point: strict age-targeted mitigations can have a powerful effect on mortality and ICU utilization, even if relative transmission rates among age groups will eventually normalize. We expect that public policy motivated by this kind of finding would have to be responsive; for example, by relaxing restrictions on larger and larger groups conservatively, while monitoring the progress of the epidemic.

Note in S2 Table in _{0} value, there are heterogeneous strategies that outperform the optimal homogeneous strategies. There are also heterogeneous strategies that do worse, sometimes much worse. These poor-performing strategies fall into one of two types:

Those with too large a relaxed population, so that the initial epidemic is not sufficiently constrained, and

Those with too small a relaxed population, so that when transmission rates resume, a second wave results (which disproportionately effects the older population).

The simple lessons for policy are twofold: first,

It is important to emphasize, however, that the extent to which strategies like those considered here are the best approach depends on what other strategies are considered feasible. For example, if the COVID-19 epidemic can be contained indefinitely (i.e., if the second wave in

Any predictive model is an oversimplification of the real world whose predictions depend on parameter values whose true values will only be known after-the-fact. The model we employ is particularly simple, and while this simplicity can be an asset when demonstrating qualitative phenomena, it also presents obvious limitations. For example:

We do not model seasonality, since the effects of seasonality on COVID-19 remain unsettled. Seasonal forcing could mean, for example, that after relaxing restrictions on some group, they may have to be reinstated as transmission rates increase.

While we do use (simple) models of known age-group contact patterns, we don’t model the effects of specific mitigations on those patterns; for example, we don’t evaluate the specific effects of things like closing schools. We also don’t attempt to model the different effects of mitigations on within-home and out-of-home contact patterns, since the ways these each contribute to empirically observed _{0} values is complicated.

There is still considerable uncertainty regarding basic parameters of COVID-19 such as its transmissibility, the infection mortality rate, and the ICU admission rate. While we have relied on expert choices for the parameters we have used, all quantitative findings we make (such as ICU utilization) are sensitive to these choices.

Like prior work modeling the COVID-19 epidemic (e.g., [

On the other hand, there are ways in which our analysis has been conservative. For example:

We have only modeled contact heterogeneity in contacts at the age-group level. Further heterogeneous clustering of contacts, or in susceptibility, could further reduce attack rates [

When modeling homogeneous mitigation strategies in _{0} value resulting from mitigations. On the other hand, for our examples of heterogeneous mitigation strategies, we have tied our hands considerably more. We simply allow ourselves to choose which age group, in intervals of 10, to release to normal transmission levels—we have just a few discrete options to choose from. In particular, if were allowed ourselves to combine mild mitigations on the relaxed group with strong mitigations on the rest, we would be able to achieve fewer mortalities and ICU utilization. Note that many of our age-targeted mitigation scenarios exhibit epidemics which end well before the 9-month mark; these curves have room to be flattened.

We have considered scenarios where not all the younger age group will be able to have relaxed transmission rates, because of cohabitation with older household members, or because of risk factors other than age. This can make our analysis worse, by increasing the average age of the required immunity herd. However, we have not taken advantage of a presumed benefit that this would confer: if effective risk-models incorporating factors beyond age could be deployed, it is plausible that the ICU admission rate for the relaxed groups could be decreased.

PONE-D-20-10135

Modeling strict age-targeted mitigation strategies for COVID-19

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Reviewer #1: The paper "Modeling strict age-targeted mitigation strategies for COVID-19"

considers an oversimplified, off-the-shelf, SIR model modified by a

social-contact matrix between age-groups. The idea is not new as the authors

let us know.

The novelty of the manuscript would then be limited to its application to the

SARS-CoV-2 pandemic. Hence, it deserves consideration in as much it

incorporates essential particularities of the current problem.

The construction of the model, its relation with the know features of

SARS-CoV-2 does not deserves much attention as the authors rush into the habits

of modelling and their own limited interest.

Nothing worth reading is going to come from ignorance of the phenomena. Even at a qualitative level: which epistemological theorem states that what it is no of my interest will have no influence in the result?

Let me list a few obvious matters that will influence the results but the

authors have not considered.

1. Behavior changes with the illness when symptoms appear.

2. Behavior changes with the social perception of risk in an epidemic.

3. The course of SARS-CoV-2 changes with age. In particular, recovery times.

4. It is suspected that mild cases are less contagious than severe cases

(before isolation).

5. The recovery time that matters is the time from onset of contagiousness to

isolation or end of the contagious period (whatever comes first). Such times

depend on age.

6. The contagious period is not exponentially distributed (without

exponentially distributed times for each compartment, there is no support for

ODE models)

7. Social contact at normal times is not the important kind of contact in terms

of the propagation of the epidemic. What is relevant is the ability to transmit

the illness.

8. A homogeneous contact (without social structure) limits any model to small

communities.

9. An ODE approach limits its scope to large numbers in each compartment.

10. The combination of (6) and (7) may limit the scope to the empty set.

11. R0 is model depending. As such, it cannot be read from the data.

12. Do people in the USA continue with their social-contacts being active

epidemiologically when hospitalized? I would really be surprised. This is just

another feature the authors built in their model without realizing it. It is

the consequence of the faulty epistemology.

The authors, upon giving a fair view to the relevance of the matters they have

ignored may very well decide that they have nothing serious to say. From my

point of view, this work will only be useful to confuse the uneducated.

I did not read beyond section 3, for it makes no sense to consider a toy model

during a period of high demand of serious modelling.

Reviewer #2: In the paper, the authors proposed a SIR-like epidemic model with contact matrix and study the effect of age-targeted mitigation strategies. It is an innovation point of the manuscript. Results are interesting and satisfactory. There are, however, still some minor problems need to be solved before publication.

1. Please give the exact value of contact matrix C when modeling mitigation strategies.

2. Please give the clear description of mitigation strategies.

3. The figures are unclear, especially the figure of ICU.

4. I cannot understand the result of figure 2 B. Why there are no infections at the beginning?

Reviewer #3: In the present work, the authors present a model for a strict age-targeted mitigation strategies for COVID-19. The model is based on a standard SIR model adapted to include an aged specific contact matrix. Also, some age-specific epidemiological parameters were included in the model. The author show how such a strategy can avoid the collapse of the ICU units as the contagion of the elderly is smooth and even lower than in the absence of such a strategy. In turn, the targeted isolation can make the quarantine more tolerable for the rest of the population.

The main results are in part trivial, as a natural result of partially isolation part of a population is preventing them from being infected. In order for this model to prove of some utility would be if it can provide robust qualitative results.

The model uses a contact matrix that is asymmetric due to the methodology used to build it. The results are based in a directed survey, where there is always a pointing and a pointed person. This is the origin of the asymmetry and not because they correspond to frequencies of interactions. It is not clear how the authors build their symmetric matrix. and where they got the information about the population pyramid.

The dynamics of the ICU is not described. There is no accurate information about the permanence of patients in ICU units.

The prevalence of risk groups among the nonisolated population is not taking into account. This information is very relevant at the moment of an accurate estimation of the occupation of ICU.

The information about the percentage of each age group ICU requirement is obtained from data collected from a different country, a different population. A simple research across reports from different countries show how scattered these percentages are.

In the last weeks, we have seen a plethora of models, with a vast majority of them presenting contradicting and out of scale results.

In the present form, the status of the present model is conjectural only, with questionable robustness. It would be irresponsible to propose a public health policy on the basis of such a feeble analysis. The author should present a stronger and more founded model.

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Modeling strict age-targeted mitigation strategies for COVID-19

PONE-D-20-10135R1

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Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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PONE-D-20-10135R1

Modeling strict age-targeted mitigation strategies for COVID-19

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