A modified age-structured SIR model for COVID-19 type viruses
AA modified age-structured SIR model for COVID-19 type viruses
Vishaal Ram a , Laura P. Schaposnik (cid:63),b ( (cid:63) ) Corresponding author: [email protected] Abstract: We present a modified age-structured SIR model based on known patterns of social contact anddistancing measures within Washington, USA. We find that population age-distribution has a significant effecton disease spread and mortality rate, and contribute to the efficacy of age-specific contact and treatmentmeasures. We consider the effect of relaxing restrictions across less vulnerable age-brackets, comparingresults across selected groups of varying population parameters. Moreover, we analyze the mitigating effectsof vaccinations and examine the effectiveness of age-targeted distributions. Lastly, we explore how our modelcan applied to other states to reflect social-distancing policy based on different parameters and metrics.Keywords: COVID-19, SIR Model, age-targeted disease control.
I. INTRODUCTION
The study of the spread of diseases and rumours withinnetworks (social and biological) in order to trace factorsthat are responsible for or contribute to their occurrencehas been done from many different perspectives. More-over, only recently have graph theory, number theory,and computer science taken researchers to several break-throughs. Back in the early 1900s, Ronald Ross pro-duced the first mathematical model of mosquito-bornepathogen transmission using mosquito spatial movementin order to reduce malaria from an area [13]. Somedecades later, William O. Kermack and Anderson G.McKendrick [12] created the
SIR model , which catego-rized people into the 3 states
Susceptible , Infectious and
Removed – the model which we shall focus on.More recently, contact networks were introduced tobetter represent a community [11]: these are adaptedto reflect certain particular characteristics of society, andthey have been of much use when doing mathematicalmodeling of epidemics. In this setting, a social network ismodelled as a graph where vertices represent individuals,and edges encode the interactions amongst people: twopeople are connected by an edge in the graph wheneverthey are related (and thus an interaction could exist).Given the recent outbreak of COVID-19, and withviews towards applications to future viral outbreaks andmarketing strategies, this paper is dedicated to the studyof contention strategies with social networks by target-ing different clusters within the network in different ways.As highlighted in [16], the importance of local clusteringin networks has been widely recognised, and not muchstudy has been done in this direction until very recently.Since evidence shows very large differences in hospital-ization and fatality rates between age groups and gen-der groups, our interest is on obtaining a modified age-structured SIR model. Very recently, a first step in ana-lyzing the role of optimal targeted lockdowns in a multi-group extension of the standard SIR model was done[1, 4], where it was found that among strategies which endwith population immunity, strict age-targeted mitigationones have the potential to greatly reduce mortalities andICU utilization for natural parameter choices [4]. More-over, the trade-offs facing policy-makers between savinglives and improving economic outcomes were analized in[1], where it is shown that better social outcomes are possible with targeted policies: “
Differential lockdownson groups with differential risks can significantly improvepolicy trade-offs, enabling large reductions in economicdamages or excess deaths or both” [1].In the present work we take different path from [1, 4]and consider an age-compartment model with a rescalingfunction completely based on the policy that Washingtonimplements, where the intensity of the social distancingpolicy is proportional to the ICU occupancy. It should benoted that a modified rescaling could be applied to otherstates, hence making our model adaptable to other set-tings, e.g. New York uses metrics including rate of changeof total infections in their policy. Moreover, we considerage-specific relaxation policy (e.g. opening schools/work)and vaccine distribution. By applying our model to pop-ulations of varied age-distribution, we see the following: • Following our rescaling function, population age-distribution is directly correlated with increasingpeak ICU occupancy and decreasing peak infectioncount. However, herd immunity threshold is un-affected by the change in population parameterswith the same proportion of the population beinginfected through the course of the epidemic. • Across all age-distributions, relaxing school andwork restrictions has the effect of infecting the sameproportion of the population across a smaller timeframe, increasing peak ICU occupancy by over to18% and 51% respectively. However, such effectsare not observed when relaxing restrictions afterthe initial peak in infections. • Administering vaccines at a constant rate loweredthe herd immunity threshold, especially amonghigh median age counties, while also reducing mor-tality rate by 28%. Moreover, strictly prioritizingvaccines to older age-brackets seems extremely ef-fective, lowering ICU occupancy and further reduc-ing mortality rate by 20% while also completelypreventing the spread of the virus in the short term.To illustrate our perspective, we study the availabledata from the state of Washington, USA, and apply ourmodified model to this dataset. Our paper is organizedas follows: we shall begin by introducing the SIR modelin Section II A, and an age structured version followingthe work in [4] in Section II B. a r X i v : . [ q - b i o . P E ] S e p II. THE (AGE-STRUCTURED) SIR MODEL
As mentioned previously, the SIR model is a simplemodel for infectious disease in which the population isdivided into three compartments: those susceptible tothe disease, those infected with the disease, and thoseremoved from the disease either through death or recov-ery. Across this paper, we shall assume that those in theremoved group are unable to be infected again.
A. The SIR model
The number of individuals in each group is given bycertain functions of time S ( t ), I ( t ), R ( t ) respectively.Moreover, the dynamics of the model are given by theset of ordinary differential equations:d S d t = − β · I · SN ; (1)d I d t = β · I · SN − γ · I ; (2)d R d t = γ · I, (3)which depend on the following parameters: • the total population N ; • the transmission rate β , measured as the averagenumber of contacts per person per time, multipliedby the probability of transmission between a in-fected and susceptible person; • and the removal rate γ , also given by 1 /D where D is the length of the period for which a person isinfectious.During the early stages in an epidemic, transmissionsbetween individuals are statistically independent, mean-ing that the probability that an infectious individual en-counters someone no longer susceptible is probabilisti-cally low. Within the model, the basic reproduction num-ber R is he number of people an individual is expectedto infect, and can be computed given the parameters ofthe SIR model as R = βγ .One should note that the R value is not a biologicalconstant as its value depends on factors such as individualcontact patterns. However, the number R of a disease isgenerally consistent among newly susceptible populationsand can be used to predict the trajectory of an epidemicor calibrate the initial conditions of a model. In partic-ular, a value of R > R value indicates faster exponential growth.For example, measles is known to be one of the mostcontagious diseases, with 12 ≤ R ≤
18, which meansthat each measles-infected person may spread the virusto 12 to 18 other individuals in a susceptible population[10]. For comparison, the CDC estimates that COVID-19 has an R value of about 5 . B. An Age-Structured SIR Model
For many diseases such as COVID-19, the effect on dif-ferent age-groups varies drastically. Therefore, we con-sider an age structured model in which we compute theage distribution of each compartment in each of the age-brackets 0-9, 10-19, . . . , 70-79, and 80+. This separation,in particular, is much more specific than the one used in[1] and thus allows us to have our results in a more refinedway.For an age-structured model, we must incorporate anage-contact matrix M describing the rate of contact be-tween each pair of age-brackets. In the present paper,we shall use the same matrix used in [4] based on datacollected by [17] for the United States, shown in Figure1 below. In this setting, the values in M are propor-tional to the total number of contacts per time betweenage-brackets, divided by the product of their populationsizes. In particular, M would be a constant matrix if in-dividuals were equally likely to contact each other acrossall age-brackets. FIG. 1: Age-Contact Matrix M . Following [4], in our age-structured SIR model we de-fine vector valued functions S ( t ), I ( t ), and R ( t ) repre-senting the age-distribution of the total individuals sus-ceptible, infectious, and removed respectively by lettingthe ith coordinate indicate the number of individuals inthe i th age-bracket for 1 ≤ i ≤
9. Then, from [4], thedynamics of the model are given by the following equa-tions: d S i d t = − β · S i N · n (cid:88) j =1 M ij · I j (4)d I i d t = β · S i N · n (cid:88) j =1 M ij · I j − γ · I i (5)d R i d t = γ · I i (6)Let vector p denote the proportion of the population ineach age-group, and let λ and v be the dominant eigen-value and corresponding eigenvector of M · diag( p ). Inthe initial state of the epidemic, the growth rate of trans-missions follows a steady state, i.e. I ∝ d I d t . It is shownin [4] that in this state, the value of R can be computedas β · λγ , with the initial infected distributed according to v . Therefore to emulate the R value of COVID-19, wecan assign β = R · γλ where γ = , indicating a 14-dayinfectious period.Consider the vectors h = { h , . . . , h } , c = { c , . . . , c } , and m = { m , . . . , m } to be the hospital-ization rate, ICU rate among hospitalizations, and mor-tality rates, respectively for each age-bracket labeled by i . Then, one can compute the vector valued functions H ( t ), C ( t ), and M ( t ) representing the age-distributionof the total individuals hospitalized, in critical care, anddeceased respectively through the following differentialequations: d H i d t = γ · h i · I i (7)d C i d t = γ · h i · c i · I i (8)d M i d t = γ · m i · I i (9)In what follows we shall use the COVID-19 estimatesfor these values from [6] shown in Figure 2 to understandthe above functions. FIG. 2: COVID-19 Age Statistics
III. SOCIAL DISTANCING IN WASHINGTONSTATE, USA.
In what remains of the manuscript, we shall pay specialattention to the COVID-19 outbreak that took place inWashington State, USA since January 2020, and use thedata available to model different Social Distancing strate-gies. The first confirmed case of the COVID-19 pandemicin the United States was announced in Washington Stateon January 21, 2020. Five weeks later, on February 29th,Washington also announced the first COVID-19 relateddeath in the country. On March 23, Governor Jay Inslee issued the first stay-at-home order which lasted until theend of May [15].On May 29th, Inslee announced a Safe Start: a fourphased county-by-county reopening plan. The plan al-lows counties to gradually relax social-distancing mea-sures based on their assessments of health care systemreadiness, testing capacity and availability, case and con-tact investigations, and ability to protect high-risk pop-ulations. One of the main factors determining a county’sreopening procedure is the percentage of ICU beds avail-able in hospitals. Therefore to model the effect of social-distancing policy it is useful to scale the contact matrix M by a value proportional to this percentage: M → λ · | C | /C max M (10)where | C | is the total number of individuals in criticalcare while C max is the ICU capacity which we set to theUS average of 34.7 per 100,000 residents. As we men-tioned before, this leads to our age-compartment model,utilizing a rescaling function completely based on the pol-icy that Washington implements where the intensity ofthe social distancing policy is proportional to the ICU oc-cupancy. We refer to λ ·| C | /C max as the mitigation factor.The constant λ determines the “strictness” of the social-distancing measures. Such reactive mitigation measureshave been done before in SIR models (e.g., see [2] withrespect to total infected count).In order to understand the implications of the constant λ , we should note that a larger λ value has the effect of“flattening the curve”, decreasing total case count whilealso slowing the rate of decline in cases. Moreover, onecan see in Figure 3 the effect of λ on the proportion ofthe population (with Washington state demographic pa-rameters) infected and in the ICU ( I and C respectively).For consistency, we use a λ value of 0 . FIG. 3: Effect of λ in SIR-model. A. Demographic Parameters
The main objective of our model is to investigate theeffect of a population’s age distribution on the transmis-sion and spread of a virus like COVID-19, which is bothhighly contagious and largely age-specific in its effect onthe population. To aid in our comparison of popula-tions, we select four sample counties of Washington statewith varying age-distribution: Jefferson, King, Ferry andAdams (outlined in red in Figures 4, 5, 6) and apply ourmodel to their demographic parameters.For each county, we use the same parameters specificto COVID-19, but adjust the initial state of the function S ( t ) based on the age-distribution of the county, whichcan be seen in Figure 4, obtained via the official govern-ment’s data in [14]. FIG. 4: Median Age.
As shown in Figure 4, there is significant variation inthe age distribution of each county with counties alongthe west coast such as Jefferson and Clallam with a me-dian age of over 50 years, while counties in central andeastern Washington such as Adams and Whitman witha median age of under 30 years. Due to this variation,we expect an epidemic such as that of COVID-19 to havesimilar variation in its effect on the population, from rateof spread to mortality rate, and thus warrant different,age-targeted mitigation measures.In order to understand the the relevance of the mortal-ity rate within our study, there are two other importantdata points to consider, which are the proportion of thepopulation in each county over the age of 60 and 80,shown in Figures 5 and 6 respectively. These two popu-lation groups represent the age-brackets most vulnerableand are generally a better indicator of the overall popu-lation mortality rate.
FIG. 5: Percent of population over 60. FIG. 6: Percent of population over 80.
IV. MODEL APPLICATION
As mentioned previously, the four counties of Wash-ington we selected for our model comparison are Adams,King, Ferry, and Jefferson. These counties have medianages of 28.3, 36.8, 49.2, and 57.9 respectively. The com-plete age distribution of the selected counties are shownin Figure 7, and in this section we shall apply our SIRmodel on each of the four selected counties.
FIG. 7: Age Distribution of Selected Counties.
In order to understand the relation between the pro-portion of the population infected and the median age ofa county, we consider Figure 8 which displays the cur-rent proportion of the population infected and in inten-sive care over time for the four selected counties. Mit-igation measures are the same as shown in 3, appliedwith λ = 0 . FIG. 8: Homogeneous Mitigation Comparison, where the model was run until the proportion of the population infected fellbelow 10 − . The
Herd Immunity Threshold (HIT) is the criticalproportion of the population that must become immunefor an epidemic to longer persist. In an SIR model, theHIT value is given by 1 − R ≈
82% for R = 5 .
7. In orderto understand the HIT for our model, we first considerthe peak and total proportion infected and in intensivecare for each of the four counties, as seen in Figure 9.
FIG. 9: County Infected/ICU Statistics
Note that in all four counties, around the same propor-tion of the population became infected while a significantproportion (about 20%) never became infected through-out the course of the epidemic. Therefore a state of herdimmunity was achieved in which a large enough propor-tion of the population achieved immunity though previ-ous infections, thereby reducing the probability of newinfections, eventually halting the spread of the disease.Since we are interested in understanding effects of mit-igation strategies for the less vulnerable population ( < <
60 years),the probability of infection is roughly the same regardlessof age-bracket and population age-distribution, as can beseen in in Figure 10.
FIG. 10: Proportion of Age-Bracket Infected
On the other hand, for the more vulnerable popula-tion ( ≥
60 years), the probability of infection increasessignificantly with the median age of the population. As aresult, counties such as Ferry and Jefferson not only havea larger vulnerable population, but also have a largerproportion of their vulnerable population infected, whichgreatly contributes to their mortality rate.
A. Effects of Age-Specific Policy
By shifting the distribution of the infected populationaway from the vulnerable population, the mortality rateof an epidemic can be reduced significantly. In what fol-lows we shall examine the effect of age-specific policyincluding partial opening of schools and workplaces thatprioritizes and targets the more vulnerable over less vul-nerable populations.We first examine the effects of relaxing school andwork restrictions. For each scenario, we choose a relaxedbracket: the part of the population unaffected by the so-cial distancing policy (that scales the contact-matrix bythe mitigation-factor).
FIG. 11: Population affected in relaxing school restrictions
For example, when relaxing school restrictions, the tar-geted population is all individuals <
30 years, meaningall contacts amongst this group (the blue group in Figure11) will not be subject to restrictions, while the remain-ing contacts (the red group) will be subject to normal restrictions given by the mitigation-factor. For relaxingwork restrictions, a similar group is relaxed, targeting allindividuals <
70 years. Note that relaxing work restric-tions is applied on top of relaxing school restrictions asthe school age-bracket is a subset of the work age-bracket.Comparing the statistics from the table in Figure 12below, to that from the table in Figure 9 presented be-fore, we find that, on average, peak infections increasedby 58% when relaxing schools and increased 160% whenrelaxing work. In both cases, total infections increasedslightly, continuing to remain around the herd immunitythreshold of 82% as expected.
FIG. 12: Age-Specific Policy Statistics
The current proportion of the population infected andin intensive care over time with fully relaxed schooland work restrictions respectively can be seen in Fig-ures 13 (a) and 13 (b). As infections among the re-laxed bracket increased drastically in proportion to therestricted bracket, the mean of the age-distribution ofthe infected population shifted towards the younger, lessvulnerable, bracket by the time the HIT was achieved.As a result, we saw that the mortality rate, on aver-age, decreased by 6 .
0% when relaxing schools and 12 . .
7% and8 .
1% reduction respectively.
FIG. 13: (a) School bracket relaxed ( <
30 years); (b) Work bracket relaxed ( <
60 years).
FIG. 14: (a) School bracket relaxed ( <
30 years) at 90 days; (b) Work bracket relaxed ( <
60 years) at 90 days.
In contrast to the above, it should also be noted that,on average, peak ICU occupancy increased by 18 . .
2% when relaxing work,significantly more than the increase in peak infections.Reducing transmissions across the younger age-bracketshas the effect of “flattening the curve”: reducing peak in-fections and ICU occupancy while infecting roughly thesame proportion of the population over a larger span oftime. Although relaxing school and work restrictions re-duced the calculated mortality rate, in practice, increas-ing peak ICU occupancy by up to 50% can put excessivestrain on hospitals that are at full capacity, leading toadditional moralities from a lack of resources needed totreat everyone requiring intensive care.We also examine the effect of relaxing school and workrestrictions after 90 days, roughly a month after the ini-tial peak in infections. Figures 14 (a) and 14 (b) displaythe current proportion of the population infected and inintensive care over time with school and work restrictionsrelaxed at 90 days respectively. The vertical blue line in-dicates when the restrictions are relaxed for the targetedage-bracket. We find that relaxing school restrictions at90 days (a) has little effect on the subsequent trajec-tory of the epidemic for all counties, with no change inpeak infections and ICU, and mortality rate increasingby an average of 3 .
5% compared to constant restrictionsin Figure 9. When relaxing work after 90 days (b), wesee a notable change in the trajectory of the epidemic incounties with a higher median age. In Jefferson county,infections reached reached a new peak of 1 . · − , iden-tical to the peak before the relaxation. ICU occupancy inthe county also increased for a period of 30 days follow-ing the relaxation. Among all counties, the total infectedincreased by an average of 3 . . B. Effects of Age-Targeted Vaccination
Vaccinations play a critical role in mitigating the ef-fects of an epidemic. By directly preventing the suscepti-ble population from contracting the disease, it is possibleto achieve herd immunity in less time and significantlyreduce peak infections and mortality rate. An importantaspect of vaccines is the strategy considered by a govern-ment or society in order to achieve the desired proportionof vaccinated population – and much research has beendone in this direction for long time know infections (e.g.,for Zika and Hepatitis B, as discussed in [3, 7]) as wellas for recent viral outbreaks such as COVID-19 (e.g, seefor example [19]).In what follows, through our modified age-structuredSIR model , we shall examine the effect of prioritizingcertain age-groups in vaccine distribution versus a ho-mogeneous distribution across all age-groups. We modelvaccinations by directly transferring individuals from thesusceptible and removed groups. In particular, if vector µ represents the number of individuals in each age-bracketvaccinated at each day, then we have the following:d S i d t = − β · S i N · n (cid:88) j =1 M ij · I j − µ i ; (11)d I i d t = β · S i N · n (cid:88) j =1 M ij · I j − γ · I i ; (12)d R i d t = γ · I i + µ i . (13)To select the distribution of vaccines, we define aweight vector ω that represents the priority of each age-bracket in our distribution. FIG. 15: (a) Relative weights of vaccine distribution; (b) ge-Targeted Vaccine Distribution Statistics
Using the weight vector, we define µ i = T · ω i · S i | ω (cid:12) S | , where T is the total number of vaccines administered ateach day and | ω (cid:12) S | is the sum of the elements of vector ω (cid:12) S . Note that when ω is constant, each member ofthe susceptible population is equally likely to receive avaccine. The United States produces enough flu vaccinesyearly for approximately half of its population [8], andthus for our model we shall set T = N/
720 where N isthe population size.For our age-targeted distributions we provide thehigher age-brackets moderate priority with vector ω M and strict priority with vector ω S . We let ω C denotethe constant weight vector for the homogeneous (control)distribution. The weight values selected are summarizedin the table of Figure 15 (a), and the model statistics foreach distribution are given in Figure 15 (b).We shall first consider the case of a homogeneous vac-cine distribution ( ω = ω C ). In this case, the infected count and ICU occupancy is shown in Figure 16 (a). Onecan see, in particular, that the administration of vaccineshad a significant mitigating effect on the epidemic, on av-erage reducing: • the peak infections by 5 . • the peak ICU occupancy by 7 . • the mortality rate was reduced by an average of28 . • the HIT was reduced, especially among high me-dian age counties, with 69 .
2% of Ferry county and59 .
2% of Jefferson county infected in total.This should be compared to the same analysis donefor the moderate vaccine distribution ( ω = ω M ) shownin Figures 16 (b). FIG. 16: (a) Homogeneous vaccine distribution ( ω = ω C ); (b) Moderate priority distribution ( ω = ω M ) One may also consider a strict priority vaccine distribu-tion ( ω = ω S ), for which the analysis is shown in Figures17 below. FIG. 17: Strict priority distribution ( ω = ω S ) By increasing the relative weight of higher age-brackets, we present a model where we administer morevaccines towards the older and more vulnerable popu-lations. As for general trends, we find that, comparedto the ω C distribution, increasing the priority of olderage-brackets one has: • an increases peak and and number of total infec-tions, • while a decreasing peak and total number of ICUoccupancy.This trend is expected as prioritizing older age-bracketsresults in a greater proportion of the younger populationsusceptible to infection who are more likely to becomeinfected and spread the virus, increasing total infections. Moreover, by reducing the proportion of the older pop-ulation susceptible, we also reduce their infections andICU occupancy, subsequently lowering the mitigationfactor. Furthermore, the age-targeted distributions werehighly effective in further reducing mortality rate, withan average 8 .
5% reduction for ω M and 15 .
7% reductionfor ω S . In particular, Adams, the low median age county,responded most effectively with a 9 .
4% and 19 .
5% reduc-tion in mortality rate for ω M and ω S respectively.To better understand how vaccinations limit thespread of the epidemic, we consider the proportion ofthe population susceptible to the virus over time inFigure 19, where the vertical lines indicate when eachproportion susceptible in each corresponding county fallsbelow (1 minus) the calculated HIT or 1 /R = 0 . FIG. 18: Number of days until herd immunity is achieved
This is the point at which the spread of the virus nolonger persists due to herd immunity, with the remain-der of the susceptible population being immune throughcontinued vaccinations. Although it results in greater to-tal infections, we find that age-targeted vaccinations areeffective in reducing the time required to achieve herdimmunity, with an average 7 .
3% and 11 .
6% reduction indays for ω M and ω S respectively. FIG. 19: Susceptible Proportion for: (a) the homogeneous priority distribution ( ω = ω C ); (b) the moderate priority distribution( ω = ω M ); (c) the strict priority distribution ( ω = ω S ) V. CONCLUSION AND SUMMARY OF OURWORK
In the present paper, we have introduced a modifiedage-structured compartmental SIR model using a func-tion that scales contacts by a factor proportional to thecurrent ICU occupancy in Washington State, USA, whichserves to emulate the phased social distancing policy im-plemented Washington State.Our modeled epidemic utilizes the same disease pa-rameters of the current COVID-19 pandemic with an R value of 5 . − R ≈ .
9% when relaxing schools and 51 .
2% when re- laxing work, significantly more than the increase inpeak infections.Although the calculated mortality rate decreased, in-creasing peak ICU occupancy by up to 50% can overloadthe healthcare capacity in practice leading to additional,preventable deaths. Moreover, we saw that(v) relaxing the school bracket after 90 days had littleeffect on the subsequent trajectory of the epidemicin all counties, with no change in peak infectionsand ICU occupancy;(vi) relaxing the work bracket at the same time hadnotable effects on high median age counties.In the particular case of Jefferson, infections reached anew peak, identical to that before the relaxation and ICUoccupancy also increased for a period of 30 days. How-ever, for all counties, the mortality rate remained mostlyunaffected.Finally, we analyze the effect of age-targeted vaccinedistribution. We model vaccinations by transferring aconstant number of individuals from the susceptible toremoved groups at each day. Under a normal homoge-neous distribution ( ω C ), the number of individuals vac-cinated in each age-bracket is proportional to the size ofits susceptible population. In contrast, in age-targeteddistributions, we apply a set of weights as shown in Fig-ure 15 (a), so that individuals in certain age-brackets aremore likely to become vaccinated, allowing us to targetvaccinations towards more vulnerable age-brackets withmoderate priority ( ω M ) and strict priority ( ω S ).On its own, administrating vaccinations homoge-neously as shown in Figure 16 had a significant mitigatingeffect on the epidemic with(vii) an average 28 .
2% reduction in mortality rate com-pared to without vaccinations.(viii) the proportion of the population infected falling be-low the expected HIT, especially among Jeffersonwith the epidemic infecting only 59 .
2% of the pop-ulation.When applying age-targeted vaccinations as shown inFigure 16 (b) and Figure 17, we found that(iix) peak infections slightly increased while peak ICUoccupancy decreased.(ix) a reducing mortality rate with an average 8 . ω M and 15 .
7% reduction for ω S compared to the homogeneous distribution, withAdams responding most effectively.This being due to a larger population of susceptible indi-viduals in the younger age-brackets who are more likelyto spread the infection while remaining less at risk forhospitalization. Finally, when plotting the proportion ofthe population susceptible, as shown in Figure 19, wefound that age-targeted vaccinations also(x) reduce the time required for the epidemic to achievethe herd immunity threshold by an average 7 .
3% for ω M and 11 .
6% for ω S .1 Final Remarks.
To conclude our work, we shall presenthere different ways in which our model might be ex-panded, as well as possible directions for future work.To provide better context for the extent of our result’simplications, we list a series of key assumptions we havemade, and which could be modified in order to make toexpand on our model;: • Our mitigation coefficient to model social distanc-ing policy is based strictly on a single parameter(ICU Occupancy). A more complex or modifiedmitigation coefficient may produce different results(e.g., one could consider economic factors, such asthose studied in [5]). • In our SIR model, we do not consider asymp-totic individuals and the possibility for re-infection(which in the case of COVID-19, one may want toconsider [9]). We have assumed that all individualswithin an age-bracket are equally likely to becomeinfected and transmit the disease.Within our work we assume independence in the policybetween different counties and do not consider the move-ment of individuals between populations, which is some-thing that would be interesting to incorporate. More-over, one should note that relaxing restrictions immedi-ately affects all targeted age-brackets and has no affecton any contacts including individuals outside of theseage-brackets. Finally, we have assume that vaccine pro-duction is constant throughout the course of the entireepidemic and we do not consider possible changes in sup-ply and demand: it will be most interesting to incorpo-rate the economic factors involved in vaccine productionwithin an age-targeted study such as ours.When considering other mitigation coefficients whichcould be used, we see the following alternatives as poten-tial paths for expanding our work further: • The infection rate ( dd t I ) is another metric used todictate policy in states such as New York and Cali-fornia, which could be considered. Percentage pos-itive tests, measured as the proportion of the pop-ulation infected ( | I | /N ), is another factor in statessuch as North Carolina and Georgia used to indi-cate the extent of disease spread. • Instead of gradual/proportional restrictions, iso-lated populations such as those of New Zealand,implemented strict lockdowns within the first caseswith the goal of eradicating the disease before anypossibility of herd immunity. Stricter policy canbe modelled by increasing the λ factor or scalingcontacts by a factor of | I | or | I | within our model. Disclaimer.
As with all mathematical models thatare applied to real world systems, our results are validonly under the model’s assumptions. As such, the goalof our research is not to convey specific public healthinformation and risks, but rather be a tool for healthstrategists for better planning and awareness withrespect to policy.
Acknowledgments.
The authors are thankful to MITPRIMES-USA for the opportunity to conduct this re-search together, and to James Unwin for inspiring con-versations. The work of Laura P. Schaposnik is par-tially supported through the NSF grants CAREER DMS1749013, and she is thankful for the flexibility that theUniversity of Illinois at Chicago, and the Mathematics,Statistics and Computer Science department in partic-ular, has given her during these difficult childcare-freemonths of the pandemic.
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