key: cord-0948782-b4s6mz76 authors: Jahedi, S.; Yorke, J. title: The advantages of the simplest pandemic models date: 2020-07-01 journal: nan DOI: 10.1101/2020.06.23.20132522 sha: e35b944fa3743ead4e68f4bd62aa9360dff4a919 doc_id: 948782 cord_uid: b4s6mz76 As a pandemic of coronavirus spreads across the globe, people debate policies to mitigate its severity. Many complex, highly detailed models have been developed to help policy setters make better decisions. However, the basis of these models is unlikely to be understood by non-experts. We describe the advantages of simple models for covid-19. We say a model is simple if its only parameter is the rate of contact between people in the population. This contact rate can vary over time, depending on choices by policy setters. Such models can be understood by a broad audience, and thus can be helpful in explaining the policy decisions to the public. They can be used to evaluate the outcomes of different policy strategies. However, simple models have a disadvantage when dealing with inhomogeneous populations. To augment the power of a simple model to evaluate complicated situations, we add what we call satellite equations that do not change the original model. For example, with the help of a satellite equation, one could know what his/her chance is of remaining uninfected through the end of epidemic. Satellite equations can model the effect of the epidemic on high-risk individuals, or death rates, or on nursing homes, and other isolated populations. To compare simple models with complex models, we introduce our slightly complex Model J. We find the conclusions of simple and complex models can be quite similar. But, for each added complexity, a modeler may have to choose additional parameter values describing who will infect whom under what conditions, choices for which there is often little rationale but that can have a big impact on predictions. Our simulations suggest that the added complexity offers little predictive advantage. either through death or recovery with at least temporary immunity. Mathematical models which are used 48 to describe individuals transitioning between the stages susceptible to exposed to infectious to removed are 49 usually called SEIR models or SIR models. For basic references on epidemic modeling refer to [1, 2, 3, 4] . 50 We refer to an infection's generation time, i.e., the mean time between being exposed and exposing 51 other individuals, as one period [5] . It seems to be approximately 1 week for covid-19, but reliable data 52 is lacking. 53 For each time period n, the average number of contacts that an infectious person has while infectious 54 is called the contact rate, βn. The contact rate varies with time, depending on interventions like social 55 distancing as well as seasonal fluctuations. Small changes in βn can result in large changes in the number 56 of cases, due to multiplicative effects [6] . In our models, we assume that infected people are immune after 57 recovery and remain immune for the duration of the simulations. 58 Model E, (E stands for exponential). When almost everyone is susceptible, we can model the 59 early stages of an outbreak by as follows, where In is the fraction of the population that is infectious in period n and βn is the disease multiplier for 61 each period. When βn is a constant, we write it as β. When β > 1, Model E has pure exponential growth, 62 and exponential decay when β < 1. For Model E, βn is the only parameter that must be chosen by the modeler. We say a model as simple 64 if the model's equation has only one parameter, namely the contact rate. Model E above and Model E+ Figure 1 : Fraction of the population that is infectious using Model E. Both curves show the fraction of the population that is infectious. One period is average time interval between successive infection generations. The red curve assumes a contact rate of β = 2. The blue curve illustrates the effect of an intervention, undertaken between periods 6 and 11, that temporarily reduces the contact rate below 1. The same data are plotted in both panels; the vertical axis is a linear scale on the left and a logarithmic scale on the right. Infectious diseases can eventually deplete the susceptible population and so do not increase exponen-79 tially forever. Hence, we should consider a more realistic model. When proposing a model epidemiologists 80 should have in mind that policymakers should understand a model so that they can make reasonable plans. There are wildly varying sources of advice available. Policymakers are less likely to adopt advice based on 82 models they do not understand. 83 In the following we have proposed two models, a simple model, Model E+, and a slightly complex 84 model, Model J. We will compare the output of these two models and we will show that Model E+ can 85 follow the outbreak as close as model Model J. But any complex model will have many parameters for 86 intra-group contact rates for which there is negligible data, so if plausible choices are used, a wide variety 87 of predictions is inevitable. For example refer to Figs. 5 and 6. Now we introduce the Model E+. 88 It's all about contact rates. 89 Model E+ below is designed for a single, "homogeneous" region; i.e., it assumes that the fraction of 90 people who are infectious is uniform throughout a region. Specifically, it assumes that at each moment in 91 time the fraction of the population that is infectious is uniform throughout the region. This hypothesis is 92 an approximation since for example, a susceptible who is living with an infectious person has an elevated 93 probability of becoming infected. But we do not know what that probability is, so we neither include that 94 nor any other non-homogeneities in our model. 95 We assume the fraction of the population who are infectious at the beginning of the simulation (n = 0) 96 should be large enough that immigration of new infectious people is no longer a significant factor in 97 transmission. While Model E is appropriate when almost everyone is susceptible, the following is more general because it includes the fraction susceptible in week n. It is accurate as long as βn · In is small. That means, it is quite unlikely for one person to be contacted by two different infectious people in the same period. In+1 ∼ Sn · βn · In, (2a) where Sn and In are the fractions of the population that are susceptible and infectious respectively in week 99 n and βn is the contact rate in week n. "Model E+", a simple SIR model is our primary model for evaluating intervention strategies. It provides reasonable simulations of an outbreak through the peak as susceptibles become depleted. It 102 corrects the defect of Eq. 2a when βn · In is large, say larger than 0.3. Then the number of contacts is 1 − e −βn·In ) ∼ 0.26. The change to 0.3 might be considered acceptable especially if the true value of β is 104 uncertain by perhaps 20%. The probability of having no contacts in a particular week. If the expected or average number of events in a time period is λ, and the events are independent, the Poisson probability that no events occur is e −λ . The events in question here are contacts in one-week periods. In week n, susceptibles average λn = βn·In infectious contacts, so according to Poisson distribution, the probability of remaining uninfected is e −βn·In , so the probability of becoming infectious is 1−exp(−βn·In). Hence In+1 = Sn· 1−exp(−βn·In) . The fraction of people in week n having a contact with an infectious person is 1 − e −βn·In and when that is small (i.e., most of the time), We use the following "Model E+" where Sn and In are the fractions of the population that are susceptible and infectious respectively in week 106 n and βn is the contact rate in week n. People exposed in period n are infectious in period n + 1 and are 107 removed after that, either becoming immune for the duration of the simulation or dying. People infectious 108 in week n + 1 were exposed in week n so Model E+ can be considered an SEIR model as well. between models, perhaps leading you to question a model's validity. Following the beginning of the lockdown, The New York Times [14] asked several modeling groups for 138 predictions of covid deaths in New York City. Four groups provided predictions for four or more weeks; 139 they are labeled "C","I","M", and "L", for Columbia University's team [8] , IHME Institution's response to 140 covid [7] , MIT University's team [10] , and Los Alamos National lab's projections [9] , in Fig. 3a . Predictions 141 "C","M", and "L" show progressive decreases in deaths. Prediction "I" shows a more dramatic decrease. Here we would like to determine what makes it different. 143 We compute βn for each prediction as follows. Model E+ concerns infectious fractions, not 144 deaths. Deaths are often used as a more accurate record of how total cases vary than verified cases. For each 145 prediction, we treat the deaths as proportional to the total infectious fraction; i.e., deathsn/In = constant 146 for week n = 1, 2, · · · . Hence once I1 is chosen, all In are known. We choose S0 = 1 and S1 = 1 − I1, so 147 Sn can be computed from Eq. 3b. 148 We don't know I1, but for each choice of I1 < 1, we can solve Eq. 3a for each βn for that prediction. That means that with the computed βs, the prediction curves will be reproduced exactly by Model E+. 150 We do not expect βn to change much during a lockdown, so we find the value of I1 for which the βn curve Why is Prediction "I" so different from the rest? Does its rapid decline in deaths simply mean the 154 contact rate was chosen extremely low? No. Fig. 3b shows that their contact rate is much higher than for 155 the other three. So why was there such a rapid drop in deaths? Our findings: The initial susceptible fraction S1 required for Prediction "I" is so low that 62% of infected and almost no one was left to become infected. An alternative hypothesis to explain Prediction "I" is that at this time when we expect the contact rate to be constant, when the other groups provide 161 predictions consistent with nearly constant contacts, the modelers of "I" predicted the contact rates, β, 162 would change quickly, dropping rapidly to near 0, thereby stopping the outbreak. No outbreak will persist 163 if the modelers assume the contact rate is near 0. Any policy setter should want to know why either the infection rates were so high, infecting 62% of 165 the remaining susceptibles in one week, or alternatively what made the contact rates drop to near 0. Running Model E+ will exactly reproduce the each of the four prediction curves, that 167 is, when Model E+ is run with the plotted βs together with I1. We use S0 = 1. If four complex models [14] reported on four simulations of deaths with projections for 4 to 6 weeks, shown here with weekly totals. The simulations included large uncertainty intervals that are not shown here. Write D n for deaths in week n. Panel (b): Contact rates β n . We assume that deaths are a constant fraction of actual cases, so D n+1 /D n = I n+1 /I n . Hence, if one has the initial fraction, I 1 , of susceptibles who are infectious and the contact rates β n , then Model E+ will exactly reproduce the curves in panel (a). See the text for a description of the computation of β and the values for I 1 s. Estimating β for the current covid-19 outbreak. Covid-19 began spreading perhaps in early week, so the contact rate β is at least 1.9. During this time China made major efforts to reduce contacts 178 among its population, making β smaller than in areas without such efforts. 179 We take β to be 2 or 3 in most of our simulations. In large cities whose transportation is dominated 180 by mass transit, the growth rate may be much higher. Two recent papers [11, 12] estimated early growth 181 rates as a factor of about 10 or more per week. It shows how the outbreak's peak depends upon the contact rate. The figure illustrates that a higher contact 184 rate causes a more severe outbreak with a higher and earlier peak. Fig. 4 also explains how to produce 185 the outbreak daily using model 1. Each point on the curves is the infectious fraction for the preceding 7 days, with the horizontal axis being the number of days. The numbers in the graph show each outbreak's peak-week infectious fraction. To get a smooth curve here and in the previous figure with daily results, we have run Model E+ seven times for each curve, as follows. Early in the outbreak, the fraction infectious increases by the factor β per week, i.e., β 1/7 per day. For each β, the simulation is run seven times with initial infectious β j/7 , j = 0, 1, . . . , 6, and the plot is shifted to the right by j days, i.e., j 7 weeks. Model E+ with "satellite" equations. We need models that can inform us about all possible 187 states in our population, and assess which ones matter for shaping the pandemic's trajectory over time. What will happen to those with pre-existing conditions or older populations or those with different contact 189 rates? How will nursing homes and long-term care facilities be affected? How may will have long-term 194 which in this case is Eq. 3a. The primary system does not depend on the new variables being reported. We 195 emphasize ability to add some complexity without compromising the intelligibility of the original system. 196 We give two examples of satellite equations. A satellite equation for predicting deaths. It is well recognized that if hospitals become over-198 loaded with seriously ill covid patients, the probability that a hospitalized patient will die is elevated. We 199 now outline a scenario that would have to be refined using actual data of hospitalized covid cases. Suppose Suppose the luncheon is in a confined space such that if B is infectious, A has a 50% chance of being 210 exposed. What are A's chances of becoming infected during the outbreak? We proceed by first making 211 some general calculations. Model E+ uses an average contact rate. To investigate an individual whose contact rate differs from the average person's, we can adapt Model E+ by adding a satellite equation. A person may have a very low contact rate, due to wearing protective gear or staying at home, or perhaps the rate is very high due to being in crowds frequently. We denote this person's contact rate in week n by γn, which can vary weekly. Let Pn denote the person's probability of not being infected by week n. We can set P0 = 1. Here we use Model E+ to determine In and Sn. We then use In in a satellite equation for determining Pn, Notice that Pn+1 can also be written as follows If a person maintains a constant γ, then γ · Ij = γ · (S0 − Sn+1). Choosing S0 = 1 yields the probability 214 of remaining uninfected at the end of the epidemic, where S final denotes the final fraction of susceptible people at the end of the period simulated. To decrease 216 the risk of getting infected, Eq. 8 suggests keeping γ low. Now we return to the life of A. We have in effect said that A has 0.5 contacts per week and is otherwise 218 isolated. That means γ = 0.5. It is likely that some of the B's that A has contact with will become infected at some point during the 220 outbreak. But A will be exposed only if some B is infectious during the week they have lunch. Suppose at or a 28% chance of becoming infected as a result of these lunches. A satellite equation for a population with no infections, e.g., a nursing home or an island. Let A denote a relatively isolated population with no cases currently, such as the resident population of a 226 nursing home or a small country or region. Suppose a population A has N visitors per infectious period 227 (which we can think of as one week), and the visitors come from a place B that is experiencing an outbreak. Assume the infectious fraction in B is In in period n. There are different types of visits. Consider the 229 "long-term" visitor, who might be staying more than one period or equivalently is a resident who returned 230 home after a long visit elsewhere. Here "long" means more than one period. A "short-term" visitor might 231 stay for a day. 232 We can show under reasonable hypotheses that such a person will on average have had half his or her 233 contacts before arrival and half after. The probability that such a visitor is infected is In. The expected 234 number of transmissions per period is (β/2) · In · N . Over the duration of the outbreak, the expected 235 number of primary introduced transmissions T is the sum as in Eq. 8. Suppose for example that during the outbreak, 2% (= 1−S final ) of the external population was infected and The purpose of a lockdown policy is to cause each person who is infected to infect fewer people. We discuss 243 the United States here but the results are applicable to many countries and to smaller regions. Before the spring 2020 lockdown period, each infected person infected on average more than 2 people. 245 We discuss policies to control diseases [?] by means other than using drugs. that there were fewer than 10 million infected people. That is based on the following. The U.S. has had 251 less than 10 5 deaths. Using a 1% death rate, we extrapolate that the number of infected people in the 252 U.S. was less than 10 million, including those whose infections were mild or non-symptomatic. The costs of the intervention to the US government is at least $2 · 10 12 , the price of the 2020 Stimulus 254 bill called the "CARES act". We estimate that when we add lost wages and the cost to the private sector, Eikenberry et al. in [15] discuss the impact of wearing face masks in preventing contacts. Suppose 261 moderately effective masks were available that when worn would prevent 60% of transmissions. If everyone 262 wore them 80% of the time, then 60% of 80% of the cases would be prevented. That is, 48% of all the 263 cases would be prevented. This is not an economics paper so we will not estimate costs of intervention 264 methods. Many of the interventions below can be thought of as mutually independent, as a first approximation. Each interrupts some fraction of contacts. If an intervention reduces the remaining infectious contacts by some factor Fj, then βn is reduced to βn · (1 − Fj). If N independent interventions are applied, then βn is reduced to We believe the implementation of each would reduce contacts by some factor 1 − Fj. The fraction Fj can 266 only be estimated, would vary over time, and would depend on the implementation. There are also additive policies. Banning daycare and closing schools both can reduce contacts, but 268 they ban different contacts, so we might say that together the fraction of contacts blocked could be written 269 F daycare + F schools . Similarly banning sports gathering and religious gatherings would likely be additive. Each of the following actions or policies would disrupt some fraction Fj of the contacts between 271 infectious and susceptible people. (FSD1) Isolate Infected individuals -those who are not seriously sick to be hospitalized rather than 273 sending them home where they might infect house-mates. (FSD2) Quarantine exposed individuals. Some factors may be important in saving lives while not falling into the above calculus. • Supporting isolation of the most at-risk individuals. • Testing for the virus must be fast and widely available in order for contact tracing to be effective. • Contact tracing. When we think of the goal of interventions to be the disruption of contacts of infectious people, we realize 301 that even limited contact tracing can play a role, such as contacting the people who were most likely to 302 be exposed. We should ask what the cost of interception and quarantining exposed individuals is. 303 3 A mildly complex "Model J". For comparison of conclusions of Model E+ with more complex models, we create our Model J by adding 305 two reasonable and common complications: contact rates that vary from group to group within the popula-306 tion and infectiousness that depends on how much earlier the infected person was exposed. They improve 307 the realism -provided we have data so that we can accurately set the parameters of the model. The 308 downside of this added complexity is that the system is harder to work with and the results are more 309 difficult to communicate to the people who are being advised. To overcome this downside, here we propose 310 an approach that can make complex models more manageable: only one constant must be determined 311 precisely after other constants. Basically, we are more likely to know that A has about twice as many 312 contacts as B than what those actual contact rates are. The elements of this model are the following. • Some people have more contacts than others. The population is partitioned into K equal-sized • Infectiousness depends on how long a person has been infected. The "infectiousness" or likelihood 323 of transmitting the infection on day j after being exposed is proportional to the constants χj. The 324 collective infectiousness of people in group i who were exposed j days earlier is proportional to ) . Hence, the collective infectiousness also depends on the contact rates. Summing over all i and j, E m (d) is proportional to The term Infec(d) is proportional to the size of the total infectious population on day d; therefore, 329 it is proportional to the level of danger to the community on day d. • The novel feature of this model is that only one parameter must be determined precisely in the 331 model. We can make Eq. 10 exact rather than proportional by multiplying it by the appropriate 332 value, J. The actual transmission rate is J is independent of m, i, and j. Below we show how to select J so that the outbreak has an initial 334 growth rate β. Typically we choose β = 3. J is time independent if there is no intervention, but 335 interventions can make J = J(d) depend on day d. In summary, the resulting Model J follows. For each group m, 1 ≤ m ≤ K, we have Tuning Model J to achieve a specific growth rate β. Suppose we are observing an outbreak that grows by a factor of β for a time period τ , and suppose we want to tune Model J so that it has the same growth rate initially. There is a value of J such that for any constant C, the following is a solution of Eq. 13a: It is independent of m, j, C, and d. we divide the population into 10 groups that can have different contact rates, and exposed people are 352 infectious for 5 days, and they are equally infectious for each of those days. Which days they are infectious 353 depends on the case, see table 1 for more details. For most cases we set the five days to be 5-9 days after 354 exposure. When there is no intervention, for example in simulations of Model J in Fig. 5 , the parameter J is 356 chosen constant (independent of d) so that initially the epidemic grows by a factor of 3 per week. That 357 corresponds to using β = 3 and τ = 7 days in Eq. 14. 358 Table 1 Figure Label color φ m infectious days S final there is more variation between the three Model J curves than there is between their average and the 362 Model E+ curve. We suggest that it is quite difficult to determine which of the three Model J simulations 363 best represents reality. We have seen in Fig. 4 that the outbreaks depend strongly on β. A small change 364 in β is more significant than whether Model E+ or Model J is used, and β has been difficult to determine 365 for actual populations. Hence we believe little is gained by using more complex models for setting policy. The cost of complexity would be a loss of intelligibility for most policy setters. Table 1 ). In both panels the red dashed curve is for Model E+ with a constant contact rate β = 3. In Fig. 6 we show two cases where the mean infectious time is either day 6 after exposure (using 373 infectious period days 4-8, Case E) or day 8 (using days 6-10, Case D). See Table 1 . In each case, Model 374 J has been calibrated though the choice of its parameter J so that the initial rate of growth is 3. Initial 375 conditions have been chosen so that during the early parts of the outbreak, the curves are aligned. Small 376 changes in the initial fraction infected shifts each curve to the left or right without otherwise changing it. Our main choice of χs assumes people are infectious for five days, days 5 through 9, and they are 378 equally infectious each of those days; that is χj = 0 except for j = 5, . . . , 9 when χj = 1. Model J is useful because we can test its sensitivity to a variety of choices of parameters. In such a 380 comparison, the exponential growth rate early in the epidemic should be the same for both models. 381 We find the differences in predictions between Model E+ and Model J are small compared with the 382 uncertainty of the parameters such as the initial growth rate or the death rate. Reasonable variations in choices of parameters yield a big impact on Model J outbreaks. In practice we don't know how much difference there is between the personal contacts of people nor do we 385 know precisely when the typical infected person is infectious. Suppose we want to model a city. We have to choose values for φs and χs. Fig. 6 shows outbreaks 387 using several choices. It would be difficult to know which choice is most appropriate. However in the right 388 panel of Fig. 6 , the fraction of the total population remaining uninfected at the end of the outbreak, S final , 389 is 15 times higher for case E (purple curve) than for case D (blue curve). Case D has a peak that is twice 390 as high as case Es. 391 The effect of Intervention 393 We model two policies to mitigate the severity of the outbreak using Model E+ and Model J. Policy I: A short-term intervention. 398 We investigate the effect of a 6-week intervention during which the effective contact rate β is reduced from 399 3 to 1.6. Such interventions in an Influenza epidemic can delay the growth of the epidemic [13] . Our 400 simulations show this delay for an early intervention. By an early intervention we mean intervention be 401 implemented more than three weeks before the uncontrolled outbreak would peak. Our simulations suggest 402 that an early intervention only delays the epidemic (blue curves in Figs. 7a and 7b) . A late intervention 403 is being implemented one week before the uncontrolled outbreak peaks. A late intervention will not have 404 a very useful impact on reducing the peak, but it will increase S final slightly (purple curves in Figs. 7a do not experience the disease will increase by more than 50%. (Golden curves in Fig. 7a and 7b) . 411 One might think that harsher restrictions will stop an outbreak. The three panels of Figure 8 show a 412 variety scenarios for interventions, all lasting from weeks 7-12, during which the contact rate is reduced. intervention. The milder interventions here allow the number of infections to grow and approach a herd 419 immunity. The severe interventions (green) cause a bigger drop in cases, which is follow by a large peak. The mild interventions result in more cases during the intervention but fewer total, because the pop- Figure 7 : Timing an intervention for optimal effect. In this simulation, the rate of contact is β = 3 before and after intervention, during an intervention β * = 1.6. The Green curves represent the uncontrolled outbreak without any intervention, the peak is about 28% and S final is about 6%. Panel (a) represents the fraction of infectious individuals and panel (b) shows the susceptible fraction of population. Here we have implemented 3 different 6-week long interventions. An early intervention (blue curves), starts at week 5 is not very interesting compared to an intermediate intervention. If the uncontrolled outbreak peaks at week n, then an intervention will be called early, intermediate, and late if it is implemented at least 4 weeks earlier than, 2 to 3 weeks earlier than and exactly one week before week n, respectively. As the figure illustrates an early intervention only delays the outbreak. A late intervention (purple curves), starts at week 8 considerably increases the fraction of people who remain uninfected at the end of outbreak. A middle intervention (golden curves) should be applied two weeks before the uncontrolled outbreak peaks. According to this figure the optimal effect, which is 50% reduction in peak and 50% increase in fraction of left susceptible, is achievable by the middle intervention. Policy II: Put a cap on the number of infectious cases. When facing a new pandemic, due to our lack of knowledge, even experts cannot predict the longevity of 425 the pandemic without a vaccine. As an article in New York Times reported [14] , most models do not suggest how long a lockdown 427 should continue, or under which conditions we can get back to our regular life routine. A common concern 428 is that a long recession will cause a huge In Fig. (9b) , we keep the number of infectious individuals lower than 2%. As Fig. (9a) illustrates by 438 applying this policy about 23% of individuals remain uninfected at the end of outbreak, which has been 439 increased by almost 8% in comparison with the otherwise uncontrolled outbreak. Tuning Model J to limit the fraction exposed daily. We will denote the fraction of the total population that is exposed on day d by E all (d). Hence, To keep E all (d) below some daily exposure rate that we call "target", we have to adjust J dynamically 441 and in practice that means interventions are needed. When E all (d) would exceed that level without 442 intervention, we decrease J(d) as follows: In Fig. (9d) by using Eq. 16 we keep the total exposed under 2% weekly. To make Model E+, which has a 444 time step of 7 days, comparable with Model J, which has a time step of one day, we plot the exposures per 445 week for both models. Note that for Model E+ exposures in week n equals the infectious in week n + 1. for disease transmission between the two groups. The following example illustrates how epidemics are chains of events whose probabilities are hard to 481 compute because each interaction modeled requires its own (unknown) contact rate. The uncertainty in 482 that contact rate exceeds the benefit of including the interaction in the model. Consider a chain of contacts. When two people live together, and one, "A", becomes infected, 484 what is the probability that the other, "B", will be infected? If an answer to this is obtained in one country, 485 will it be valid for others? New York City apartments might be different from Arizona ranches. Without 486 considerable accurate data, there is a large range of possible parameters values that must be investigated 487 when making predictions with no way to choose between them. Suppose the probability of the housemate ,"B", being infected is between P1 and P2 with P1 < P2. The "uncertainty factor" can be said to be P2/P1. If "B" now is infectious and might ride on a standing-room-only bus or train, how many will become 491 exposed as a result? Suppose the number is between N1 and N2 with N1 < N2. The uncertainty factor is 492 about N2/N1. Hence the possibly infectious person "B" might take a ride on crowded public transportation. (We may have a good estimate as to how often that happens.) The uncertainty factor of how many people 494 are likely to be infected is P2/P1 · N2/N1 by "B" during the ride. Next suppose that some uncertain fraction F2/F1 of the people on the bus go to a football game or 496 similar crowded event. Each infected person is likely to infect an uncertain number M2/M1 of attendees. Again the uncertain factors multiply. The uncertainty factor for this chain of events could be estimated as (P2/P1) · (N2/N1) · (F2/F1) · 499 (M2/M1). These ever-compounding uncertainties will contribute to errors in the average contact rate β. Since the above chain of infections involves 3 stages of transmissions, it creates uncertainty in β 3 if we are 501 using this chain as part of an estimation of β. Technically, one might argue that we should be discussing standard deviations or variances of each 503 event of the chain. That would require assigning a probability distribution to each step. Then the variance 504 of the log of the number of people infected in such a chain of events is the sum of the variances of logarithms. But we prefer to keep the discussion less technical. Tuning complex models so that it gets "reasonable" predictions. Of course all the uncer-507 tainties should be tuned so that the resulting β is in agreement with the data. Our Model J is designed 508 with that in mind. To minimize the problem of tuning, we use a single parameter J which is introduced Adding features to complex models. It is attractive to include many features when modeling an 520 outbreak. Each added feature enhances the appearance of reality. There are many possible refinements. Models can split the population into many small groups, perhaps by age, sex, their locations, or population 522 density. Models may also include gatherings for sports, music, movies, religion, politics, or holiday and 523 beach festivities. How many people at such a gathering will be exposed by one infectious person? It 524 depends on the type of gathering, and in any case the answers are not known. Before a model's predictions are released, its numbers will be adjusted so that the epidemic's growth 529 rate will be in agreement with the observed growth rate. The current growth rate is hard to estimate 530 because of the lack of consistent random screening data. However, the current growth rate is the aspect 531 for which there the most data is available, even if that data is inadequate. It cannot be derived from 532 collections of model features for which there is less data. 533 We believe that the fewer the features a model has, the more intelligible it and its assumptions are. 534 We find that the difference between Model E+ and Model J is small compared with changes in β. Our approach. To run a simulation using Model E+, Eq. 3a, a modeler chooses I0 and β. Those 536 who set policy can choose what β should be, then can choose and vary policies aimed at achieving the 537 target β. 538 We do not predict the future of an outbreak. Instead, we only discuss what will happen for different 539 choices of β. We select β directly. That will determine when the outbreak peaks, if a large enough fraction 540 of the population is infected. The policy maker makes choices that determine βn. The contact rate will 541 vary week by week as interventions change the contact rate. Estimating βn. We believe the most important factors in predicting the severity of an outbreak are the 544 contact rate, the hospitalization rate, and the death rate. A higher contact rate leads to a higher peak. Modeling interactions between people will not reveal the death rate. The contact rate β can be estimated 546 from data. The overall contact rate of the population is the easiest interaction parameter to evaluate. Policies I and II. Suppose the goal is to reduce the susceptible population so as to approach herd 548 immunity. We offered two different control policies. In policy I we only apply lockdown restrictions only 549 for a short period of time and then we relax the restrictions. All our simulation is done only for 6-week 550 long interventions. Of course a longer intervention will have a better effect on reducing the unpleasant effect of the outbreak, but also a higher economic cost. The benefit of policy I might be to provide enough time for supplying the medical establishment. If that 553 is the goal, it would be best to apply inexpensive interventions early, like requiring business personnel to 554 wear masks and applying social distancing. In the United States, during the covid outbreak, nonessential 555 businesses would not have been closed (high economic cost) if the use of masks (low cost) was recommended. According to our simulations, an excellent control for policy I is a middle intervention which should be 557 implemented two to three weeks before the uncontrolled outbreak peaks. It can reduce the peak by 50% 558 and increase the people who remain uninfected at the end of outbreak by 50%. A harsh intervention will 559 be more likely to be followed by a larger peak than a milder intervention. When we face a situation in an outbreak where we lack healthcare staff and lack space in our hospitals, it 561 is especially important to keep the total fraction of individuals who are infectious under a target percentage. Then we suggest implementing policy II in which authorities change the social-distancing policies weekly 563 to maintain the maximum number of infectious individuals under the bearable target. Interventions, their cost and their effectiveness. When choosing between interventions, it would 565 be advantageous to evaluate each intervention and estimate how many contacts will be disrupted by its 566 implementation and the cost per interrupted contact. The costs include jobs lost, businesses being closed 567 temporarily, or permanently. Interrupting contacts with people at high risk has extra importance. Complex 568 models and simple models with satellite equations are designed to take into account some of these types 569 of contacts. The data for such may not allow us to use the model to determine β, but it may be beneficial Infectious Diseases of Humans: Dynamics and Control Virus Dynamics: Mathematical Principles of Immunology And Virology Mathematical Biology: I. An Introduction, Third Edition A Note on Generation Times in Epidemic Models Recurrent outbreaks of measles, chickenpox and mumps. Seasonal variations in contact rates The Institute for Health Metrics and Evaluation (IHME), Covid-19 Projections Mailman School of Public Health, Epidemiology. Forecasts of Covid-19 cases Los Alamos National laboratory. COVID-19 Confirmed and Forecasted Case Data Massachusetts Institute of Technology, DELPHI Epidemiological Case Predictions The Effect of Large-Scale Anti-Contagion Policies on the Coronavirus (COVID-19) Pandemic Imperial College COVID-19 Response Team Estimating the Effects of Non-Pharmaceutical Interventions on COVID-19 in Europe Best practice assessment of disease modelling for infectious disease outbreaks What 5 Coronavirus Models Say the Next Month Will Look Like? To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic Special Report: The Simulations Driving the World's Response to COVID-19 Acknowledgments. We thank Ishan Saha, Eric Kostelich, Aleksey Zimin, Louise Raphael and James Watmough for their input.