The susceptible-infectious-recovered (SIR) models (1) , widely used in predictions of epidemic spread, predict that social distancing will "flatten the curve", i.e., reduce the total number of cases and delay the peak of new daily cases. The peak in distress is reduced by a modest fraction; but the misery lengthens in approximately inverse proportion. With this image highlighted in popular media, concerns about the economic devastation projected to be 25 associated with such measures applied in response to COVID-19 has many individuals and governments advocating that social distancing measures be lifted early (2) to shift the peak forward and end the economic, educational, and social disruption sooner.
In the real world, countries have applied different mitigation measures (3) (4) (5) to control the COVID-19 pandemic, with varied success. These natural experiments have provided an 30 opportunity to compare model projections of the effect of different containment measures against real world data.
A review of the seminal paper for SIR models reveals a fundamental flaw early in the model formulation. The paper states that "…the chance of an infection is proportional to the 35 number infected on one hand, and to the number not yet infected on the other" (1, p. 703) . The authors argue that the change in the number of infections at any time is proportional to the number of people infected, multiplied by the number of people not yet infected. Though seemingly logical, the statement implies that people who have not yet been infected associate only with infected people and vice versa. However, the interactions between a recovered person and a susceptible or infected person must also be accounted for because they "use up" interactions that could have occurred between an infected person and a susceptible person. Due to the omission of these "unsuccessful" interactions in the model formulations, the equations that describe the change in infected, recovered and susceptible people (Equations 1, 2 and 3) in the 5 SIR model and its variants lack any terms related to the recoveries.
One consequence of this omission can be seen in Equation 3 where the change in the recovered population as a function of time is depicted as linearly related to the total size of the infected population alone. This cannot be correct because the members of the infectious population that recover at any specific time were infected at a time in the past separated from the 10 current time by an interval called the recovery time. Given the various nonlinear relationships embedded in the model, it is illogical that the change in the recovered population, would be directly proportional to the total size of the currently infected population.
To determine the degree to which these flaws affect the model performance, we tested the capability of an SIR model (6) to predict observed trends in the COVID pandemic using the 15 following equations: 
where S = number of people susceptible to infection at time t; I = number of people infected at 25 time t; R = number of people recovered at time t; N p = total number of people in the population; β = rate of contact and transmission; t r = time of infectiousness; and γ = rate of recoveries = 1/t r .
In Figure 1A , we see that the SIR model projects that the time at which the total number of cases levels off is delayed as social distancing increases (represented by decreasing β ), while the ultimate total case number remains quite similar. Increasing social distancing also delays and 30 reduces (here by about half) the peak of new infections, while resulting in a broader peak (Fig.  1B) .
To further enhance the analysis of the model, we introduce here a new quantity called the rate of change operator (RCO, Fig. 1C ); the logarithm of the ratio of the change in daily cases to the time-associated total of cases (6.) This formulation produces a new function that summarizes that have comparable population densities but that implemented mitigation measures with different timings and intensities (3) (4) (5) 8) . In particular, New Zealand and South Korea had much earlier and stronger social distancing measures than Italy and Sweden.
In contrast to the SIR model predictions (Fig. 1) , the country data ( Fig. 2) show that stronger social distancing measures are associated with an earlier and lower peak in new 5 infections, an earlier levelling off at a lower number of total cases, and a steeper RCO slope. All three trends in Fig. 2 demonstrate that the SIR model (Fig 1A-C) is not merely inaccurate but actually projects epidemic data to trend in the opposite direction to the reported data. The SIR model simply does not match real world experience. All variants of the SIR model suffer from this fault and predict that the peak in daily cases will be delayed as social distancing increases. 10 Because the faults in the SIR models stem from their mathematical foundations, no amount of curve fitting will enable them to predict the observed trends. Rather, a new model formulation, built on the proper foundation, is needed.
Taking a new approach, we developed a new model using basic principles incorporating 15 the proper population interactions (6) . Because the model was developed using insight gained by casting the number of infected and recovered individuals as fractions of the total case count, we call the model the fractional epidemic (FE) model. The model is described by the following equations:
25 I(t) represents the currently infectious people, R(t) people who have been infected but are no longer contagious, and N(t) = R(t) + I(t). K T is a property of the disease that is a constant representing disease transmissibility, P c is the instantaneous number of contacts each individual in N(t) has with the entire population, and Figure 3A-C depicts the qualitative behavior of the FE model. With increased social distancing (higher K 1 ), the total number of daily cases is lower and attained earlier (Fig. 3A) , the peak in new cases is lower and occurs earlier (Fig. 3B) , and the slope of the RCO curve is 35 steeper (Fig. 3C) . Unlike the SIR model trends ( Figure 1A -C), the FE model trends ( Figure 3A -C) are identical to those exhibited by the real-life COVID-19 data in Figure 2 . Whereas the SIR models fail to predict the proper impact of social distancing, the FE model predicts the trends seen in the real data and, in particular, predicts that stronger social distancing produces an earlier peak of daily cases.
As a demonstration of the predictive capability of the FE model, we tested the model quantitatively on data from several countries by first estimating two country-specific constants, K 1 and K 2 . These were found by using the RCO expression (Equation 6) to transform each 5 country's daily total cases to a new time series.
Shortly after intervention measures were imposed in each country, the resulting RCO time series (Fig. 4) have distinctive segments where the curves appear to become straight lines. The parameters K 1 and K 2 , tabulated in in Table 1 , were then estimated by fitting a linear expression to a small, early (nine data point) portion of these straight segments. The date ranges 10 for these nine data point portions are listed in the table and these data points are highlighted in blue in Figures 4-6 . The derived values for K 1 and K 2 were then used to predict the balance of the epidemic data that followed (grey dots in Figs. 5 and 6).
Using this approach, the FE model correctly predicted the total cases with an R 2 > 0.97 (Fig. 5 , grey dots) for all but one of six countries for the 3 weeks following the 9-day data 15 segment used to determine the two constants in the model ( Table 1 ). The model also predicted daily new cases ( Fig. 6 ) for these countries with an R 2 range of 0.61-0.79 and the projected peak of new cases (Fig. 6 , grey dots) was very close to the observed peak for all countries.
The FE model projections in Figures 5 and 6 are not fits to the full-length of the data shown. Rather, a short portion of the epidemic data, with specified characteristics, was excised 20 for the purposes of determining coefficients of the model that were then used to determine the rest of the curves.
In an additional demonstration of the FE model's veracity, we tested the assumption that ‫ܭ‬ ் is a property of the disease; and therefore, should be the same for each country. Equation 71 (6) shows that the model parameters, expressed in a function, f(N(t)), should be linearly 25 proportional to time with a constant of proportionality −K T . The fit of Equation 71 to the country data from Table 2 has an R 2 = 0.95 and a slope (which is equal to −K T ) of 0.19. This excellent result, along with Figs. 3, 5 and 6, demonstrates that the FE model correctly characterizes epidemic dynamics from multiple countries in a unified way; something SIR models simply cannot duplicate. 30 To illustrate why the FE model predicts epidemic dynamics while SIR models do not, we cast the FE model in terms of S, I and R equations:
Additional insight is gained by using equation 22 (6) , to rewrite Equation 8:
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is the effect of the recovered population's interactions on the infection rate. The ratio of the recovered population to the total number of cases is the rate transmission slows; and, coupled with the behavior of the population 5 represented by K 1 , this controls the epidemic dynamics.
Because the recovered population interactions appear on the righthand side of Equations 8-11 either explicitly or implicitly (via
), the FE model correctly projects the observed data from multiple countries in a unified way. In contrast, no terms for the recovered population interactions appear on the right-hand sides of Equations 1-3; and, therefore, the SIR models fail 10 to predict the trends in the observed data. This is seen explicitly in Equation 9 which contains the effect of the recovered population that is missing in Equation 3.
Quantitative mathematical relationships, derived within the FE model (6) , characterize the dynamics of an epidemic and illustrate that strong and early intervention is critical. This is 15 highlighted by Equation 46 (6) which projects the ultimate number of individuals infected in an epidemic, N ∞ , to be exponentially dependent on the number of people a with which a person interacts. The country data provide vivid examples. Both South Korea and New Zealand enacted strong and early interventions compared to other countries (3, 4) . This is reflected by their K 1 values (Table 1) . These strong interventions led to earlier peaks in new cases and to far fewer 20 cases than in other countries ( Fig. 2) : the peak number of new cases in South Korea and New Zealand was 90-99% lower than in other countries, compelling evidence that strong intervention led to exponentially more favorable outcomes.
In a further illustration of the importance of early intervention, actions in the USA began to have an effect around March 26 (Fig. 4B ); the number of active cases on that date was 46,442 25 (Table 1 ). Using the values of K 1 and K 2 in Table 1 , Equation 46 predicts that the ultimate number of cases should have been approximately 1.4 million. In contrast, had the intervention been implemented and sustained starting on March 10, when there were 62 times fewer (754) cases (7), the model predicts that the ultimate number of cases would also have been 62 times lower, or 23,500. Earlier action could have reduced the ultimate number of projected cases by 30 more than 98%.
The projected estimate of ~1.4 million total US cases would only have occurred if the effectiveness of the interventions launched on March 26 had been sustained. Unfortunately, a a marked reduction in effective interventions occurred widely in the USA in mid-April, well before the official reopening of the economy (9). This caused a second surge in new cases and a 35 new, third surge has been caused by additional reopening activities in early fall.
The FE model also provides an estimate of the time to the peak of new cases, t max . Using Equation 54 (6) and the values of K 1 and K 2 from Table 1 , the predicted peak in new cases in the USA would have occurred near March 24 if the intervention had begun on March 10. Instead, the USA reached an initial peak on April 11 (7), 18 days later, and it was much higher (Fig. 6B ).
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The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint Epidemic acceleration, the instantaneous potential to change the pace of the epidemic, can be determined at any point in the epidemic and depends on the social containment actions in effect at that time (Equation 55 (6) ). Two countries with identical numbers of cases on a given day can, in fact, have different accelerations on the same day, and exhibit different dynamics immediately after that day. 5 For example, South Korea and New Zealand ( Table 1 ). However, since South Korea has a much higher population density than New Zealand (Table 2) , it had a much higher number 10 of interactions when the interventions were imposed, and therefore, a higher rate of acceleration. Indeed, the rate of change of new cases was higher in South Korea than in New Zealand; and, the later number of cases in South Korea was higher than New Zealand's ( Fig. 2A, B) .
The lessons are clear: early, and strong interventions, especially in countries with indigenously high levels of social interaction, are necessary to stop an epidemic in the initial 15 stages; and reopening actions, enacted too early, can reignite the epidemic dramatically increasing the number of cases. The astonishing magnitude of the effects, driven by only days of delay, derive from the doubly exponential nature of the underlying relationships.
The FE model can also be used to design measures to end an epidemic in an advanced 20 stage. The management plan is built by first using Equation 12 to predict how many days a given level of intervention, ‫ܭ‬ ଵ , takes to reduce the new daily cases by a target fraction:
is the time to the desired reduction and ‫ܦ‬ ௧ is the target as a fraction of the current level of new daily cases. 25 For example, if a country were to target a 90% reduction of new cases per day (e.g. from 50,000 to 5000 cases per day, (7)).
In this example, since భ ൌ ܲ , K 1 = 0.2 characterizes a lockdown in which people in the country can each only have one plausibly infectious contact with one specific person for the containment duration. This does not mean they cannot contact anyone other than the one person; but they must use care, using masks and proper distancing, to ensure there is no plausibly 35 infectious contact with anyone other than the one person.
Continuing the planning example: Once the initial 90% reduction is achieved, a reasonable next step might be to relax the social containment to a level allowing a viable economy while preventing the epidemic from erupting again. The level of ‫ܭ‬ ଵ necessary to achieve a chosen target can be found using Equation 12. If an additional 90% reduction in new 40 cases per day is desired and a period of 90 days is tolerable for that reduction, then the new level . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint of ‫ܭ‬ ଵ needed is ~0.025. This equates to allowing each person to be in contact with 7 specific people, in an infectable way, for 90 days. [This is 3 times less stringent than the original US shutdown level in April, 2020.] Thus, with a well-planned approach, a country can reduce its new daily cases by 99% in approximately 100 days enabling the country to control, and essentially end the epidemic, while maintaining economic viability. 5 If even
is too restrictive, an alternative, lower ‫ܭ‬ ଵ can be chosen, but it must be large enough to avoid a new outbreak A bound for the new value of ‫ܭ‬ ଵ , low enough to prevent an outbreak and continue to decrease the new cases per day, can be found using Equation 56 . Of course, a lower ‫ܭ‬ ଵ will extend the time to the additional 90% reduction, but this may be palatable if the initial containment was strong enough. 10 The progress of interventions is easily monitored using the RCO, as the curve for South Korea illustrates. (Fig. 4A ) Had they maintained the implemented level of distancing measures, the data would have followed the initial slope. However, the actual data departed from the slope, heralding the failures in (or relaxations of) social distancing, which were later documented to have occurred during the indicated time frame (3) (circled data, Figs. 2F and 4A). Because it 15 summarizes the epidemic dynamics, the RCO can be used to continuously determine if implemented measures are effective or need to be altered.
The RCO, with Equations 12 and 56, form a simple, yet powerful toolbox for policy makers to kraft measures for controlling an epidemic. With only the time series of the case data and a calculator or spreadsheet, specific goals, and the necessary quantitative behavior of the 20 populace to achieve those goals, can be determined.
The data on the current COVID-19 pandemic are widely accessible from a variety of sources, updated daily. The unfolding panorama provides a test bed for models used to predict outcomes and the effects of various interventions. Because different countries have employed 25 different containment strategies (3) (4) (5) 8) , the world is conducting an epidemiological experiment on a grand scale.
The foundations of current epidemiological modelling are the SIR models (1). In use since 1927, they exist in many variants, both deterministic and stochastic, and their behavior is widely known. It is startling, then, that when the classic SIR model is tested using the currently 30 available data from the COVID-19 pandemic, it fails the most basic test for any model: it projects trends that are the opposite of those easily visible in the data from tens of countries.
Myths die hard. The false "flattening of the curve" myth may well have caused country leaders, especially those most concerned with economic performance, to see social distancing as producing only a modest reduction in the horror of an epidemic peak while significantly 35 prolonging economic disruption. Flattening the curve, as seen in SIR analyses, projects an extension of the epidemic at a high number of cases; however, this notion is completely incorrect
The FE model, a fundamental departure from the SIR models, developed from basic principles and more realistic assumptions, incorporates the effect of the recovered population's interactions with the susceptible and infected populations and accurately projects the epidemic 40 trends of many countries in a unified way. The model makes clear that short and sharp social distancing produces rapid truncation of epidemic upward trends thereby shortening, not lengthening the time needed to bring the epidemic under control. Second, an indicator of the rate . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint of change in epidemic dynamics (the RCO) allows direct observation of the effectiveness of intervention measures and provides policymakers with an opportunity to react before a new outbreak gains momentum.
Every country and economy can use the FE model to plan and implement the highest level of social distancing measures deemed sustainable to quickly reduce case numbers to levels 5 at which case identification, contract tracing, testing and isolation can be maintained, allowing the return of nearly normal social interactions while minimizing economic consequences.. The ultimate insight from the model is one of hope: the path of an epidemic is not an uncontrollable force of nature nor is epidemic control inevitably the road to economic ruin. Rather, the afflicted population can, through their behavior, choose to control their destiny.
Methodology, Validation. T. Reichert: Data Investigation. Competing interests: Authors declare no competing interests. Data and materials availability: The datasets generated and analyzed in the present study are available from the corresponding author on reasonable request.
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The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint 6). K 2 = 1 for all plots. The FE model projects trends identical to those in the country-reported 5 data (Fig. 2) .
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The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint Figure 4 were used for the parameter fits. R 2 > 0.97 for all countries except South Korea (R 2 = 0.80). The FE model was calibrated using data from the date ranges listed in Table 1 . is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
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The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint Parameters from linear fit of rate of change operator (RCO) data in Figure 4 : K 1 , slope; K 2 , . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
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Supplementary Materials for 5 The Fractional Theory of Epidemics: Managing and Ending an Epidemic T. Duclos, T. Reichert.
Correspondence to: tedduclos56@gmail.com 10
The SIR Model
The SIR model, with non-time varying parameters ߚ and ߛ , is described by the following equations: A solution to Equations 1-4 is easily obtained using straightforward computational 30 methods. Fig. 1 depicts one such solution, computed using a simple Euler method to solve differential Equations 1-4 with a time step of 1 day.
The development of equations 1-4 in (1) relied on the basic assumption that the rate of new infections in the susceptible population (S) is proportional to the product of the susceptible and infected populations (I). On its surface, this assumption seems reasonable. After all, infected 35 people only infect susceptible people. However, it ignores that over time, the population of recovered individuals (R) grows and interacts with infected and susceptible people. Since the recovered population grows substantially during the epidemic, interactions between recovered people and the rest of the population significantly alter the epidemic dynamics.
In addition, Equation 3 states that the recovery rate, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
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in the SIR model raise serious doubts about the validity of the model and suggest we need a more complete model to account for these deficiencies. 5 The FE Model
We begin developing a new model by first separating individuals who were infected into two categories: those who are currently infectious, defined as I(t), and those who have had the infection, but are no longer contagious, defined as R(t). The population, R(t), is also referred to as the recovered population because although they may still be ill, they are no longer infectious. 10 We define people already infected during the epidemic as the sum of these two populations:
The goal of developing the new model is to find an expression for N(t) and for the total number of people who will be infected during the epidemic, defined as (Note: Equations 1-12 were introduced in the body of the paper, and the numbering begun in the body has been continued in the supplementary materials. Therefore, some equations in this section may appear to be numbered out of order.)
As we develop the model, we note the following:
30 Thus, we can write the following expression:
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, whose solution is
Likewise, the solutions for R(t) are 
During an epidemic, if every contact made by a person within the population N(t) could result in an infection, then the disease progression can be represented by the following expression: has within the entire population. We use the . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint words, specific infectable, because we mean that each person only interacts with the same people (quantity = ܲ ) in a way they can transmit the disease for the time under consideration. Based on this definition, therefore, the number of interactions between the population .
(30) Since interactions among the population during different stages of the epidemic are expected to be relatively stable, we initially assume that the contact rate, ܲ , is a constant and therefore ܲ is also a constant. Additionally, for simplicity in the expressions, we define ‫ܨ‬ ሺ ‫ݐ‬ ሻ as the fraction of
Using these definitions, we can write the following expression using Equation 29:
Equation 32 can then be written as the following difference equation:
15 Equation 33 can also be rewritten in the following manner: 
We should also note that
In the next time step, applying Equation 35 to Equation 37, we find the following expression:
noting that
Equation 39 can be simplified to
Subsequent application of the same logic leads to the recurrence relationship . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
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Rearranging terms and defining ݊ ∆ ‫ݐ‬ ൌ ‫ݐ‬ , we can write the following:
If we now allow ∆ ‫ݐ‬ ՜ 0 and therefore,
, and also note that
, we get the following expression for
or, more generally,
Comparing Equation 43 to Equation 29, we can see that
Returning to ܲ , we will now further explain this parameter. As defined, ܲ is the number of specific infectable contacts a member of the population N(t) has within the entire population and 10 is a function of the population's behavior. Initially, we assume this to be a function of population density and that people cover a constant average effective area per unit of time. We define this area as the effective area rate, ‫ܣ‬ ଵ . This quantity is assumed to be a constant because the fraction of the population an average person interacts with on any given day presumably remains stable for a given time period. 15 Using these assumptions, we can write an expression for , where ‫ܣ‬ ଵ is the effective specific area traversed by a person. In this case, "specific" has the same meaning as it has for ܲ . That is, each person only traverses the 30 same area for the duration of the time under consideration. We also call ‫ܣ‬ ଵ the "effective area" because the population will not be evenly dispersed throughout a given region within a country 10 . If we take this into account, then
From the preceding discussion, we can now write an expression for
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The copyright holder for this this version posted November 12, 2020. from Equation 29, we arrive at Equations 5 and 6 (presented in the Main Text):
Equation 5 can also be rewritten as
where
is the time between the start of the epidemic and the time at which the first data point is measured in the epidemic. Equation 6 is the equation that was fit to the country data in Figures 2 and 4 , used in the RCO projections of Figures 1 and 3 , and to find the parameters in Table 1 .
The expression
ሻ is a convenient operator in that it transforms Equation 5
into a linear equation. We call this expression, 
Taking the limit of Equation 7 as
, we can also obtain the following expression for 25 the total number of individuals who will be infected by the epidemic:
Equation 46 introduces an important and rather abstract concept, not previously discussed. Mathematically, it demonstrates that the number of infections to be expected, have an exponential effect on the final number of infections that will come to be. This means that small changes in population behavior dramatically affect the epidemic's outcome. It also means that the eventual number of cases produced by the epidemic is not foreordained; but rather a strong function of interventions introduced. 35 Additional insight into the model is gained from the following relationship, derived from Equations 5, 7, and 46:
(47) In words, the form of Equation 47 is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint , and (9) 
The left-hand side of Equation 11 is the ratio of the rate of change in the number of new infections per person currently infected. The first two terms on the furthest right-hand side of 
, can be interpreted intuitively. The epidemic acceleration is linearly related to the rate of change in the infections. That is, when there are more daily new infections than daily recoveries, the epidemic is accelerating and expanding. Conversely, when there are fewer daily new infections than daily recoveries, the 20 epidemic is slowing and will end if the containment measures are kept in place long enough to extinguish the outbreak. This leads to the inescapable conclusion that the goal of all containment measures must be to limit the creation of new infections as fast as possible to increase the rate of deceleration to the highest level possible.
The , the desired fraction of the current new cases as:
Using Equations 5 and 7, we arrive at the following expression:
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The copyright holder for this this version posted November 12, 2020. ; https://doi.org/10.1101/2020.11.08.20227793 doi: medRxiv preprint If ‫ݐ‬ ‫ب‬ ‫ݐ‬ Equations 68 and 69 predict the epidemic's progression before and after interventions are instituted. They demonstrate that the dynamics of the epidemic will depend on prior containment measures, as shown by the appearance of the expression ‫‬ ‫ܭ‬ ଵ ሺ ‫ݐ‬ ሻ ݀ ‫ݐ‬ ௧ ௧ బ in both equations. This 5 means that as long as the initial assumptions of immunity remain valid and no new infections are introduced from outside the area, then when interventions are relaxed, outbreaks can still grow nearly exponentially, but the growth of second and subsequent waves will not grow as fast as the initial outbreak. This is a consequence of the fact that ‫ܭ‬ ଵ 0 at all times. Since ‫ܭ‬ ଵ is proportional to the inverse of the effective area traversed by an individual, it is proportional to 10 the number of social interactions, and social interactions can never be less than zero.
If the immunity wanes or new infection from the outside are introduced to a previously uninfected area, then ‫ܭ‬ ଵ may become less than zero for a limited period of time. If, in fact, ‫ܭ‬ ଵ does become less than zero, this indicates that either immunity is waning in the population or that new infections have come to a previously unaffected area and a major outbreak is occurring. 15 The new outbreak can be modelled using Equation 7 or 67 by adding another term to the equation, using a time offset to the time of the outbreak start and choosing an appropriate ‫ܭ‬ ଵ to represent the behavior of the population in the outbreak area. Any number of these terms can be added to model multiple outbreaks. Regardless of how the outbreak is modelled, a negative ‫ܭ‬ ଵ requires immediate action, within days, from policy makers to strengthen intervention measures 20 and prevent the outbreak from overwhelming prior progress in controlling the epidemic. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint
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